This context only exacerbated my disappointment with the "Divisor" reenactment.
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But instead the calculation uses a much smaller divisor — currently 0.14602128057775.
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Last year it became part of a new national park, Sierra del Divisor.
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According to The Wall Street Journal, the Dow's divisor as of June 29, 2017, was 5003.
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Together with some 21959 people, I was going to reenact Lygia Pape's 260 performance "Divisor" ("Divider," 21967).
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You can do that by using an absolute reference to the cell that contains the constant divisor.1.
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Type the name of the cell that contains the divisor, adding a "$" before both the letter and number.
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The current Dow divisor is 0.14602, which means that a 1-point move in any Dow stock moves the Dow by 6.8484 points.
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Because 13 is prime, it must be a divisor of one of those factors, and in fact we see that 26 = 2 × 83.
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The overall level of the Dow does not change when its components do, because the divisor used to calculate the index is adjusted.
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"Divisor" ("Divider") became one of Pape's most significant artworks; she restaged it several times in 1968 in much more glamorous parts of Rio.
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The price-weighted index takes company share prices against a divisor, so moves in higher-priced companies have a more meaningful point impact for the index.
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The index is calculated by simply adding up the stock prices of the constituents and dividing by the "Dow divisor" to adjust for stock splits over time.
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Staged during the military dictatorship, "Divisor" gave people agency to move through public space en masse during a time when protests were suppressed and the streets surveilled.
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Keep cycling through this basic routine, adjusting n at each step and ratcheting up the divisor, until the result of the division is less than 2—then you're done.
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To accompany its retrospective on Lygia Pape, the Met Breuer organized a reenactment of the artist's performance "Divisor," where up to 2365 people parade the streets under a giant sheet.
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We were given instructions about our route via a megaphone, and Met staff members led the performance with plaques written "Divisor" across them, as though we were a pack of tourists.
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The divisor has been adjusted over the decades to account for stock splits, spin-offs, mergers and stock dividends, to assure that numbers from before and after yield apples-to-apples comparisons.
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In order to absorb the changes when companies are added or a stock is split, that sum of the 30 share prices is divided by the Dow divisor, which is currently 0.14602.
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Because once we got going, participants went relatively quiet, walking at a steady pace, when "Divisor" is designed for people to move animatedly within it, and play with the fabric that connects them.
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I think of the sheet blanketing our bodies in "Divisor," and how it literally highlighted our movements — illuminating the connections, divisions, and stories we make every day by just walking on the streets.
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The Dow Jones divisor, which helps determine what each "point" is worth for the 30 companies that are represented in the index, changes occasionally to reflect stock splits and the addition and removal of companies from the index.
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Peru's Ministry of Energy and Mines and Perupetro had planned to develop oil zones within the Sierra del Divisor national park, home to indigenous groups living in voluntary isolation which were recognized by the government in 2018 and 2019, according to government sources.
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While much of her art is interactive and therefore limiting to see today in a museum, the Met did take some good faith efforts, including staging a reenactment of her piece "Divisor," for which dozens of people processed under a giant white sheet down Madison Avenue.
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Therefore, (f) is well-defined. Any divisor of this form is called a principal divisor. Two divisors that differ by a principal divisor are called linearly equivalent. The divisor of a meromorphic 1-form is defined similarly.
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An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. If there are no nontrivial zero divisors in , then is a domain.
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For a non-singular projective variety, a result of Kodaira states that this is equivalent to a divisor in the class being the sum of an ample divisor and an effective divisor.
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Given an effective Cartier divisor D, there are two equivalent ways to associate Weil divisor [D] to it.
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A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
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One key divisor on a compact Riemann surface is the canonical divisor. To define it, one first defines the divisor of a nonzero meromorphic 1-form along the lines above. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two nonzero meromorphic 1-forms yield linearly equivalent divisors. Any divisor in this linear equivalence class is called the canonical divisor of X, KX. The genus g of X can be read from the canonical divisor: namely, KX has degree 2g − 2.
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An algebraic cycle is a higher-codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
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The extended Euclidean algorithm for the greatest common divisor of two integers and is certifying: it outputs three integers (the divisor), , and , such that . This equation can only be true of multiples of the greatest common divisor, so testing that is the greatest common divisor may be performed by checking that divides both and and that this equation is correct.
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The functions half and third curry the divide function with a fixed divisor. The divisor function also forms a closure by binding the variable `d`.
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In fact, if is a divisor of such that , then is a divisor of such that . If one tests the values of in increasing order, the first divisor that is found is necessarily a prime number, and the cofactor cannot have any divisor smaller than . For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of that is not smaller than and not greater than . There is no need to test all values of for applying the method.
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A divisor of a global meromorphic 1-form is called the canonical divisor (usually denoted K). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor). The symbol \deg(D) denotes the degree (occasionally also called index) of the divisor D, i.e. the sum of the coefficients occurring in D. It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class. The number \ell(D) is the quantity that is of primary interest: the dimension (over \C) of the vector space of meromorphic functions h on the surface, such that all the coefficients of (h) + D are non-negative.
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A boundary Q-divisor on a variety is a Q-divisor D of the form ΣdiDi where the Di are the distinct irreducible components of D and all coefficients are rational numbers with 0≤di≤1. A logarithmic pair, or log pair for short, is a pair (X,D) consisting of a normal variety X and a boundary Q-divisor D. The log canonical divisor of a log pair (X,D) is K+D where K is the canonical divisor of X. A logarithmic 1-form on a log pair (X,D) is allowed to have logarithmic singularities of the form d log(z) = dz/z along components of the divisor given locally by z=0.
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A simple example of an output-sensitive algorithm is given by the division algorithm division by subtraction which computes the quotient and remainder of dividing two positive integers using only addition, subtraction, and comparisons: def divide(number: int, divisor: int) -> Tuple[int, int]: """Division by subtraction.""" if not divisor: raise ZeroDivisionError if number < 1 or divisor < 1: raise ValueError( f"Positive integers only for " f"dividend ({number}) and divisor ({divisor})." ) q = 0 r = number while r >= divisor: q += 1 r -= divisor return q, r Example output: >>> divide(10, 2) (5, 0) >>> divide(10, 3) (3, 1) This algorithm takes Θ(Q) time, and so can be fast in scenarios where the quotient Q is known to be small. In cases where Q is large however, it is outperformed by more complex algorithms such as long division.
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To calculate the DJIA, the sum of the prices of all 30 stocks is divided by a divisor, the Dow Divisor. The divisor is adjusted in case of stock splits, spinoffs or similar structural changes, to ensure that such events do not in themselves alter the numerical value of the DJIA. Early on, the initial divisor was composed of the original number of component companies; this initially made the DJIA a simple arithmetic average. The present divisor, after many adjustments, is less than one (meaning the index is larger than the sum of the prices of the components).
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W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 107, and Cohen showed that any such number must have at least three different prime factors. showed that there are no odd harmonic divisor numbers smaller than 1024. Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×109, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.
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In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle -- it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical.
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Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph G is called the period of G.
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However, divisor methods such as the current method do not.
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In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that , or equivalently if the map from to that sends to is not injective (one to one). Similarly, an element of a ring is called a right zero divisor if there exists a nonzero such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor.
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In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted \gcd (x,y). For example, the gcd of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include greatest common factor ('gcf), etc...Some authors present ' as synonymous with greatest common divisor.
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Let U be the open subset where X is regular; then the canonical bundle KU is a line bundle. The restriction from the divisor class group Cl(X) to Cl(U) is an isomorphism, and (since U is smooth) Cl(U) can be identified with the Picard group Pic(U). As a result, KU defines a linear equivalence class of Weil divisors on X. Any such divisor is called the canonical divisor KX. For a normal scheme X, the canonical divisor KX is said to be Q-Cartier if some positive multiple of the Weil divisor KX is Cartier. (This property does not depend on the choice of Weil divisor in its linear equivalence class.) Alternatively, normal schemes X with KX Q-Cartier are sometimes said to be Q-Gorenstein.
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One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; in this case, it is not necessarily true that (using the previous notation) x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R. When R is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says that a is a divisor of b, or that b is a multiple of a, and one writes a \mid b . Elements a and b of an integral domain are associates if both a \mid b and b \mid a . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes. Notes: These definitions make sense in any magma R, but they are used primarily when this magma is the multiplicative monoid of a ring.
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Huxley also improved the known bound on the Dirichlet divisor problem.
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In arithmetic operations it holds the addend, subtrahend, multiplicand, or divisor.
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An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called right regular or right cancellable.
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Later rules included resistance factors, such as draught or freeboard. These resistance factors could either be subtracted from the speed factors or used as divisors of the speed factors. Some rules thus took the form of fractions—some "trivial", where the divisor was merely a constant, and others "non-trivial", where the divisor was a resistance factor. The Union Rule was a trivial fraction (the divisor being "150") and the Universal Rule non-trivial (the divisor being 5 times the cube root of the draught).
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The other significant invariant of D is its degree d, which is the sum of all its coefficients. A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.Hartshorne p.296 Clifford's theorem states that for an effective special divisor D, one has: :\ell(D)- 1 \le d/2, and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.
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Laphria divisor is a species of robber flies in the family Asilidae.
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S contains the greatest common divisor]: PRINT S DONE: HALT, END, STOP.
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Integers divisible by 2 are called even, and integers not divisible by 2 are called odd. 1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non- trivial divisor (or strict divisorFoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois). A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
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A Cartier divisor is effective if its local defining functions fi are regular (not just rational functions). In that case, the Cartier divisor can be identified with a closed subscheme of codimension 1 in X, the subscheme defined locally by fi = 0. A Cartier divisor D is linearly equivalent to an effective divisor if and only if its associated line bundle O(D) has a nonzero global section s; then D is linearly equivalent to the zero locus of s. Let X be a projective variety over a field k.
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The quotient of the Cartier divisors by the principal divisors is a subgroup of the divisor class group, isomorphic to the Picard group of invertible sheaves on Spec(A). Example: in the ring k[x,y,z]/(xy–z2) the divisor class group has order 2, generated by the divisor y=z, but the Picard subgroup is the trivial group.Hartshorne, GTM52, Example 6.5.2, p.
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Currying is the process of changing a function so that rather than taking multiple inputs, it takes a single input and returns a function which accepts the second input, and so forth. In this example, a function that performs division by any integer is transformed into one that performs division by a set integer. >>> def divide(x, y): ... return x / y >>> def divisor(d): ... return lambda x: divide(x, d) >>> half = divisor(2) >>> third = divisor(3) >>> print(half(32), third(32)) 16.0 10.666666666666666 >>> print(half(40), third(40)) 20.0 13.333333333333334 While the use of anonymous functions is perhaps not common with currying, it still can be used. In the above example, the function divisor generates functions with a specified divisor.
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A composite divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors.
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The municipality contains 32% of the Serra do Divisor National Park, created in 1989.
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17 is divided into 3 groups of 5, with 2 as leftover. Here, the dividend is 17, the divisor is 5, the quotient is 3, and the remainder is 2 (which is strictly smaller than the divisor 5), or more symbolically, 17 = (5 × 3) + 2. In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in such a way that produces a quotient and a remainder smaller than the divisor. A fundamental property is that the quotient and the remainder exist and are unique, under some conditions.
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Division by zero — where the divisor is zero — is usually left undefined in elementary arithmetic.
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Sierra del Divisor is a mountain range located in the border between Peru and Brazil.
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Divisibility properties can be determined in two ways, depending on the type of the divisor.
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Set the control lever to Subtraction/Division and set the divisor onto the input sliders. While keeping the carriage lifted, reset the display registers, set the dividend, right justified, using the corresponding knobs and shift the carriage so that the highest number in the dividend corresponds to the highest number in the divisor. Lower the carriage then turn the execution lever as many times as required until the number situated above the divisor is less than the divisor, then shift the carriage once to the left and repeat this operation until the carriage is back to its default position and the number in the accumulator is less than the divisor, then the quotient will be in the operations counter and the remainder will be what is left over in the accumulator.
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Conservation units in the west of Acre. 9\. Serra do Divisor National Park The Serra do Divisor National Park is divided between the municipalities of Rodrigues Alves (13.45%), Porto Walter (26.99%), Marechal Thaumaturgo (4.73%), Mâncio Lima (31.71%) and Cruzeiro do Sul (23.12%) in the state of Acre. It has an area of . The park is bounded to the west by the border with Peru, which runs along the Serra Divisor mountain range.
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Stock splits do not affect the divisor since they do not affect market capitalization. When a company is dropped and replaced by another with a different market capitalization, the divisor needs to be adjusted in such a way that the value of the S&P; 500 index remains constant. All divisor adjustments are made after the close of trading and after the calculation of the closing value of the S&P; 500 index.
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The divisor is a tool used by the S&P; to ensure the index only represents changes in market driven price movements. When stocks are added or deleted, the divisor is adjusted to maintain the same market value of the index. Similarly, in the event of non- market driven price movements (e.g. corporate actions, company inclusion or exclusion from the index), the divisor is adjusted to remove the effects of these actions on index value.
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His dissertation, "New Application of Pfeiffer's method for Dirichlet's divisor problem", caused a stir in 1922.
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In the second sense, a quotient is simply the ratio of a dividend to its divisor.
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Other applications of multi-modular arithmetic include polynomial greatest common divisor, Gröbner basis computation and cryptography.
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On a projective scheme X over a field, Kleiman's criterion implies that ampleness is an open condition on the class of an R-divisor (an R-linear combination of Cartier divisors) in N^1(X), with its topology based on the topology of the real numbers. (An R-divisor is defined to be ample if it can be written as a positive linear combination of ample Cartier divisors.Lazarsfeld (2004), Definition 1.3.11.) An elementary special case is: for an ample divisor H and any divisor E, there is a positive real number b such that H+aE is ample for all real numbers a of absolute value less than b.
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Kurenniemi presented his theory of mathematical music in the article "Harmonioiden teoria" ("Theory of harmonies", 1985) and "Musical harmonies are divisor sets" (1988), in which he defines harmony as a function of the divisor set of an integer. Harmony can be read by interpreting the successive numbers as intervals. Harmonies are symmetrical, that is, their interval relations remain constant regardless of whether the divisor set is read from beginning to end or vice versa. Correspondingly, all intervals and chords can be expanded into a harmony by calculating the greatest common factor of the set, f, and its smallest common divisor, s, resulting in the harmony as H(s/f).
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The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder. Synonyms for the GCD include the greatest common factor (GCF), the highest common factor (HCF), the highest common divisor (HCD), and the greatest common measure (GCM). The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b), although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD.
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It is equivalent to require that around each x, there exists an open affine subset such that , where f is a non-zero divisor in A. The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves. There is a good theory of families of effective Cartier divisors. Let be a morphism. A relative effective Cartier divisor for X over S is an effective Cartier divisor D on X which is flat over S. Because of the flatness assumption, for every S'\to S, there is a pullback of D to X \times_S S', and this pullback is an effective Cartier divisor.
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Zerah calculated in his head that it was not and has divisor 641. The other divisor is 6,700,417 and can easily be found using a calculator. His father capitalized on his boy's talents by taking Zerah around the country and eventually abroad, demonstrating the boy's exceptional abilities.
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Balanced ternary division is analogous to that of binary and decimal. However, 0.510 = 0.1111...bal3 or 1.TTTT...bal3. If the dividend over the plus or minus half divisor, the trit of the quotient must be 1 or T. If the dividend is between the plus and minus of half the divisor, the trit of the quotient is 0. The magnitude of the dividend must be compared with that of half the divisor before setting the quotient trit.
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Macaulay proved that the -ideal generated by these principal minors is a principal ideal, which is generated by the greatest common divisor of these minors. As one is working with polynomials with integer coefficients, this greatest common divisor is defined up its sign. The generic Macaulay resultant is the greatest common divisor which becomes , when, for each , zero is substituted for all coefficients of P_i, except the coefficient of x_i^{d_i}, for which one is substituted.
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However, in this case there are additional constraints on the divisor beyond having zero sum of multiplicities..
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The reserve will also support regional conservation, as it is part of a bi-national biological corridor with the Sierra del Divisor National Park in Peru and the Sierra del Divisor National Park and the Extractive reserves in Brazil - Alto Juruá Extractive Reserve and Alto Tarauacá Extractive Reserve.
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Although this always happens eventually, the resulting Greatest common divisor (GCD) is a divisor of n other than 1. This may be n itself, since the two sequences might repeat at the same time. In this (uncommon) case the algorithm fails, and can be repeated with a different parameter.
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In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta- function. It is therefore an algebraic subvariety of A of dimension dim A − 1\.
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In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are :1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 .
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In mathematics, the Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. The concept is named after Rolf Nevanlinna.
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A corresponding statement for the case that k is not a divisor of n is an open mathematical problem.
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The "Riemann hypothesis" statement depends on the fact that the largest proper divisor of can be no larger than .
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To divide by a multidigit divisor, this process is used: #The dividend is set into the accumulator, and the divisor is set into the operand dials. #The input section is moved with the end crank until the lefthand digits of the two numbers line up. #The operation crank is turned and the divisor is subtracted from the accumulator repeatedly until the left hand (most significant) digit of the result is 0. The number showing on the multiplier dial is then the first digit of the quotient.
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This is true for every common divisor of a and b. However, unlike other common divisors, the greatest common divisor is a member of the set; by Bézout's identity, choosing u = s and v = t gives g. A smaller common divisor cannot be a member of the set, since every member of the set must be divisible by g. Conversely, any multiple m of g can be obtained by choosing u = ms and v = mt, where s and t are the integers of Bézout's identity.
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Division can be carried out by successive subtractions: the quotient is the number of time the divisor can be subtracted from the dividend, and the remainder is what is left rest after all the possible subtractions. This process, which can be very long, may be made efficient if instead of the divisor we subtract multiple of the divisor, and computations are easier if we restrict to multiple by a power of 2. In facts, this is what we do in the long division method.
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Therefore, a fudge factor called the "Divisor" is used to ensure that the index value only changes when stock prices change, not whenever market capitalisation changes. For example, if a company increases its market capitalisation by issuing new shares, the Divisor is adjusted so that the ASX 200 index value does not change.
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If R and S are rings, a rng homomorphism whose image contains a non-zero-divisor maps 1R to 1S.
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Let and be polynomials with coefficients in an integral domain , typically a field or the integers. A greatest common divisor of and is a polynomial that divides and , and such that every common divisor of and also divides . Every pair of polynomials (not both zero) has a GCD if and only if is a unique factorization domain. If is a field and and are not both zero, a polynomial is a greatest common divisor if and only if it divides both and , and it has the greatest degree among the polynomials having this property.
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The simplest linear Diophantine equation takes the form , where , and are given integers. The solutions are described by the following theorem: :This Diophantine equation has a solution (where and are integers) if and only if is a multiple of the greatest common divisor of and . Moreover, if is a solution, then the other solutions have the form , where is an arbitrary integer, and and are the quotients of and (respectively) by the greatest common divisor of and . Proof: If is this greatest common divisor, Bézout's identity asserts the existence of integers and such that .
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When the constraint of all the squares being different sizes is relaxed, a squared square such that the side lengths of the smaller squares do not have a common divisor larger than 1 is called a "Mrs. Perkins's quilt". In other words, the greatest common divisor of all the smaller side lengths should be 1. The Mrs.
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The Euclidean algorithm may be applied to some noncommutative rings such as the set of Hurwitz quaternions. Let and represent two elements from such a ring. They have a common right divisor if and for some choice of and in the ring. Similarly, they have a common left divisor if and for some choice of and in the ring.
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Riemann further proved the Riemann singularity theorem, identifying the multiplicity of a point p = class(D) on Wg − 1 as the number of linearly independent meromorphic functions with pole divisor dominated by D, or equivalently as h0(O(D)), the number of linearly independent global sections of the holomorphic line bundle associated to D as Cartier divisor on C.
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Now multiply each digit of the divisor by the new digit of the quotient and subtract that from the left-hand segment of the dividend. Where the subtrahend and the dividend segment differ, cross out the dividend digit and write if necessary the subtrahend digit and next vertical empty space. Cross out the divisor digit used.
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Explicitly, the first Chern class c_1(L) is the divisor (s) of any nonzero rational section s of L.Lazarsfeld (2004), Example 1.1.5.
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Divisor function σ0(n) up to n = 250 Sigma function σ1(n) up to n = 250 Sum of the squares of divisors, σ2(n), up to n = 250 Sum of cubes of divisors, σ3(n) up to n = 250 In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
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Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime.
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In this case, the extended Euclidean algorithm may be used. See polynomial greatest common divisor#Bézout's identity and extended GCD algorithm for details.
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In algebraic geometry a normal crossings singularity is a point in an algebraic variety that is locally isomorphic to a normal crossings divisor.
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If Σniai is a divisor linearly equivalent to 0, then ΠE(x,ai)ni is a meromorphic function with given poles and zeros.
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The computational complexity of the computation of greatest common divisors has been widely studied. If one uses the Euclidean algorithm and the elementary algorithms for multiplication and division, the computation of the greatest common divisor of two integers of at most bits is O(n^2). This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication. However, if a fast multiplication algorithm is used, one may modify the Euclidean algorithm for improving the complexity, but the computation of a greatest common divisor becomes slower than the multiplication.
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By the fundamental theorem of arithmetic, every integer greater than 1 has a unique (up to the order of the factors) factorization into prime numbers, which are those integers which cannot be further factorized into the product of integers greater than one. For computing the factorization of an integer , one needs an algorithm for finding a divisor of or deciding that is prime. When such a divisor is found, the repeated application of this algorithm to the factors and gives eventually the complete factorization of . For finding a divisor of , if any, it suffices to test all values of such that and .
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However, irreducibility depends on the ambient field, and a polynomial may be irreducible over and reducible over some extension of . Similarly, divisibility by a square depends on the ambient field. If an irreducible polynomial over is divisible by a square over some field extension, then (by the discussion above) the greatest common divisor of and its derivative is not constant. Note that the coefficients of belong to the same field as those of , and the greatest common divisor of two polynomials is independent of the ambient field, so the greatest common divisor of and has coefficients in .
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Since a and b are both multiples of g, they can be written a = mg and b = ng, and there is no larger number G > g for which this is true. The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c.
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In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or .Lam (2001), p. 3 (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.
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A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers.
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Its known range lies within the remote Serra do Divisor National Park in Brazil and the adjacent Sierra del Divisor National Park in Peru, but it is believed to be common there. Because of its apparently stable population, the International Union for the Conservation of Nature has listed it as a species of least concern despite its relatively restricted range.
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In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the algebra is called a Sklyanin algebra. The notion is studied in the context of noncommutative projective geometry.
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The Euclidean domains and the UFD's are subclasses of the GCD domains, domains in which a greatest common divisor of two numbers always exists.
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We can construct another solution by setting X′ = X(49Y3 + 81Z3), Y′ = −Y(32X3 + 81Z3), Z′ = Z(32X3 − 49Y3) and omitting the common divisor.
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In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
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If , the GCD is 0. However, some authors consider that it is not defined in this case. The greatest common divisor of and is usually denoted "". The greatest common divisor is not unique: if is a GCD of and , then the polynomial is another GCD if and only if there is an invertible element of such that :f=u d and :d=u^{-1} f.
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In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.
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Division, denoted by the symbols \div or /, is essentially the inverse operation to multiplication. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For distinct positive numbers, if the dividend is larger than the divisor, the quotient is greater than 1, otherwise it is less than 1 (a similar rule applies for negative numbers).
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Naturally, some of the most interesting examples of such identities result when considering the average order summatory functions over an arithmetic function f(n) defined as a divisor sum of another arithmetic function g(n). For particular examples of divisor sums involving special arithmetic functions and special Dirichlet convolutions of arithmetic functions can be found on the following pages: here, here, here, here, and here.
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Visualisation of using the binary GCD algorithm to find the greatest common divisor (GCD) of 36 and 24. Thus, the GCD is 22 × 3 = 12. The binary GCD algorithm, also known as Stein's algorithm, is an algorithm that computes the greatest common divisor of two nonnegative integers. Stein's algorithm uses simpler arithmetic operations than the conventional Euclidean algorithm; it replaces division with arithmetic shifts, comparisons, and subtraction.
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A well-known example arises in the estimation of the population variance by sample variance. For a sample size of n, the use of a divisor n − 1 in the usual formula (Bessel's correction) gives an unbiased estimator, while other divisors have lower MSE, at the expense of bias. The optimal choice of divisor (weighting of shrinkage) depends on the excess kurtosis of the population, as discussed at mean squared error: variance, but one can always do better (in terms of MSE) than the unbiased estimator; for the normal distribution a divisor of n + 1 gives one which has the minimum mean squared error.
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65284/594 using galley division The completed problem 65284/594 using "modern" long division for comparison Set up the problem by writing the dividend and then a bar. The quotient will be written after the bar. Steps: :(a1) Write the divisor below the dividend. Align the divisor so that its leftmost digit is directly below the dividend's leftmost digit (if the divisor is 594, for instance, it would be written an additional space to the right, so that the "5" would appear below the "6", as shown in the illustration). :(a2) Dividing 652 by 594 yields the quotient 1 which is written to the right of the bar.
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Cross out 1,3, and 8 of the dividend and write 5 and 7 above. Cross out the 9 of the divisor. The resulting dividend is 574.
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Starting with a large trial divisor, it performed division of 262,144 by repeated subtraction then checked if the remainder was zero. If not, it decremented the trial divisor by one and repeated the process. Google released a tribute to the Manchester Baby, celebrating it as the "birth of software". In the late 1950s and early 1960s, a popular innovation was the development of computer languages such as Fortran, COBOL and BASIC.
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Attention should be paid on how it is working in the language chosen; i. e. if it is giving back the decimal rest or the integer rest in order to get proper results. 11 is used as divisor because a container number has 11 letters and digits in total. In step 1 the numbers 11, 22 and 33 are left out as they are multiples of the divisor.
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Let (, ) be a pair of amicable numbers with , and write and where is the greatest common divisor of and . If and are both coprime to and square free then the pair (, ) is said to be regular , otherwise it is called irregular or exotic. If (, ) is regular and and have and prime factors respectively, then is said to be of type . For example, with , the greatest common divisor is and so and .
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The key trichotomy among compact Riemann surfaces X is whether the canonical divisor has negative degree (so X has genus zero), zero degree (genus one), or positive degree (genus at least 2). For example, this determines whether X has a Kähler metric with positive curvature, zero curvature, or negative curvature. The canonical divisor has negative degree if and only if X is isomorphic to the Riemann sphere CP1.
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Implementation of this method has eliminated debates about the proper divisor for district size; any divisor that gives 435 members has the same apportionment. It created other problems however, because, given the fixed-size House, each state's congressional delegation changes as a result of population shifts, with various states either gaining or losing seats based on census results. Each state is then responsible for designing the shape of its districts.
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A significant problem in algebraic geometry is to analyze which line bundles are ample, since that amounts to describing the different ways a variety can be embedded into projective space. One answer is Kleiman's criterion (1966): for a projective scheme X over a field, a line bundle (or R-divisor) is ample if and only if its class in N^1(X) lies in the interior of the nef cone.Lazarsfeld (2004), Theorem 1.4.23. (An R-divisor is called ample if it can be written as a positive linear combination of ample Cartier divisors.) It follows from Kleiman's criterion that, for X projective, every nef R-divisor on X is a limit of ample R-divisors in N^1(X).
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In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that : ax + by = \gcd(a, b). This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor. also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.
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The validity of the Euclidean algorithm can be proven by a two-step argument. In the first step, the final nonzero remainder rN−1 is shown to divide both a and b. Since it is a common divisor, it must be less than or equal to the greatest common divisor g. In the second step, it is shown that any common divisor of a and b, including g, must divide rN−1; therefore, g must be less than or equal to rN−1. These two conclusions are inconsistent unless rN−1 = g. To demonstrate that rN−1 divides both a and b (the first step), rN−1 divides its predecessor rN−2 : since the final remainder rN is zero.
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If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset. A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
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In algebraic geometry, the normal degree of a rational curve C on a surface is defined to be –K.C–2 where K is the canonical divisor of the surface.
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The choice of which version of a method to call may be based either on a single object, or on a combination of objects. The former is called single dispatch and is directly supported by common object-oriented languages such as Smalltalk, C++, Java, Objective-C, Swift, JavaScript, and Python. In these and similar languages, one may call a method for division with syntax that resembles dividend.divide(divisor) # dividend / divisor where the parameters are optional.
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In the natural sciences, including physiology and engineering, a specific quantity generally means a physical quantity normalized "per unit" of something (often mass); the name signals a division of the subject quantity by a parametizing quantity that may or may not be named. If the divisor quantity is named, the name is usually placed before "specific" in the term (i.e., thrust specific fuel consumption). Named and unnamed divisor quantities are given for the terms below.
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The only exceptions in Fibonacci case for n up to 12 are: :F(1)=1 and F(2)=1, which have no prime divisors :F(6)=8 whose only prime divisor is 2 (which is F(3)) :F(12)=144 whose only prime divisors are 2 (which is F(3)) and 3 (which is F(4)) The smallest primitive prime divisor of F(n) are :1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, ... Carmichael's theorem says that every Fibonacci number, apart from the exceptions listed above, has at least one primitive prime divisor. If n > 1, then the nth Pell number has at least one prime divisor that does not divide any earlier Pell number. The smallest primitive prime divisor of nth Pell number are :1, 2, 5, 3, 29, 7, 13, 17, 197, 41, 5741, 11, 33461, 239, 269, 577, 137, 199, 37, 19, 45697, 23, 229, 1153, 1549, 79, 53, 113, 44560482149, 31, 61, 665857, 52734529, 103, 1800193921, 73, 593, 9369319, 389, 241, ...
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In algebraic geometry a simple normal crossings singularity is a point in an algebraic variety, the latter having smooth irreducible components, that is locally isomorphic to a normal crossings divisor.
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A Hall divisor(also called a unitary divisor) of an integer n is a divisor d of n such that d and n/d are coprime. The easiest way to find the Hall divisors is to write the prime factorization for the number in question and take any product of the multiplicative terms (the full power of any of the prime factors), including 0 of them for a product of 1 or all of them for a product equal to the original number. For example, to find the Hall divisors of 60, show the prime factorization is 22·3·5 and take any product of {3,4,5}. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.
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Two divisors that differ by a principal divisor are called linearly equivalent. On a compact Riemann surface, the degree of a principal divisor is zero; that is, the number of zeros of a meromorphic function is equal to the number of poles, counted with multiplicity. As a result, the degree is well- defined on linear equivalence classes of divisors. Given a divisor D on a compact Riemann surface X, it is important to study the complex vector space of meromorphic functions on X with poles at most given by D, called H0(X, O(D)) or the space of sections of the line bundle associated to D. The degree of D says a lot about the dimension of this vector space.
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Sylow's test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n. Proof: If n is a prime-power, then a group of order n has a nontrivial centerSee the proof in p-group, for instance. and, therefore, is not simple.
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The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder.
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In commutative algebra, one major focus of study is divisibility among polynomials. If is an integral domain and and are polynomials in , it is said that divides or is a divisor of if there exists a polynomial in such that . One can show that every zero gives rise to a linear divisor, or more formally, if is a polynomial in and is an element of such that , then the polynomial () divides . The converse is also true.
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However, all but finitely many terms in a nonsingular EDS admit a primitive prime divisor. J. H. Silverman. Wieferich's criterion and the abc-conjecture. J. Number Theory, 30(2):226-237, 1988.
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Division of two real numbers results in another real number (when the divisor is nonzero). It is defined such that a/b = c if and only if a = cb and b ≠ 0.
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574 − 4×9 = 538\. Cross out the 7 and 4 of the dividend and write 3 and 8 above them. Cross out the 4 of the divisor. The resulting dividend is 538.
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When is the field of rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition.
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Fourier division or cross division is a pencil-and-paper method of division which helps to simplify the process when the divisor has more than two digits. It was invented by Joseph Fourier.
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If F(X) = a_0 + a_1 X + \dots + a_n X^n is a polynomial with integer coefficients, then F is called primitive if the greatest common divisor of all the coefficients a_0, a_1, \dots, a_n is 1; in other words, no prime number divides all the coefficients. Proof: Clearly the product f(x).g(x) of two primitive polynomials has integer coefficients. Therefore, if it is not primitive, there must be a prime p which is a common divisor of all its coefficients.
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Here, "greatest" means "having a maximal degree" or, equivalently, being maximal for the preorder defined by the degree. Given a greatest common divisor of two polynomials, the other greatest common divisors are obtained by multiplication by a nonzero constant (that is, all greatest common divisors of and are associated). In particular, two polynomials that are not both zero have a unique greatest common divisor that is monic (leading coefficient equal to ). The extended Euclidean algorithm allows computing (and proving) Bézout's identity.
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This is the case if and only if the greatest common divisor of the polynomial and its derivative is not a constant. Thus for testing if a polynomial is square-free, it is not necessary to consider explicitly any field extension nor to compute the roots. In this context, the case of irreducible polynomials requires some care. A priori, it may seem that being divisible by a square is impossible for an irreducible polynomial, which has no non-constant divisor except itself.
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The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend—before making the remainder negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20, before the remainder becomes negative: : 20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0, while : 20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0\. In this sense, a quotient is the integer part of the ratio of two numbers.
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Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written as: :\frac ab This can be read out loud as "a divided by b" or "a over b". A way to express division all on one line is to write the dividend, then a slash, then the divisor, as follows: :a/b This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters. A handwritten or typographical variation — which is halfway between these two forms — uses a solidus (fraction slash) but elevates the dividend and lowers the divisor, as follows: : Any of these forms can be used to display a fraction.
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Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided by 4 is −3 with remainder 3.
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Third, S(x) is the greatest common divisor of P(x) and P'(x). Fourth, the denominator of the remaining integral Y(x) can be calculated from the equation P(x) = S(x)\,Y(x).
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The sign rules for division are the same as for multiplication. For example, :, :, and :. If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative.
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Division is performed in a similar fashion. To divide 46785399 by 96431, the bars for the divisor (96431) are placed on the board, as shown in the graphic below. Using the abacus, all the products of the divisor from 1 to 9 are found by reading the displayed numbers. Note that the dividend has eight digits, whereas the partial products (save for the first one) all have six. So the final two digits of 46785399, namely the '99', are temporarily ignored, leaving the number 467853.
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In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.
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In algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a convex body in Euclidean space associated to a divisor (or more generally a linear system) on a variety. The convex geometry of a Newton–Okounkov body encodes (asymptotic) information about the geometry of the variety and the divisor. It is a large generalization of the notion of the Newton polytope of a projective toric variety. It was introduced (in passing) by Andrei Okounkov in his papers in the late 1990s and early 2000s.
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Let R be a ring,In this article, rings are assumed to have a 1. and let a and b be elements of R. If there exists an element x in R with , one says that a is a left divisor of b in R and that b is a right multiple of a.Bourbaki, p. 97 Similarly, if there exists an element y in R with , one says that a is a right divisor of b and that b is a left multiple of a.
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Euclid's lemma applies to . That is, if divides , and is coprime with , then is divides . Here, coprime means that the monic greatest common divisor is . Proof: By hypothesis and Bézout's identity, there are , , and such that and .
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This implies that x_1 is a common root of P(Q(x)) and P(x). Its is therefore a root of the greatest common divisor of these two polynomials. It follows that this greatest common divisor is a non constant factor of P(x). Euclidean algorithm for polynomials allows computing this greatest common factor. For example, if one know or guess that: P(x)=x^3 -5x^2 -16x +80 has two roots that sum to zero, one may apply Euclidean algorithm to P(x) and P(-x).
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The condition of having no fixed prime divisor is certainly effectively checkable in a given case, since there is an explicit basis for the integer-valued polynomials. As a simple example, : x^2 + 1 has no fixed prime divisor. We therefore expect that there are infinitely many primes : n^2 + 1 This has not been proved, though. It was one of Landau's conjectures and goes back to Euler, who observed in a letter to Goldbach in 1752 that n^2 + 1 is often prime for n up to 1500.
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There are several ways to find the greatest common divisor of two polynomials. Two of them are: #Factorization of polynomials, in which one finds the factors of each expression, then selects the set of common factors held by all from within each set of factors. This method may be useful only in simple cases, as factoring is usually more difficult than computing the greatest common divisor. #The Euclidean algorithm, which can be used to find the GCD of two polynomials in the same manner as for two numbers.
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Since multiplication is not commutative, there are two versions of the Euclidean algorithm, one for right divisors and one for left divisors. Choosing the right divisors, the first step in finding the by the Euclidean algorithm can be written :\rho_0 = \alpha - \psi_0\beta = (\xi - \psi_0\eta)\delta, where represents the quotient and the remainder. This equation shows that any common right divisor of and is likewise a common divisor of the remainder . The analogous equation for the left divisors would be :\rho_0 = \alpha - \beta\psi_0 = \delta(\xi - \eta\psi_0).
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The two hemispheres are divided by the divisor. Privolva never sees Earth (Volva), Subvolva sees Volva as their moon. Volva goes throughout the same phases as the actual Moon. The daemon continues the descriptions of Subvolva and Privolva.
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Chow coordinates can generate the smallest field of definition of a divisor. The Chow coordinates define a point in the projective space corresponding to all forms. The closure of the possible Chow coordinates is called the Chow variety.
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Any prime number is clearly cyclic. All cyclic numbers are square-free.For if some prime square p2 divides n, then from the formula for φ it is clear that p is a common divisor of n and φ(n).
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This algorithm computes not only the greatest common divisor (the last non zero ), but also all the subresultant polynomials: The remainder is the -th subresultant polynomial. If , the -th subresultant polynomial is . All the other subresultant polynomials are zero.
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Another theorem commonly referred to as Krull's theorem: :::Let R be a Noetherian ring and a an element of R which is neither a zero divisor nor a unit. Then every minimal prime ideal P containing a has height 1.
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Sierra del Divisor National Park () is a national park in the Amazon rainforest of Peru, established in 2015. It covers an area of in the provinces of Coronel Portillo, in the region of Ucayali and Ucayali, in the region of Loreto.
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Also, in ring theory, an element is called a "zero divisor" only if it is nonzero and for a nonzero element . Thus, there are no zero divisors among the integers (and by definition no zero divisors in an integral domain).
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Thus, one can find two numbers x and y, with x2 − y2 divisible by n and again with probability at least one half we get a factor of n by finding the greatest common divisor of n and x − y.
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Porto Walter ( or ) is a municipality located in the west of the Brazilian state of Acre. Its population is 8170 and its area is 6,136 km². The municipality contains 27% of the Serra do Divisor National Park, created in 1989.
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The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that H^0(C,L) eq 0 has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles).Hartshorne (1977), Lemma IV.1.2. In particular, every basepoint-free line bundle on a curve has nonnegative degree.
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In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class. They are named for Pasquale del Pezzo who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space , which are the del Pezzo surfaces of degree at least 3.
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A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60. More generally, an a-by-b rectangle can be covered with square tiles of side length c only if c is a common divisor of a and b. For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60.
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The Baby had a 32-bit word length and a memory of 32 words (1 kibibit). As it was designed to be the simplest possible stored-program computer, the only arithmetic operations implemented in hardware were subtraction and negation; other arithmetic operations were implemented in software. The first of three programs written for the machine calculated the highest proper divisor of 218 (262,144), an algorithm that would take a long time to execute—and so prove the computer's reliability—by testing every integer from 218 downwards, as division was implemented by repeated subtraction of the divisor.
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A Riemann surface is a 1-dimensional complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a compact Riemann surface X is the free abelian group on the points of X. Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients. The degree of a divisor on X is the sum of its coefficients. For any nonzero meromorphic function f on X, one can define the order of vanishing of f at a point p in X, ordp(f).
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The arithmetic billiard for the numbers 15 and 40: the greatest common divisor is 5, the least common multiple is 120. In recreational mathematics, arithmetic billiards provide a geometrical method to determine the least common multiple and the greatest common divisor of two natural numbers by making use of reflections inside a rectangle whose sides are the two given numbers. This is an easy example of trajectory analysis of dynamical billiards. Arithmetic billiards have been discussed as mathematical puzzles by Hugo Steinhaus and Martin Gardner, and are known to mathematics teachers under the name 'Paper Pool'.
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As a simple example, in a function using division, the programmer may prove that the divisor will never equal zero, preventing a division by zero error. Let's say, the divisor 'X' was computed as 5 times the length of list 'A'. One can prove, that in the case of a non-empty list, 'X' is non-zero, since 'X' is the product of two non-zero numbers (5 and the length of 'A'). A more practical example would be proving through reference counting that the retain count on an allocated block of memory is being counted correctly for each pointer.
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But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form qα where α ≡ 1 (mod 4). Therefore, for a perfect number M, τ(M) is even and the average of the divisors is the product of M with the unit fraction 2/τ(M); thus, M is a harmonic divisor number. Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers.
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This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization. The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and all factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram. Here is an example: : 48 = 2 × 2 × 2 × 2 × 3, : 180 = 2 × 2 × 3 × 3 × 5, sharing two "2"s and a "3" in common: :400px : Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720 : Greatest common divisor = 2 × 2 × 3 = 12 This also works for the greatest common divisor (gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection.
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Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of 'chunking' a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself. By allowing one to subtract more multiples than what the partial remainder allows at a given stage, more flexible methods, such as the bidirectional variant of chunking, can be developed as well. More systematic and more efficient (but also more formalised, more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the multiplication tables can divide two integers with pencil and paper using the method of short division, if the divisor is small, or long division, if the divisor is larger.
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For instance, he required every non-zero- divisor to have a multiplicative inverse.Fraenkel, p. 144, axiom R8). In 1921, Emmy Noether gave the modern axiomatic definition of (commutative) ring and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.
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The largest remainder method (also known as Hare–Niemeyer method, Hamilton method or as Vinton's method) is one way of allocating seats proportionally for representative assemblies with party list voting systems. It contrasts with various highest averages methods (also known as divisor methods).
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Two kinds of star heptagons (heptagrams) can be constructed from regular heptagons, labeled by Schläfli symbols {7/2}, and {7/3}, with the divisor being the interval of connection. 200px Blue, {7/2} and green {7/3} star heptagons inside a red heptagon.
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Short division does not use the slash (/) or division sign (÷) symbols. Instead, it displays the dividend, divisor, and quotient (when it is found) in a tableau. An example is shown below, representing the division of 500 by 4. The quotient is 125.
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These divisors are not subsets of a set, making the divisors of n a Boolean algebra that is not concrete according to our definitions. However, if we represent each divisor of n by the set of its prime factors, we find that this nonconcrete Boolean algebra is isomorphic to the concrete Boolean algebra consisting of all sets of prime factors of n, with union corresponding to least common multiple, intersection to greatest common divisor, and complement to division into n. So this example while not technically concrete is at least "morally" concrete via this representation, called an isomorphism. This example is an instance of the following notion.
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It may occur that all terms of a sum are products and that some factors are common to all terms. In this case, the distributive law allows factoring out this common factor. If there are several such common factors, it is worth to divide out the greatest such common factor. Also, if there are integer coefficients, one may factor out the greatest common divisor of these coefficients. For example, :6x^3y^2 + 8x^4y^3 - 10x^5y^3 = 2x^3y^2(3 + 4xy -5x^2y), since 2 is the greatest common divisor of 6, 8, and 10, and x^3y^2 divides all terms.
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Multiplicative number theory deals primarily in asymptotic estimates for arithmetic functions. Historically the subject has been dominated by the prime number theorem, first by attempts to prove it and then by improvements in the error term. The Dirichlet divisor problem that estimates the average order of the divisor function d(n) and Gauss's circle problem that estimates the average order of the number of representations of a number as a sum of two squares are also classical problems, and again the focus is on improving the error estimates. The distribution of primes numbers among residue classes modulo an integer is an area of active research.
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Montserrat Teixidor i Bigas is a professor of Mathematics at Tufts University in Medford, Massachusetts. She specializes in Algebraic geometry especially Moduli of Vector Bundles on Curves.People Montserrat Teixidor i Bigas In 1986, she earned her PhD at the University of Barcelona in 1986 where she wrote her dissertation, "Geometry of linear systems on algebraic curves", under the supervision of Gerard Eryk Welters.Mathematics Genealogy Project She worked in the Department of pure mathematics at the University of Liverpool, UK, where in 1988 she wrote "The divisor of curves with a vanishing theta-null",The divisor of curves with a vanishing theta-null which was published in Compositio Mathematica.
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A primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect number that has no semiperfect proper divisor. The first few primitive semiperfect numbers are 6, 20, 28, 88, 104, 272, 304, 350, ... There are infinitely many such numbers. All numbers of the form 2mp, with p a prime between 2m and 2m+1, are primitive semiperfect, but this is not the only form: for example, 770. There are infinitely many odd primitive semiperfect numbers, the smallest being 945, a result of Paul Erdős: there are also infinitely many primitive semiperfect numbers that are not harmonic divisor numbers.
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Any such must clearly be a multiple of , but it must also be coprime to (if it had any common divisor with , then would be a larger common divisor of and ). Conversely, any multiple of which is coprime to will satisfy . We can generate such numbers by taking the numbers less than coprime to and multiplying each one by (these products will of course each be smaller than , as required). This in fact generates all such numbers, as if is a multiple of coprime to (and less than ), then will still be coprime to , and must also be smaller than , else would be larger than .
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The condition of having no fixed prime divisor is purely local (depending just on primes, that is). In other words, a finite set of irreducible integer-valued polynomials with no local obstruction to taking infinitely many prime values is conjectured to take infinitely many prime values.
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In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a divisor of a local ring to be principal. It was introduced independently by in answer to a question of Grothendieck and by C. P. Ramanujam in an appendix to a paper by , and was generalized by .
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The word "aliquot" comes ultimately from the Latin word meaning "some, several". In mathematics "aliquot" means "an exact part or divisor", reflecting the fact that the length of an aliquot string forms an exact division of the length of longer strings with which it vibrates sympathetically.
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The space A_k is freely generated by the triples (X, f, \alpha), where X is a smooth, k-dimensional complex manifold, f:\; X \mapsto M a holomorphic map, and \alpha is a rational k-form on X, with first order poles on a divisor with normal crossing.
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Calculating a greatest common divisor is an essential step in several integer factorization algorithms,, pp. 225–349 such as Pollard's rho algorithm,, pp. 369–371 Shor's algorithm, Dixon's factorization method and the Lenstra elliptic curve factorization. The Euclidean algorithm may be used to find this GCD efficiently.
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In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs,, p. 105; , p. 94. and divisor graphs.
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Long division in binary is again similar to its decimal counterpart. In the example below, the divisor is 1012, or 5 in decimal, while the dividend is 110112, or 27 in decimal. The procedure is the same as that of decimal long division; here, the divisor 1012 goes into the first three digits 1102 of the dividend one time, so a "1" is written on the top line. This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit (a "1") is included to obtain a new three-digit sequence: 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 \----- 0 0 1 The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted: 1 0 1 ___________ 1 0 1 ) 1 1 0 1 1 − 1 0 1 \----- 1 1 1 − 1 0 1 \----- 1 0 Thus, the quotient of 110112 divided by 1012 is 1012, as shown on the top line, while the remainder, shown on the bottom line, is 102.
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In fact, it is a surface with a well-understood divisor class group and simplest cases share with Del Pezzo surfaces the property of being a rational surface. But many problems of contemporary mathematics remain open, notably (for those examples which are not rational) the question of unirationality.
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His 'matrix divisor' (vector bundle avant la lettre) Riemann–Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles. The Weil conjecture on Tamagawa numbers proved resistant for many years. Eventually the adelic approach became basic in automorphic representation theory.
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Multiplication in a finite field is multiplication modulo an irreducible reducing polynomial used to define the finite field. (I.e., it is multiplication followed by division using the reducing polynomial as the divisor--the remainder is the product.) The symbol "•" may be used to denote multiplication in a finite field.
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The GCD can be visualized as follows. Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly. The sides of the rectangle can be divided into segments of length c, which divides the rectangle into a grid of squares of side length c. The greatest common divisor g is the largest value of c for which this is possible. For illustration, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5). The GCD of two numbers a and b is the product of the prime factors shared by the two numbers, where a same prime factor can be used multiple times, but only as long as the product of these factors divides both a and b. For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the greatest common divisor of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors.
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In number theory, Carmichael's theorem, named after the American mathematician R.D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind Un(P,Q) with relatively prime parameters P, Q and positive discriminant, an element Un with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12)=U12(1, -1)=144 and its equivalent U12(-1, -1)=-144. In particular, for n greater than 12, the nth Fibonacci number F(n) has at least one prime divisor that does not divide any earlier Fibonacci number. Carmichael (1913, Theorem 21) proved this theorem. Recently, Yabuta (2001) gave a simple proof.
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In a 2011 preprint, Fedor Soloviev showed that the pentagram map has a Lax representation with a spectral parameter, and proved its algebraic-geometric integrability. This means that the space of polygons (either twisted or ordinary) is parametrized in terms of a spectral curve with marked points and a divisor. The spectral curve is determined by the monodromy invariants, and the divisor corresponds to a point on a torus—the Jacobi variety of the spectral curve. The algebraic-geometric methods guarantee that the pentagram map exhibits quasi-periodic motion on a torus (both in the twisted and the ordinary case), and they allow one to construct explicit solutions formulas using Riemann theta functions (i.e.
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For a divisor D on a projective variety X over a field k, the k-vector space H0(X, O(D)) has finite dimension. The Riemann–Roch theorem is a fundamental tool for computing the dimension of this vector space when X is a projective curve. Successive generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, give some information about the dimension of H0(X, O(D)) for a projective variety X of any dimension over a field. Because the canonical divisor is intrinsically associated to a variety, a key role in the classification of varieties is played by the maps to projective space given by KX and its positive multiples.
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When a complex manifold X is blown up along a submanifold Z, the blow up locus Z is replaced by an exceptional divisor E and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an \epsilon-neighborhood of the blow up locus, followed by the collapse of the boundary by the Hopf map. Blowing up a symplectic manifold is more subtle, since the symplectic form must be adjusted in a neighborhood of the blow up locus in order to continue smoothly across the exceptional divisor in the blow up. The symplectic cut is an elegant means of making the neighborhood- deletion/boundary-collapse process symplectically rigorous.
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A divisor on a Riemann surface C is a formal sum \textstyle D = \sum_P m_P P of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining L(D) as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative coefficient, with at least that multiplicity. The dimension of L(D) is finite, and denoted \ell(D). The linear system of divisors attached to D is the corresponding projective space of dimension \ell(D)-1.
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Môa River () is a river of Acre state in western Brazil. It is a tributary of the Juruá River. The Môa River flows through the northern part of the Serra do Divisor National Park, and forms part of its north eastern boundary. It continues east until it joins the Juruá River.
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The Nasdaq Composite is a capitalization-weighted index; its price is calculated by taking the sum of the products of closing price and index share of all of the securities in the index. The sum is then divided by a divisor which reduces the order of magnitude of the result.
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He published important contributions to Fermat's last theorem, for which he proved the cases n = 5 and n = 14, and to the biquadratic reciprocity law. The Dirichlet divisor problem, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers.
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In number theory, a hemiperfect number is a positive integer with a half- integral abundancy index. For a given odd number k, a number n is called k-hemiperfect if and only if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to × n.
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Richert made contributions to additive number theory, Dirichlet series, Riesz summability, the multiplicative analog of the Erdős–Fuchs theorem, estimates of the number of non-isomorphic abelian groups, and bounds for exponential sums. He proved the exponent 15/46 for the Dirichlet divisor problem, a record that stood for many years.
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In mathematics, Ruffini's rule is a practical way for paper-and-pencil computation of the Euclidean division of a polynomial by a binomial of the form . It was described by Paolo Ruffini in 1804. Ruffini's rule is a special case of synthetic division when the divisor is a linear factor.
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Linnik obtained numerous results concerning infinitely divisible distributions. In particular, he proved the following generalisation of Cramér's theorem: any divisor of a convolution of Gaussian and Poisson random variables is also a convolution of Gaussian and Poisson. He has also coauthored the book on the arithmetics of infinitely divisible distributions.
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Wheel factorization with n=2x3x5=30. No primes will occur in the yellow areas. Wheel factorization is an improvement of the trial division method for integer factorization. The trial division method consists of dividing the number to be factorized successively by the first integers (2, 3, 4, 5, ...) until finding a divisor.
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In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor D with H0(O(D)) ≠ 0 and (D, D) = 0\. constructed some examples. They are surfaces of class VII, so are non-Kähler and have Kodaira dimension −∞.
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By definition this is fractions. Similarly, there are fractions with denominator 10 (, , , ), fractions with denominator 5 (, , , ), and so on. In detail, we are considering the fractions of the form where is an integer from 1 to inclusive. Upon reducing these to lowest terms, each fraction will have as its denominator some divisor of .
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Case where only the last digit(s) matter This applies to divisors that are a factor of a power of 10. This is because sufficiently high powers of the base are multiples of the divisor, and can be eliminated. For example, in base 10, the factors of 101 include 2, 5, and 10.
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To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisor D on X is ample, meaning that the associated line bundle O(D) is ample. The intersection number D\cdot C can be defined as the degree of the line bundle O(D) restricted to C. In the other direction, for a line bundle L on a projective variety, the first Chern class c_1(L) means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L. On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if h^0(X,L\otimes O(-x-y))=h^0(X,L)-2 for all k-rational points x,y in X.Hartshorne (1977), Proposition IV.3.1. Let g be the genus of X. By the Riemann–Roch theorem, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample.
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A semigroup S that is a homomorphic image of a subsemigroup of T is said to be a divisor of T. The Krohn–Rhodes theorem for finite semigroups states that every finite semigroup S is a divisor of a finite alternating wreath product of finite simple groups, each a divisor of S, and finite aperiodic semigroups (which contain no nontrivial subgroups).Holcombe (1982) pp. 141–142 In the automata formulation, the Krohn–Rhodes theorem for finite automata states that given a finite automaton A with states Q and input set I, output alphabet U, then one can expand the states to Q' such that the new automaton A' embeds into a cascade of "simple", irreducible automata: In particular, A is emulated by a feed-forward cascade of (1) automata whose transitions semigroups are finite simple groups and (2) automata that are banks of flip-flops running in parallel.The flip-flop is the two-state automaton with three input operations: the identity (which leaves its state unchanged) and the two reset operations (which overwrite the current state by a resetting to a particular one of the two states).
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The only such numbers are the triangular numbers with only one nontrivial odd divisor, because for those numbers, according to the bijection described earlier, the odd divisor corresponds to the triangular representation and there can be no other polite representations. Thus, polite numbers whose only polite representation starts with 1 must have the form of a power of two multiplied by an odd prime. As Jones and Lord observe, there are exactly two types of triangular numbers with this form: #the even perfect numbers 2n − 1(2n − 1) formed by the product of a Mersenne prime 2n − 1 with half the nearest power of two, and #the products 2n − 1(2n + 1) of a Fermat prime 2n + 1 with half the nearest power of two. .
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Bézout's identity works for univariate polynomials over a field exactly in the same ways as for integers. In particular the Bézout's coefficients and the greatest common divisor may be computed with the extended Euclidean algorithm. As the common roots of two polynomials are the roots of their greatest common divisor, Bézout's identity and fundamental theorem of algebra imply the following result: :For univariate polynomials f and g with coefficients in a field, there exist polynomials a and b such that af + bg = 1 if and only if f and g have no common root in any algebraically closed field (commonly the field of complex numbers). The generalization of this result to any number of polynomials and indeterminates is Hilbert's Nullstellensatz.
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Then (a, b, c) may be written as a = gx, b = gy, and c = gz where the three numbers (x, y, z) are pairwise coprime. In other words, the greatest common divisor (GCD) of each pair equals one :GCD(x, y) = GCD(x, z) = GCD(y, z) = 1 If (a, b, c) is a solution of Fermat's equation, then so is (x, y, z), since the equation :an \+ bn = cn = gnxn \+ gnyn = gnzn implies the equation : xn \+ yn = zn. A pairwise coprime solution (x, y, z) is called a primitive solution. Since every solution to Fermat's equation can be reduced to a primitive solution by dividing by their greatest common divisor g, Fermat's Last Theorem can be proven by demonstrating that no primitive solutions exist.
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In the theory of algebraic curves, Brill–Noether theory, introduced by , is the study of special divisors, certain divisors on a curve C that determine more compatible functions than would be predicted. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors. The condition to be a special divisor D can be formulated in sheaf cohomology terms, as the non-vanishing of the H1 cohomology of the sheaf of the sections of the invertible sheaf or line bundle associated to D. This means that, by the Riemann–Roch theorem, the H0 cohomology or space of holomorphic sections is larger than expected. Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor ≥ −D on the curve.
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A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity, whence the terminology. Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain and thus Bézout domains are GCD domains.
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The improved understanding of dynamical systems coming from chaos theory helped shed light on what was termed the small denominator problem or small divisor problem. It was observed in the 19th century (by Poincaré, and perhaps earlier), that sometimes 2nd and higher order terms in the perturbative series have "small denominators". That is, they have the general form \psi_n V\phi_m / (\omega_n -\omega_m) where \psi_n, V and \phi_m are some complicated expressions pertinent to the problem to be solved, and \omega_n and \omega_m are real numbers; very often they are the energy of normal modes. The small divisor problem arises when the difference \omega_n -\omega_m is small, causing the perturbative correction to blow up, becoming as large or maybe larger than the zeroth order term.
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Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor. A blowup can also be called monoidal transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map.
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A more general version of Frobenius's theorem states that if C is a conjugacy class with h elements of a finite group G with g elements and n is a positive integer, then the number of elements k such that kn is in C is a multiple of the greatest common divisor (hn,g).
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Solution: Take the total number as the dividend, and 1 package plus 10 cash as the divisor. # "Out of 3485 ounces of silk how many pieces of satin can be made, 5 ounces being required for each piece?" Answer: 697. Solution: Multiply the number of ounces by 2 and go back by one row.
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The ideal is generated by the unique monic element in C of minimum degree, the generator polynomial g. This must be a divisor of x^n-1. It follows that every cyclic code is a polynomial code. If the generator polynomial g has degree d then the rank of the code C is n-d.
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For any given divisor, only finitely many different remainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, ..., 73\. If at any point in the division the remainder is 0, the expansion terminates at that point. Then the length of the repetend, also called “period”, is defined to be 0.
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It's 9. multiply 0'27 by 9: 0'2 4 3 subtract: ——————— 9'7 5 What times 7 ends in 5? It's 5. multiply 0'27 by 5: 0'1 3 5 subtract: ——————— 9'8 4 repetition of original makes 592' minus sixteen twenty-sevenths Division works when the divisor and the base have no factors in common except 1.
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Then the result may be roll-normalized by checking whether the first digit equals the first digit after the quote. Likewise for subtraction. For both addition and subtraction, quote notation is superior to the other two notations. Multiplication in numerator-denominator notation is two integer multiplications, finding a greatest common divisor, and then two divisions.
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The Serra do Divisor National Park () is a national park on the westernmost point of Brazil, in the state of Acre, near the Peruvian border. It also has the highest point in that state, reaching 609 meters above sea level. It has been nominated by the Brazilian government as a Tentative World Heritage Site since 1998.
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Eläintarha () is a large park in central Helsinki, Finland. The name "eläintarha" means "zoo". The park's location acts as a divisor between the districts of Töölö to the west, and Hakaniemi and Kallio to the east. The southern half of the park includes two bays of the Baltic Sea: Töölönlahti to the west, and Eläintarhanlahti to the east.
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Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers". In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.
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This implies that the submatrix of the m + n − 2i first rows of the column echelon form of Ti is the identity matrix and thus that si is not 0. Thus Si is a polynomial in the image of \varphi_i, which is a multiple of the GCD and has the same degree. It is thus a greatest common divisor.
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In this case, the greatest common divisor of 2u and u2 \+ 3v2 is 3. That implies that 3 divides u, and one may express u = 3w in terms of a smaller integer, w. Since u is divisible by 4, so is w; hence, w is also even. Since u and v are coprime, so are v and w.
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The concept of algorithm has existed since antiquity. Arithmetic algorithms, such as a division algorithm, was used by ancient Babylonian mathematicians c. 2500 BC and Egyptian mathematicians c. 1550 BC. Greek mathematicians later used algorithms in the sieve of Eratosthenes for finding prime numbers, and the Euclidean algorithm for finding the greatest common divisor of two numbers.
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If is a multiple of , then for some integer , and is a solution. On the other hand, for every pair of integers and , the greatest common divisor of and divides . Thus, if the equation has a solution, then must be a multiple of . If and , then for every solution , we have :, showing that is another solution.
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A triangle with integer sides and integer area has sides in arithmetic progression if and only if the sides are (b – d, b, b + d), where :b=2(m^2+3n^2)/g, \, :d=(m^2-3n^2)/g, \, and where g is the greatest common divisor of m^2-3n^2, 2mn, and m^2+3n^2.
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A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60. More generally, an a-by-b rectangle can be covered with square tiles of side-length c only if c is a common divisor of a and b. Let g = gcd(a, b).
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A person can calculate division with a slide rule by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.
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Demonstration, with Cuisenaire rods, of the number 72 being powerful An Achilles number is a number that is powerful but not a perfect power. A positive integer is a powerful number if, for every prime factor of , is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful.
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8 Colton proved that no refactorable number is perfect. The equation \gcd(n,x) = \tau(n) has solutions only if n is a refactorable number, where \gcd is the greatest common divisor function. Let T(x) be the number of refactorable numbers which are at most x. The problem of determining an asymptotic for T(x) is open.
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Moustached Tamarin in Serra do Divisor National Park, Acre, Brazil Moustached tamarins are inhabitants of tropical rainforests in Brazil, Bolivia and Peru. They live in arid, upland forests in the Amazonian lowland, mostly occupying higher tree branches. They are arboreal, diurnal and precocial. Tamarins walk and run on all fours, similar to squirrels and use their claws for stability.
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Japanese Zen Buddhist prayer beads (Juzu). Prayer beads (, , (yeomju), ) are also used in many forms of Mahayana Buddhism, often with a lesser number of beads (usually a divisor of 108). In Pure Land Buddhism, for instance, 27-bead malas are common. These shorter malas are sometimes called "prostration rosaries" because they are easier to hold when enumerating repeated prostrations.
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Rodrigues Alves () is a municipality located in the west of the Brazilian state of Acre. Its population is 17 945 according to the 2017 estimates and its area is 3,305 km². Conservation units in the west of the state of Acre, Brazil The municipality contains 13.45% of the Serra do Divisor National Park, created in 1989.
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The converse of Lagrange's theorem states that if is a divisor of the order of a group , then there exists a subgroup where . We will examine the alternating group , the set of even permutations as the subgroup of the Symmetric group . :. so the divisors are . Assume to the contrary that there exists a subgroup in with .
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Every nontrivial reverse divisor must be either 1/4 or 1/9 of its reversal. The number of -digit nontrivial reverse divisors is 2F(\lfloor(d-2)/2\rfloor) where F(i) denotes the th Fibonacci number. For instance, there are two four-digit reverse divisors, matching the formula 2F(\lfloor(d-2)/2\rfloor)=2F(1)=2.
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A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful.
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In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor GCD(a1,a2) is equal to 1, and such that for n > 2 there are no primes in the sequence of numbers calculated from the formula :an = an − 1 + an − 2. The first primefree sequence of this type was published by Ronald Graham in 1964.
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Therefore, the standard form of Schinzel's hypothesis H is that if Q defined as above has no fixed prime divisor, then all f_i(n) will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials f_i(x) with positive leading coefficients. As proved by Schinzel and Sierpiński in page 188 of it is equivalent to the following: if Q defined as above has no fixed prime divisor, then there exists at least one positive integer n such that all f_i(n) will be simultaneously prime, for any choice of irreducible integral polynomials f_i(x) with positive leading coefficients. If the leading coefficients were negative, we could expect negative prime values; this is a harmless restriction. There is probably no real reason to restrict to integral polynomials, rather than integer-valued polynomials.
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It is the largest invariant factor of the Cartan matrix of the block, and occurs with multiplicity one. Also, the power of p dividing the index of the defect group of a block is the greatest common divisor of the powers of p dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of p dividing the degrees of the ordinary irreducible characters in that block. Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the p-part of a group element g is in the defect group of a given block, then each irreducible character in that block vanishes at g. This is one of many consequences of Brauer's second main theorem.
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In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
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Since every unit vector can be thought of as a point on a unit sphere, and since a versor can be thought of as the quotient of two vectors, a versor has a representative great circle arc, called a vector arc, connecting these two points, drawn from the divisor or lower part of quotient, to the dividend or upper part of the quotient.
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Every rational number may be expressed in a unique way as an irreducible fraction , where and are coprime integers and . This is often called the canonical form of the rational number. Starting from a rational number , its canonical form may be obtained by dividing and by their greatest common divisor, and, if , changing the sign of the resulting numerator and denominator.
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10th century al-Uqlidis division Sunzi division al Khwarizmi division of 825AD was identical to Sunzi division algorithm. 11th century Kushyar ibn Labban division, a replica of Sunzi division The animation on the left shows the steps for calculating . #Place the dividend, 309, in the middle row and the divisor, 7, in the bottom row. Leave space for the top row.
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He co-edited Analytic Number Theory, a tome about prime numbers, divisor problems, Diophantine equations, and other topics related to analytic number theory, including Diophantine approximations, and the theory of zeta and L-functions. His other book, Algebraic And Analytic Aspects Of Zeta Functions And L-Functions, a compilation of lectures at the French-Japanese Winter School, was published in 2010.
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In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many quantities have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".
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There are several properties that may be familiar from ring theory, or from associative algebras, which are not always true for non-associative algebras. Unlike the associative case, elements with a (two- sided) multiplicative inverse might also be a zero divisor. For example, all non-zero elements of the sedenions have a two-sided inverse, but some of them are also zero divisors.
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Output CRT Three programs were written for the computer. The first, consisting of 17 instructions, was written by Kilburn, and so far as can be ascertained first ran on 21 June 1948. It was designed to find the highest proper factor of 218 (262,144) by trying every integer from 218 − 1 downwards. The divisions were implemented by repeated subtractions of the divisor.
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The Archdiocese of Rouen distributed a prayer to request Father Jacques's intercession. The prayer makes reference to the circumstances of his murder, including his unmasking of Satan, the divisor and his death in the habits of prayer. Archbishop Lebrun announced on 1 February 2019 that the diocesan inquiry for the beatification process would be solemnly closed on 9 March 2019.
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In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of X \cap U is generated by a regular sequence of length r. A regular embedding of codimension one is precisely an effective Cartier divisor.
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In 1998 Gerhard Frey firstly proposed using trace zero varieties for cryptographic purpose. These varieties are subgroups of the divisor class group on a low genus hyperelliptic curve defined over a finite field. These groups can be used to establish asymmetric cryptography using the discrete logarithm problem as cryptographic primitive. Trace zero varieties feature a better scalar multiplication performance than elliptic curves.
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For proving that there is no solution, one may reduce the equation modulo . For example, the Diophantine equation :x^2+y^2=3z^2, does not have any other solution than the trivial solution . In fact, by dividing and by their greatest common divisor, one may suppose that they are coprime. The squares modulo 4 are congruent to 0 and 1.
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Given an integer , called a modulus, two integers are said to be congruent modulo , if is a divisor of their difference (i.e., if there is an integer such that ). Congruence modulo is a congruence relation, meaning that it is an equivalence relation that is compatible with the operations of addition, subtraction, and multiplication. Congruence modulo is denoted: :a \equiv b \pmod n.
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Since the remainders decrease with every step but can never be negative, a remainder rN must eventually equal zero, at which point the algorithm stops. The final nonzero remainder rN−1 is the greatest common divisor of a and b. The number N cannot be infinite because there are only a finite number of nonnegative integers between the initial remainder r0 and zero.
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8128 is the integer following 8127 and preceding 8129. It is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 127, 27 – 1, with 26 (27 – 1) yielding 8128. Also related to its being a perfect number, 8128 is a harmonic divisor number.
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In a 1784 publication, Euler studied the function further, choosing the Greek letter to denote it: he wrote for "the multitude of numbers less than , and which have no common divisor with it".L. Euler, Speculationes circa quasdam insignes proprietates numerorum, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115.
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Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, a, the dividend, and b, the divisor, such that b ≠ 0, there are unique integers q, the quotient, and r, the remainder, such that a = bq + r and 0 ≤ r < , where denotes the absolute value of b.
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Integers are not closed under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches: # Say that 26 cannot be divided by 11; division becomes a partial function.
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Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see Modulo operation for the details. Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
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In (11,1) dimensions, the minimum number of components would be 64. The superfield C being a cocycle of the ordinary 4-differential cohomology on Calabi-Yau varieties of moduli spaces of line bundles which under decomposition into various cup product associated with a divisor of the CY4, yields intermediate Jacobians and Artin-Mazur formal groups of degrees of maximum three (0,1,2).
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As in all division problems, a number called the dividend is divided by another, called the divisor. The answer to the problem would be the quotient, and in the case of Euclidean division, the remainder would be included as well. Using short division, one can solve a division problem with a very large dividend by following a series of easy steps.
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For every integer and every prime , there is a natural number such that is divisible by precisely when divides . This number is a divisor of either or . The proof of this number theoretical property was first given in a paper by Shuxiang Goh and N. J. Wildberger. It involves considering the projective analogue to quadrance in the finite projective line .
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This construction has been generalized, in algebraic geometry, to the notion of a divisor. There are different definitions of divisors, but in general they form an abstraction of a codimension-one subvariety of an algebraic variety, the set of solution points of a system of polynomial equations. In the case where the system of equations has one degree of freedom (its solutions form an algebraic curve or Riemann surface), a subvariety has codimension one when it consists of isolated points, and in this case a divisor is again a signed multiset of points from the variety. The meromorphic functions on a compact Riemann surface have finitely many zeros and poles, and their divisors can again be represented as elements of a free abelian group, with multiplication or division of functions corresponding to addition or subtraction of group elements.
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This equivalence is also used for computing greatest common divisors of polynomials, although the Euclidean algorithm is defined for polynomials with rational coefficients. In fact, in this case, the Euclidean algorithm requires one to compute the reduced form of many fractions, and this makes the Euclidean algorithm less efficient than algorithms which work only with polynomials over the integers (see Polynomial greatest common divisor).
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For instance, Levania does not seem to move while the Earth is seen to move just as Earth does not seem to move when on Earth but the Moon is seen to move. This is an example of Kepler defending Copernicus' diurnal rotation. The inhabitants at the divisor see the planets different from the rest of the Moon. Mercury and Venus specifically seem bigger to them.
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The tasks are mathematically equivalent, putting states in the place of parties and population in place of votes. In some countries, the Jefferson system is known by the names of local politicians or experts who introduced them locally. For example, it is known in Israel as the Bader–Ofer system. Jefferson's method uses a quota (called a divisor), as in the largest remainder method.
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29÷7 equals 4 remainder 1. Place the quotient, 4, on top, leaving the divisor in place. Place the remainder in the middle row in place of the dividend in this step. The result is the quotient is 44 with a remainder of 1 The Sunzi algorithm for division was transmitted in toto by al Khwarizmi to Islamic country from Indian sources in 825AD.
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What is the greatest common divisor of 54 and 24? The number 54 can be expressed as a product of two integers in several different ways: : 54 \times 1 = 27 \times 2 = 18 \times 3 = 9 \times 6. \, Thus the divisors of 54 are: 1, 2, 3, 6, 9, 18, 27, 54. \, Similarly, the divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
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The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even and odd apply only to integers. But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor..
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In the special case where X is a projective variety, holomorphic line bundles are in bijection with linear equivalences class of divisors, and given a divisor D on X with associated line bundle O(D), the class c1(O(D)) is Poincaré dual to the homology class given by D. Thus, this establishes the usual formulation of the Hodge conjecture for divisors in projective varieties.
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The first step of Fermat's proof is to factor the left-hand sideRibenboim, pp. 11–14. : (x2 \+ y2)(x2 − y2) = z2 Since x and y are coprime (this can be assumed because otherwise the factors could be cancelled), the greatest common divisor of x2 \+ y2 and x2 − y2 is either 2 (case A) or 1 (case B). The theorem is proven separately for these two cases.
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If the abelian variety has a principal polarization then the form on is skew symmetric which implies that the order of is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of is a square (if it is finite).
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In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).
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Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned. 1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.
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Over a field, every nonzero polynomial is associated to a unique monic polynomial. Given two polynomials, and , one says that divides , is a divisor of , or is a multiple of , if there is a polynomial such that . A polynomial is irreducible if it is not the product of two non-constant polynomials, or equivalently, if its divisors are either constant polynomials or have the same degree.
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In abstract algebra, the total quotient ring,Matsumura (1980), p. 12 or total ring of fractions,Matsumura (1989), p. 21 is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings R that may have zero divisors. The construction embeds R in a larger ring, giving every non-zero-divisor of R an inverse in the larger ring.
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A cyclic redundancy check (CRC) is a non- secure hash function designed to detect accidental changes to digital data in computer networks. It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the divisor in a polynomial long division over a finite field, taking the input data as the dividend. The remainder becomes the result.
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Bézout's identity provides yet another definition of the greatest common divisor g of two numbers a and b. Consider the set of all numbers ua + vb, where u and v are any two integers. Since a and b are both divisible by g, every number in the set is divisible by g. In other words, every number of the set is an integer multiple of g.
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Effective Cartier divisors are those which correspond to ideal sheaves. In fact, the theory of effective Cartier divisors can be developed without any reference to sheaves of rational functions or fractional ideal sheaves. Let X be a scheme. An effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk Ix is principal.
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Division in overscore notation is problematic because it requires a sequence of subtractions, which are problematic in overscore notation. Division in quote notation proceeds just like multiplication in quote notation, producing the answer digits from right to left, each one determined by the rightmost digit of the current difference and divisor (trivial in binary). For division, quote notation is superior to both overscore and numerator-denominator notations.
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Lecture Notes in Mathematics. 163. Berlin-Heidelberg-New York: Springer-Verlag. Let X be a complex manifold, D ⊂ X a divisor, and ω a holomorphic p-form on X−D. If ω and dω have a pole of order at most one along D, then ω is said to have a logarithmic pole along D. ω is also known as a logarithmic p-form.
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Marechal Thaumaturgo (, English: Marshall Thaumaturgo) is a municipality located in the west of the Brazilian state of Acre. Its population is 13,061 and its area is 7,744 km². The municipality contains 5% of the Serra do Divisor National Park, created in 1989. It contains most of the Alto Juruá Extractive Reserve, created in 1990, and 5% of the Alto Tarauacá Extractive Reserve, created in 2000.
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Mathematical analysis of this division-like process reveals how to select a divisor that guarantees good error-detection properties. In this analysis, the digits of the bit strings are taken as the coefficients of a polynomial in some variable x—coefficients that are elements of the finite field GF(2), instead of more familiar numbers. The set of binary polynomials is a mathematical ring.
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For C of genus 0 there is one such divisor class, namely the class of -P, where P is any point on the curve. In case of higher genus g, assuming the field over which C is defined does not have characteristic 2, the theta characteristics can be counted as :22g in number if the base field is algebraically closed. This comes about because the solutions of the equation on the divisor class level will form a single coset of the solutions of :2D = 0. In other words, with K the canonical class and Θ any given solution of :2Θ = K, any other solution will be of form :Θ \+ D. This reduces counting the theta characteristics to finding the 2-rank of the Jacobian variety J(C) of C. In the complex case, again, the result follows since J(C) is a complex torus of dimension 2g.
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In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it is now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases. Given a non-constant morphism :φ: C1 -> C2 of algebraic curves, write Ji for the Jacobian variety of Ci. Then from φ construct the corresponding morphism :ψ: J1 -> J2, which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor.
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The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension at least 4 and a smooth ample divisor Y in X, the restriction Pic(X) → Pic(Y) is an isomorphism. For example, if Y is a smooth complete intersection variety of dimension at least 3 in complex projective space, then the Picard group of Y is isomorphic to Z, generated by the restriction of the line bundle O(1) on projective space. Grothendieck generalized Lefschetz's theorem in several directions, involving arbitrary base fields, singular varieties, and results on local rings rather than projective varieties. In particular, if R is a complete intersection local ring which is factorial in codimension at most 3 (for example, if the non-regular locus of R has codimension at least 4), then R is a unique factorization domain (and hence every Weil divisor on Spec(R) is Cartier).
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Demonstration, with Cuisenaire rods, of the perfection of the number 6 For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself, as can be seen from the definitions. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k. Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d.
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In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then :f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of f and g have disjoint support (which can be removed). In the case of the projective line, this can be proved by manipulations with the resultant of polynomials.
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The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics. The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954.
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A number whose prime factors are not of the form 4k+1 cannot be the hypotenuse of a primitive integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle. The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first n square numbers using only n+o(n) additions.
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The first of these generalizes chip-firing from Laplacian matrices of graphs to M-matrices, connecting this generalization to root systems and representation theory. The second considers chip-firing on abstract simplicial complexes instead of graphs. The third uses chip-firing to study graph-theoretic analogues of divisor theory and the Riemann–Roch theorem. And the fourth applies methods from commutative algebra to the study of chip-firing.
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In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The modulo operation is the operation that produces such a remainder when given a dividend and divisor.
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The default choice for the average is a simple moving average, but other types of averages can be employed as needed. Exponential moving averages are a common second choice. Usually the same period is used for both the middle band and the calculation of standard deviation.Since Bollinger Bands use the population method of calculating standard deviation, the proper divisor for the sigma calculation is n, not n − 1.
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A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G. In other words, it is a subgroup whose order is coprime to its index. If π is a set of primes, then a Hall π-subgroup is a subgroup whose order is a product of primes in π, and whose index is not divisible by any primes in π.
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The division operation yields a real number, but fails when the divisor is zero. If we were to write a function that performs division, we might choose to return 0 on this invalid input. However, if the dividend is 0, the result is 0 too. This means there is no number we can return to uniquely signal attempted division by zero, since all real numbers are in the range of division.
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RSA and Diffie–Hellman use modular exponentiation. In computer algebra, modular arithmetic is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of polynomial greatest common divisor, exact linear algebra and Gröbner basis algorithms over the integers and the rational numbers.
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By definition, a and b can be written as multiples of c : a = mc and b = nc, where m and n are natural numbers. Therefore, c divides the initial remainder r0, since r0 = a − q0b = mc − q0nc = (m − q0n)c. An analogous argument shows that c also divides the subsequent remainders r1, r2, etc. Therefore, the greatest common divisor g must divide rN−1, which implies that g ≤ rN−1.
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Bézout's identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b. In other words, it is always possible to find integers s and t such that g = sa + tb. The integers s and t can be calculated from the quotients q0, q1, etc. by reversing the order of equations in Euclid's algorithm.
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Bayer was born in Barcelona on February 13, 1946. Before becoming a mathematician, she was certified as a piano teacher by the Municipal Conservatory of Barcelona in 1967. She graduated from the University of Barcelona in 1968, and completed her Ph.D. there in 1975. Her dissertation, Extensiones maximales de un cuerpo global en las que un divisor primo descompone completamente, was jointly supervised by Rafael Mallol Balmaña and Jürgen Neukirch.
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In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by , and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a commutative affine algebraic group, giving nontrivial examples of Chevalley's structure theorem.
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The pair of integers and such that is unique, in the sense that there can't be other pair of integers that satisfies the same condition in the Euclidean division theorem. In other words, if we have another division of by , say with , then we must have that :. To prove this statement, we first start with the assumptions that : : : : Subtracting the two equations yields :. So is a divisor of .
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It is also useful to consider the normal schemes X for which the canonical divisor KX is Cartier. Such a scheme is sometimes said to be Q-Gorenstein of index 1. (Some authors use "Gorenstein" for this property, but that can lead to confusion.) A normal scheme X is Gorenstein (as defined above) if and only if KX is Cartier and X is Cohen–Macaulay.Kollár & Mori (1998), Corollary 5.69.
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Uwe Storch Uwe Storch (born 12 July 1940, Leopoldshall- Lanzarote, 17 September 2017) was a German mathematician. His field of research was commutative algebra and analytic and algebraic geometry, in particular derivations, divisor class group, resultants. Storch studied mathematics, physics and mathematical logic in Münster and in Heidelberg. He got his PhD 1966 under the supervision of Heinrich Behnke with a thesis on almost (or Q) factorial rings.
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In this document a two-part set of fractions was written in terms of Eye of Horus fractions which were fractions of the form and remainders expressed in terms of a unit called ro. The answers were checked by multiplying the initial divisor by the proposed solution and checking that the resulting answer was 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 5 ro , which equals 1..
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Let us give some details of that computation: By a generic point of the cubic F goes 6 lines. Let s be a point of S and let Ls be the corresponding line on the cubic F. Let Cs be the divisor on S parametrizing lines that cut the line Ls. The self-intersection of Cs is equal to the intersection number of Cs and Ct for t a generic point.
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The Riemann–Roch theorem for a compact Riemann surface of genus g with canonical divisor K states :\ell(D)-\ell(K-D) = \deg(D) - g + 1. Typically, the number \ell(D) is the one of interest, while \ell(K-D) is thought of as a correction term (also called index of specialityStichtenoth p.22Mukai pp.295–297) so the theorem may be roughly paraphrased by saying :dimension − correction = degree − genus + 1.
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Quantitative ecological inventory of terra firme and várzea tropical forest on the Rio Xingu, Brazilian Amazon. Brittonia 38(4): 369-393 the Rio Falsino (Amapá), (Roraima), the Rio Moa and Serra do Divisor National Park (Acre). These expeditions resulted in several notable papers on allelopathy,Campbell, D. G., P. M. Richardson & A. R. Rosas. 1989. Field screening for allelopathy in tropical forest trees, particularly Duroia hirsuta, in the Brazilian Amazon.
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The Calkin–Wilf tree, drawn using an H tree layout. The Calkin–Wilf tree may be defined as a directed graph in which each positive rational number occurs as a vertex and has one outgoing edge to another vertex, its parent. We assume that is in simplest terms; that is, the greatest common divisor of and is 1. If , the parent of is ; if , the parent of is .
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In principle, greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18, 84), we find the prime factorizations 18 = 2 · 32 and 84 = 22 · 3 · 7, and since the "overlap" of the two expressions is 2 · 3, gcd(18, 84) = 6\. In practice, this method is only feasible for small numbers; computing prime factorizations in general takes far too long. Here is another concrete example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180\. First, find the prime factorizations of the two numbers: : 48 = 2 × 2 × 2 × 2 × 3, : 180 = 2 × 2 × 3 × 3 × 5. What they share in common is two "2"s and a "3": :300pxGustavo Delfino, "Understanding the Least Common Multiple and Greatest Common Divisor", Wolfram Demonstrations Project, Published: February 1, 2013.
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This unit is defined as 1/360,000 of a centimeter and thus there are 914,400 EMUs per inch, and 12,700 EMUs per point. This unit was chosen so that integers can be used to accurately represent most dimensions encountered in a document. Floating point cannot accurately represent a fraction that is not a sum of powers of two and the error is magnified when the fractions are added together many times, resulting in misalignment. As an inch is exactly 2.54 centimeters, or 127/50, 1/127 inch is an integer multiple of a power-of-ten fraction of the meter (2×10−4 m). To accurately represent (with an integer) 1 μm = 10−6 m, a divisor of 100 is further needed. To accurately represent the point unit, a divisor of 72 is needed, which also allows divisions by 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 to be accurate.
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Euclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because remainder EA is less than DC. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. On the right Nicomachus's example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300). In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's Topics in Algebra and Serge Lang's Algebra, use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder.
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The number 9! is the lowest factorial which is multiple of 810, so the proper factor 811 is found in this step. The factor 139 is not found this time because p−1 = 138 = 2 × 3 × 23 which is not a divisor of 9! As can be seen in these examples we do not know in advance whether the prime that will be found has a smooth p+1 or p−1.
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Also found in Oevres completes Ser. 1, vol. 5, pp. 434-42. In his book History of Mathematical Notations, Cajori titled the section "Negative numerals". For completeness, ColsonJohn Colson (1726) "A Short Account of Negativo- Affirmativo Arithmetik", Philosophical Transactions of the Royal Society 34:161–173. Available as Early Journal Content from JSTOR uses examples and describes addition (pp 163,4), multiplication (pp 165,6) and division (pp 170,1) using a table of multiples of the divisor.
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Serra do Divisor, Acre, Brazil Vegetation: Rapid ecological assessment survey in 1991 characterised 10 forest types within the Park and recorded biodiversity. Most of the area is covered by open rainforest with palm trees or bamboos, dense and open sub-mountain rainforests, and dense and open alluvial rainforests (Periodically-inundated forests). All forest types show quite differentiated structure, flora and tree species dominance. Open forest grows on poorly drained, wet or inundated soils.
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The trouble starts when X is no longer integral. Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf.
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In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor.
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His Habilitationsschrift developed a generalization of the elementary divisor theory to infinite matrices, continuing ideas of Ulm's teacher Toeplitz. It was submitted in Münster in 1936 and refereed by Heinrich Behnke, Gottfried Köthe, F. K. Schmidt, and B. L. van der Waerden. Ulm's promotion was delayed, apparently, due to his anti- Nazi views. From 1935 until his retirement in 1974 Ulm worked at the University of Münster, as an assistant, docent, and eventually professor (1968).
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By assumption all coefficients in the product are divisible by p, leading to a contradiction. Therefore, the coefficients of the product can have no common divisor and are thus primitive. \square The proof is given below for the more general case. Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial in Z[X]; this explains the need for "non-constant" in the statement.
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The Manú poison frog (Ameerega macero) is a species of frogs in the family Dendrobatidae found in southern Peru and Brazil. It can be found in the drainages of the Manú, Urubamba, Upper Purus and Ucayali Rivers. It can also be found in Serra do Divisor National Park and Alto Juruá Extractive Reserve. Its natural habitats are lowland tropical moist forests and montane forests, in particular bamboo forests, at elevations of 150–1,450 m.
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Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".Hardy & Wright, intro. to Ch. XVI An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime- counting functions.
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The discriminant of a polynomial over a field is zero if and only if the polynomial has a multiple root in some field extension. The discriminant of a polynomial over an integral domain is zero, if and only if the polynomial and its derivative have a non-constant common divisor. In characteristic 0, this is equivalent to saying that the polynomial is not square-free (i.e., divisible by the square of a non-constant polynomial).
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This frequency, divided by 216 (the largest divisor the 8253 is capable of) produces the ≈18.2 Hz timer interrupt used in MS-DOS and related operating systems. In the original IBM PCs, Counter 0 is used to generate a timekeeping interrupt. Counter 1 (A1=0, A0=1) is used to trigger the refresh of DRAM memory. The last counter (A1=1, A0=0) is used to generate tones via the PC speaker.
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Conservation units in the proposal included the Serra do Divisor National Park, Alto Tarauacá Extractive Reserve, Alto Juruá Extractive Reserve, Macauã National Forest, Chico Mendes Extractive Reserve, Jaci Paraná Extractive Reserve, Bom Futuro National Forest, Pacaás Novos National Park, Guaporé Biological Reserve and Corumbiara State Park. As of 2010 the proposed corridor would contain 30 indigenous territories and 19 conservation units in Acre along the border with Peru, covering 45.66% of the state of Acre.
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Per mille, (from Latin , "in each thousand"), is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille. The associated sign is written , which looks like a percent sign with an extra zero in the divisor. The term occurs so rarely in English that major dictionaries do not agree on the spelling and some major dictionaries such as Macmillan do not even contain an entry.
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Because it is the dimension of a vector space, the correction term \ell(K-D) is always non-negative, so that :\ell(D) \ge \deg(D) - g + 1. This is called Riemann's inequality. Roch's part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus g, K has degree 2g-2, independently of the meromorphic form chosen to represent the divisor.
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A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.
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This contradicts the common meaning of the words that are used, as denominator refers to fractions, and two fractions do not have any greatest common denominator (if two fractions have the same denominator, one obtains a greater common denominator by multiplying all numerators and denominators by the same integer). Historically, other names for the same concept have included greatest common measure.. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below).
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The two terms are subtracted, which leaves 8212999. The same steps are repeated: the number is truncated to six digits, the partial product immediately less than the truncated number is chosen, the row number is written as the next digit of the quotient, and the partial product is subtracted from the difference found in the first repetition. The process is shown in the diagram. The cycle is repeated until the result of subtraction is less than the divisor.
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In a uniform matroid U{}^r_n, the circuits are the sets of exactly r+1 elements. Therefore, a uniform matroid is Eulerian if and only if r+1 is a divisor of n. For instance, the n-point lines U{}^2_n are Eulerian if and only if n is divisible by three. The Fano plane has two kinds of circuits: sets of three collinear points, and sets of four points that do not contain any line.
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Soon after their development by Genaille, the rulers were adapted to a set of rods that can perform division. The division rods are aligned similarly to the multiplication rods, with the index rod on the left denoting the divisor, and the following rods spelling out the digits of the dividend. After these, a special "remainder" rod is placed on the right. The quotient is read from left to right, following the lines from one rod to the next.
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That is, given two polynomials and in , there is a unique pair of polynomials such that , and either or . This makes a Euclidean domain. However, most other Euclidean domains (excepts integers) do not have any property of uniqueness for the division nor an easy algorithm (such as long division) for computing the Euclidean division. The Euclidean division is the basis of the Euclidean algorithm for polynomials that computes a polynomial greatest common divisor of two polynomials.
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Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.
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If two numbers have no prime factors in common, their greatest common divisor is 1 (obtained here as an instance of the empty product), in other words they are coprime. A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors. Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility.
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Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity. For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b.
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For instance, one applies vector hashing to each 16-word block of the string, and applies string hashing to the \lceil k/16 \rceil results. Since the slower string hashing is applied on a substantially smaller vector, this will essentially be as fast as vector hashing. # One chooses a power-of-two as the divisor, allowing arithmetic modulo 2^w to be implemented without division (using faster operations of bit masking). The NH hash-function family takes this approach.
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The first thousand values of . The points on the top line represent when is a prime number, which is In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1.
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A related statement, due to Sándor Kovács, is that knowing one ample divisor A in Pic(X) determines the whole cone of curves of X. Namely, suppose that X has Picard number \rho\geq 3. If the set of roots \Delta is empty, then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements u\in\Delta with A\cdot u>0.
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Also, the polynomials are always supposed to be square free. There are two reasons for that. Firstly Yun's algorithm for computing the square- free factorization is less costly than twice the cost of the computation of the greatest common divisor of the polynomial and its derivative. As this may produce factors of lower degrees, it is generally advantageous to apply root- isolation algorithms only on polynomials without multiple roots, even when this is not required by the algorithm.
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In algebra, the polynomial remainder theorem or little Bézout's theorem (named after Étienne Bézout) is an application of Euclidean division of polynomials. It states that the remainder of the division of a polynomial f(x) by a linear polynomial x-r is equal to f(r) . In particular, x-r is a divisor of f(x) if and only if f(r)=0,Larson, Ron (2014), College Algebra, Cengage Learning a property known as the factor theorem.
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Approximations were subsequently employed, largely owing to the Didot point's unwieldy conversion to metric units (the divisor of its conversion ratio has the prime factorization of ). In 1878 Hermann Berthold defined 798 points as being equal to 30 cm, or 2660 points equalling 1 meter: that gives around to the point. A more precise number,, sometimes is given; this is used by TeX as the unit. This has become the standard in Germany and Central and Eastern Europe.
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To add two numbers in numerator-denominator notation, for example (+a/b) + (–c/d) , requires the following steps. • sign comparison to determine if we will be adding or subtracting; in our example, the signs differ so we will be subtracting • then 3 multiplications; in our example, a×d , b×c , b×d • then, if we are subtracting, a comparison of a×d to b×c to determine which is subtrahend and which is minuend, and what is the sign of the result; let's say a×d < b×c so the sign will be – • then the addition or subtraction; b×c – a×d and we have –(b×c – a×d)/(b×d) • finding the greatest common divisor of the new numerator and denominator • dividing numerator and denominator by their greatest common divisor to obtain a normalized result Normalizing the result is not necessary for correctness, but without it, the space requirements quickly grow during a sequence of operations. Subtraction is almost identical to addition. Adding two numbers in overscore notation is problematic because there is no right end to start at.
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In this section, we show that factoring over Q (the rational numbers) and over Z (the integers) is essentially the same problem. The content of a polynomial p ∈ Z[X], denoted "cont(p)", is, up to its sign, the greatest common divisor of its coefficients. The primitive part of p is primpart(p)=p/cont(p), which is a primitive polynomial with integer coefficients. This defines a factorization of p into the product of an integer and a primitive polynomial.
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However, the total increase in a manufacturer's average fuel economy rating due to dual-fueled vehicles cannot exceed 1.2mpg. Section 32906 reduces the increase due to dual-fueled vehicles to 0 through 2020. Electric vehicles are also incentivized by the 0.15 fuel divisor, but are not subject to the 1.2 mpg cap like dual-fuel vehicles. Manufacturers are also allowed to earn CAFE "credits" in any year they exceed CAFE requirements, which they may use to offset deficiencies in other years.
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Although every concrete Boolean algebra is a Boolean algebra, not every Boolean algebra need be concrete. Let n be a square-free positive integer, one not divisible by the square of an integer, for example 30 but not 12. The operations of greatest common divisor, least common multiple, and division into n (that is, ¬x = n/x), can be shown to satisfy all the Boolean laws when their arguments range over the positive divisors of n. Hence those divisors form a Boolean algebra.
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Partition theory is ubiquitous in mathematics with connections to the representation theory of the symmetric group and the general linear group, modular forms, and physics. Thus, Subbarao's conjectures, though seemingly simple, will generate fundamental research activity for years to come. He also researched special classes of divisors and the corresponding analogues of divisor functions and perfect numbers, such as those arising from the exponential divisors ("e-divisors") which he defined. Many other mathematicians have published papers building on his work in these subjects.
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The Serra do Divisor National Park was created by decree 97.839 on 16 June 1989 to protect and preserve sample of ecosystems, ensure preservation of its natural resources, and allow controlled use by the public, education and scientific research. The consultative council was created on 5 July 2002. The management plan was approved on 24 December 2002. In 2013, Rainforest Trust launched a campaign to fund the establishment of another national park in the same area on the Peruvian side of the border.
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In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. Specifically, given a number a and a non-zero number b, if another number c times b equals a, that is: :c \times b = a then a divided by b equals c. That is: :\frac ab = c For instance, :\frac 63 = 2 since :2 \times 3 = 6. In the above expression, a is called the dividend, b the divisor and c the quotient.
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The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis HN) does. That requires an extra polynomial F(x) , which in the Goldbach problem would just be x , for which :N − F(n) is required to be a prime number, also. This is cited in Halberstam and Richert, Sieve Methods. The conjecture here takes the form of a statement when N is sufficiently large, and subject to the condition :Q(n)(N − F(n)) has no fixed divisor > 1\.
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Cauchy's theorem implies that for any prime divisor of the order of , there is a subgroup of whose order is —the cyclic group generated by the element in Cauchy's theorem. Cauchy's theorem is generalised by Sylow's first theorem, which implies that if is the maximal power of dividing the order of , then has a subgroup of order (and using the fact that a -group is solvable, one can show that has subgroups of order for any less than or equal to ).
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15th monohedral convex pentagonal type, discovered in 2015 In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a regular pentagon, 108°, is not a divisor of 360°, the angle measure of a whole turn. However, regular pentagons can tile the hyperbolic plane and the sphere; the latter produces a tiling topologically equivalent to the dodecahedron.
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Accessed 14 March 2013. East of Mannerheimintie, the Helsinki tram lines 2, 7A and 7B run along Nordenskiöldinkatu, and east of Reijolankatu also many bus lines going through Pasila. The western end of the street is located in Taka-Töölö. Its eastern part acts as a neighbourhood divisor, as Eläintarha is part of Taka-Töölö, but the Laakso Hospital lot and the southern part of the Central Park are part of Laakso, and the Aurora Hospital is in Länsi-Pasila.
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Room modes are the collection of resonances that exist in a room when the room is excited by an acoustic source such as a loudspeaker. Most rooms have their fundamental resonances in the 20 Hz to 200 Hz region, each frequency being related to one or more of the room's dimensions or a divisor thereof. These resonances affect the low-frequency low-mid-frequency response of a sound system in the room and are one of the biggest obstacles to accurate sound reproduction.
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After substituting this root in the basis, the second coordinates of this solution is a root of the greatest common divisor of the resulting polynomials that depends only on this second variable, and so on. This solving process is only theoretical, because it implies GCD computation and root-finding of polynomials with approximate coefficients, which are not practicable because of numeric instability. Therefore, other methods have been developed to solve polynomial systems through Gröbner bases (see System of polynomial equations for more details).
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Harold Mortimer Edwards, Jr. (born August 6, 1936) is an American mathematician working in number theory, algebra, and the history and philosophy of mathematics. He was one of the co-founding editors, with Bruce Chandler, of The Mathematical Intelligencer. He is the author of expository books on the Riemann zeta function, on Galois theory, and on Fermat's Last Theorem. He wrote a book on Leopold Kronecker's work on divisor theory providing a systematic exposition of that work—a task that Kronecker never completed.
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LADSPA exists primarily as a header file written in the programming language C. There are many audio plugin standards and most major modern software synthesizers and sound editors support a variety. The best known standard is probably Steinberg's Virtual Studio Technology. LADSPA is unusual in that it attempts to provide only the "Greatest Common Divisor" of other standards. This means that its scope is limited, but it is simple and plugins written using it are easy to embed in many other programs.
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The Euclidean algorithm can be visualized in terms of the tiling analogy given above for the greatest common divisor. Assume that we wish to cover an a-by-b rectangle with square tiles exactly, where a is the larger of the two numbers. We first attempt to tile the rectangle using b-by-b square tiles; however, this leaves an r0-by-b residual rectangle untiled, where r0 < b. We then attempt to tile the residual rectangle with r0-by-r0 square tiles.
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Although the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers), it may be generalized to the real numbers, and to other mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory.
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With either choice, the process is repeated as above until the greatest common right or left divisor is identified. As in the Euclidean domain, the "size" of the remainder must be strictly smaller than , and there must be only a finite number of possible sizes for , so that the algorithm is guaranteed to terminate. Most of the results for the GCD carry over to noncommutative numbers. For example, Bézout's identity states that the right can be expressed as a linear combination of and .
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In a diophantine equation, there are two kinds of variables: the parameters and the unknowns. The diophantine set consists of the parameter assignments for which the diophantine equation is solvable. A typical example is the linear diophantine equation in two unknowns, :a_1x_1 + a_2x_2 = a_3, where the equation is solvable when the greatest common divisor of a_1, a_2 divides a_3. The values for a_1, a_2, a_3 that satisfy this restriction are a diophantine set and the equation above is its diophantine definition.
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It adjoins the Serra do Divisor National Park on the northwest, and the Riozinho da Liberdade and Alto Tarauacá extractive reserves on the northeast. It also adjoins the Arara do Rio Amônia and Kampa do Rio Amônea indigenous territories on the west and the Jaminawa/Arara do Rio Bagé, Kaxinawa do Baixo Jordão, Kaxinawa do Rio Jordão and Kaxinawa/Ashaninka do Rio Breu indigenous territories to the east. The reserve would be included in the proposed Western Amazon Ecological Corridor.
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On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same. The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves.Dieudonné (1985), section VI.6. The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain.
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Prime numbers have been studied throughout recorded history. Euclid devoted one book of the Elements to the theory of primes; in it he proved the infinitude of the primes and the fundamental theorem of arithmetic, and presented the Euclidean algorithm for finding the greatest common divisor of two numbers. In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. But most further development of the theory of primes in Europe dates to the Renaissance and later eras.
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An integer is square- free if and only if q_i=1 for all . An integer greater than one is the th power of another integer if and only if is a divisor of all such that q_i eq 1. The use of the square-free factorization of integers is limited by the fact that its computation is as difficult as the computation of the prime factorization. More precisely every known algorithm for computing a square- free factorization computes also the prime factorization.
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Some NP-problems are not known to be either NP-complete or in P. These problems (e.g. factoring, graph isomorphism, parity games) are suspected to be difficult. Similarly there are problems in P that are not known to be either P-complete or NC, but are thought to be difficult to parallelize. Examples include the decision problem forms of finding the greatest common divisor of two numbers, and determining what answer the extended Euclidean algorithm would return when given two numbers.
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Euclidean division is based on the following result, which is sometimes called Euclid's division lemma. Given two integers and , with , there exist unique integers and such that : and :, where denotes the absolute value of . In the above theorem, each of the four integers has a name of its own: is called the , is called the , is called the and is called the . The computation of the quotient and the remainder from the dividend and the divisor is called or — in case of ambiguity — .
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140 is an abundant number and a harmonic divisor number. It is the sum of the squares of the first seven integers, which makes it a square pyramidal number, and in base 10 it is divisible by the sum of its digits, which makes it a Harshad number. 140 is an odious number because it has an odd number of ones in its binary representation. The sum of Euler's totient function φ(x) over the first twenty-one integers is 140.
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66 (more specifically 66.667) megahertz (MHz) is a common divisor for the front side bus (FSB) speed, overall central processing unit (CPU) speed, and base bus speed. On a Core 2 CPU, and a Core 2 motherboard, the FSB is 1066 MHz (~16 × 66 MHz), the memory speed is usually 666.67 MHz (~10 × 66 MHz), and the processor speed ranges from 1.86 gigahertz (GHz) (~66 MHz × 28) to 2.93 GHz (~66 MHz × 44), in 266 MHz (~66 MHz × 4) increments.
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On > one day alone, 59 different motions to fix a divisor were made in a House > containing but 242 members. The values ranged from 30,000 to 140,000 with > more than half between 50,159 and 62,172. But the Senate had tired of this > approach and proposed instead an apportionment of 223 members using > Webster's method. In the House John Quincy Adams urged acceptance of the > method but argued vehemently for enlarging the number of members, as New > England's portion was steadily dwindling.
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A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by , who found six cases in characteristic 3 (in which case 6K= 0) and eight in characteristic 2 (in which case 6K or 4K vanishes).
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If there is a remainder in a place value decimal rod calculus division, both the remainder and the divisor must be left in place with one on top of another. In Liu Hui's notes to Jiuzhang suanshu (2nd century BCE), the number on top is called "shi" (实), while the one at bottom is called "fa" (法). In Sunzi Suanjing, the number on top is called "zi" (子) or "fenzi" (lit., son of fraction), and the one on the bottom is called "mu" (母) or "fenmu" (lit.
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The two-dimensional Fourier transform of a line through the origin, is a line orthogonal to it and through the origin. The divisor is thus zero for all but a single dimension, by consequence, the optical transfer function can only be determined for a single dimension using a single line-spread function (LSF). If necessary, the two-dimensional optical transfer function can be determined by repeating the measurement with lines at various angles. The line spread function can be found using two different methods.
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A common fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further. A more basic way to show division is to use the obelus (or division sign) in this manner: :a \div b. This form is infrequent except in basic arithmetic. The obelus is also used alone to represent the division operation itself, for instance, as a label on a key of a calculator.
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To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The R-divisors modulo numerical equivalence form a real vector space N^1(X) of finite dimension, the Néron–Severi group tensored with the real numbers.Lazarsfeld (2004), Example 1.3.10. (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.) An R-divisor is called nef if it has nonnegative degree on every curve.
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In other words, the GCD is unique up to the multiplication by an invertible constant. In the case of the integers, this indetermination has been settled by choosing, as the GCD, the unique one which is positive (there is another one, which is its opposite). With this convention, the GCD of two integers is also the greatest (for the usual ordering) common divisor. However, since there is no natural total order for polynomials over an integral domain, one cannot proceed in the same way here.
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If two of the three numbers (a, b, c) can be divided by a fourth number d, then all three numbers are divisible by d. For example, if a and c are divisible by d = 13, then b is also divisible by 13. This follows from the equation : bn = cn − an If the right-hand side of the equation is divisible by 13, then the left-hand side is also divisible by 13. Let g represent the greatest common divisor of a, b, and c.
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More generally, if the greatest common divisor of the moves played so far is g, then only finitely many multiples of g can remain to be played, and after they are all played then g must decrease on the next move. Therefore, every game of sylver coinage must eventually end. When a sylver coinage game has only a finite number of remaining moves, the largest number that can still be played is called the Frobenius number, and finding this number is called the coin problem.
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As specified in the Election to Members of the Constituent Assembly Act (2007), party list representation will be calculated using a result divisor method, the Sainte-Laguë method. The seats for first- past-the-post elections remained at 240, making the total number 601 seats instead of the earlier 497."Nepali leaders agree to hold CA election by mid- April 2008", Xinhua (People's Daily Online), 16 December 2007. The word "republic" was also included, but will have to be confirmed by the Constituent Assembly.
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He began his academic career at the Tata Institute of Fundamental Research, Mumbai as a Visiting Fellow in 1983. Srinivas has worked mainly in algebraic geometry specialising in the study of algebraic cycles on singular algebraic varieties. He has also worked on the interface with commutative algebra: on projective modules, divisor class groups, unique factorization domains, and Hilbert functions and multiplicity. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in 2003, the highest science award in India, in the mathematical sciences category.
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This is systematically used (explicitly or implicitly) in all implemented algorithms (see Polynomial greatest common divisor and Factorization of polynomials). Gauss's lemma, and all its consequences that do not involve the existence of a complete factorization remain true over any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring.
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P is known to contain many natural problems, including the decision versions of linear programming, calculating the greatest common divisor, and finding a maximum matching. In 2002, it was shown that the problem of determining if a number is prime is in P.Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781–793. The related class of function problems is FP. Several natural problems are complete for P, including st-connectivity (or reachability) on alternating graphs.
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Plus and minuses. An obelus used as a variant of the minus sign in an excerpt from an official Norwegian trading statement form called «Næringsoppgave 1» for the taxation year 2010. Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, "a divided by b" can written as: :\frac ab which can also be read out loud as "divide a by b" or "a over b".
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31–33 (1988) The marks on two adjacent spokes flank the digit 0 inscribed on this wheel. On four of the known machines, above each wheel, a small quotient wheel is mounted on the display bar. These quotient wheels, which are set by the operator, have numbers from 1 to 10 inscribed clockwise on their peripheries (even above a non-decimal wheel). Quotient wheels seem to have been used during a division to memorize the number of times the divisor was subtracted at each given index.
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The polynomial remainder theorem may be used to evaluate f(r) by calculating the remainder, R. Although polynomial long division is more difficult than evaluating the function itself, synthetic division is computationally easier. Thus, the function may be more "cheaply" evaluated using synthetic division and the polynomial remainder theorem. The factor theorem is another application of the remainder theorem: if the remainder is zero, then the linear divisor is a factor. Repeated application of the factor theorem may be used to factorize the polynomial.
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A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples.
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The story of continued fractions begins with the Euclidean algorithm,300 BC Euclid, Elements - The Euclidean algorithm generates a continued fraction as a by- product. a procedure for finding the greatest common divisor of two natural numbers m and n. That algorithm introduced the idea of dividing to extract a new remainder - and then dividing by the new remainder repeatedly. Nearly two thousand years passed before Rafael Bombelli1579 Rafael Bombelli, L'Algebra Opera devised a technique for approximating the roots of quadratic equations with continued fractions in the mid-sixteenth century.
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In comparison, the Webster/Sainte-Laguë method, a divisor method, reduces the reward to large parties, and it generally has benefited middle- size parties at the expense of both large and small parties. The axiomatic properties of the D'Hondt method were studied and they proved that the D'Hondt method is the unique consistent, monotone, stable, and balanced method that encourages coalitions. A method is consistent if it treats parties which received tied vote equally. By monotonicity, the number of seats provided to any state or party will not decrease if the house size increases.
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In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint- free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi- ampleness is a kind of "nonnegativity". More strongly, a line bundle on X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space.
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More precisely, if the multiplication of two integers of bits takes a time of , then the fastest known algorithm for greatest common divisor has a complexity O\left(T(n)\log n\right). This implies that the fastest known algorithm has a complexity of O\left(n\,(\log n)^2\log\log n\right). Previous complexities are valid for the usual models of computation, specifically multitape Turing machines and random-access machines. The computation of the greatest common divisors belongs thus to the class of problems solvable in quasilinear time.
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In mathematics, a superperfect number is a positive integer n that satisfies :\sigma^2(n)=\sigma(\sigma(n))=2n\, , where σ is the divisor summatory function. Superperfect numbers are a generalization of perfect numbers. The term was coined by D. Suryanarayana (1969). The first few superperfect numbers are : :2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... . To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.
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A number that belongs to a singleton club, because no other number is "friendly" with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary . For a prime number p we have σ(p) = p + 1, which is co-prime with p.
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As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID). That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b, then there are elements x and y in R such that ax + by = d. The reason is that the ideal Ra+Rb is principal and equal to Rd. An integral domain in which Bézout's identity holds is called a Bézout domain.
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Thus finding more repeated strings narrows down the possible lengths of the keyword, since we can take the greatest common divisor of all the distances. The reason this test works is that if a repeated string occurs in the plaintext, and the distance between corresponding characters is a multiple of the keyword length, the keyword letters will line up in the same way with both occurrences of the string. For example, consider the plaintext: crypto is short for cryptography. "" is a repeated string, and the distance between the occurrences is 20 characters.
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In September 1992, a year after Estonia had regained its independence from the Soviet Union, elections to the Riigikogu took place according to the Constitution of Estonia adopted in the summer of the same year. According to the 1992 constitution, the Riigikogu has 101 members. The present Riigikogu was elected on March 3, 2019. The main differences between this system and a pure political representation, or proportional representation, system are the established 5% national threshold, and the use of a modified D'Hondt formula (the divisor is raised to the power 0.9).
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150 of the seats are regular district seats. These are awarded based on the election results in each county, and are unaffected by results in other counties. Nineteen of the seats (one for each county) are leveling seats, awarded to parties who win fewer seats than their share of the national popular vote otherwise entitles them to. A modification of the Sainte-Lague method, where the first quotient for each party is calculated using a divisor of 1.4 instead of 1, is used to allocate both the constituency and leveling seats.
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The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of the Euclidean algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions.
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Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time. One closely related problem drops the assumption of properness of the divisor, and will be called the improper Znám problem hereafter. One solution to the improper Znám problem is easily provided for any k: the first k terms of Sylvester's sequence have the required property. showed that there is at least one solution to the (proper) Znám problem for each k ≥ 5.
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The Krohn–Rhodes complexity (also called group complexity or just complexity) of a finite semigroup S is the least number of groups in a wreath product of finite groups and finite aperiodic semigroups of which S is a divisor. All finite aperiodic semigroups have complexity 0, while non-trivial finite groups have complexity 1. In fact, there are semigroups of every non-negative integer complexity. For example, for any n greater than 1, the multiplicative semigroup of all (n+1)×(n+1) upper-triangular matrices over any fixed finite field has complexity n (Kambites, 2007).
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In a fraction, the number of equal parts being described is the numerator (from Latin ', "counter" or "numberer"), and the type or variety of the parts is the denominator (from Latin ', "thing that names or designates"). As an example, the fraction amounts to eight parts, each of which is of the type named "fifth". In terms of division, the numerator corresponds to the dividend, and the denominator corresponds to the divisor. Informally, the numerator and denominator may be distinguished by placement alone, but in formal contexts they are usually separated by a fraction bar.
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A4 is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of , there does not necessarily exist a subgroup of G with order d: the group , of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group. For all , An has no nontrivial (that is, proper) normal subgroups. Thus, An is a simple group for all .
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In mathematics, a Sylvester matrix is a matrix associated to two univariate polynomials with coefficients in a field or a commutative ring. The entries of the Sylvester matrix of two polynomials are coefficients of the polynomials. The determinant of the Sylvester matrix of two polynomials is their resultant, which is zero when the two polynomials have a common root (in case of coefficients in a field) or a non-constant common divisor (in case of coefficients in an integral domain). Sylvester matrices are named after James Joseph Sylvester.
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In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i.e., the sum of all positive divisors of n). A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of k-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of k being 1, 2, 1, 6, 3, 1, 12, ... .
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For this reason, Doppler DF systems normally mount their antennas on a small disk that is spun at high speed using an electric motor. Performing the same calculation using an antenna mounted to a diameter disk spinning at 1000 Hz results in: : Which is easily detected. Nevertheless, such a rotation speed, 60,000 rpm, demands precision systems. Because the antennas have to move at very high speeds, this technique is only really useful for higher frequency signals where the antennas can be shorter and the higher Fc produces a larger divisor.
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In the finite Desarguesian planes PG(2,pn), the subplanes have orders which are the orders of the subfields of the finite field GF(pn), that is, pi where i is a divisor of n. In non-Desarguesian planes however, Bruck's theorem gives the only information about subplane orders. The case of equality in the inequality of this theorem is not known to occur. Whether or not there exists a subplane of order M in a plane of order N with M2 \+ M = N is an open question.
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The Manuripi-Heath Amazonian Wildlife National Reserve is located in the southernmost area of this region in Bolivia covering of dense tropical forest. Several extractive reserves, the largest being Chico Mendes Extractive Reserve and Alto Juruá Extractive Reserve, are actively managed in Brazil. Other protected areas include national parks (Serra do Divisor National Park, Madidi National Park, Isoboro Secure National Park, Bahuaja- Sonene National Park), national forests, Rio Acre Ecological Station, Antimari State Forest, Apurimac Reserve Zone, among others. Most protected areas suffer from insufficient administration and patrol.
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It is known that the nth Mersenne number is prime only if n is prime. Fermat's little theorem implies that if is prime, then Mp−1 is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime, ::A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if p2 divides Mq. Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free.
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The Chow variety Chow(d,P3) is a projective algebraic variety which parametrizes degree d curves in P3. It is constructed as follows. Let C be a curve of degree d in P3, then consider all the lines in P3 that intersect the curve C. This is a degree d divisor DC in G(2, 4), the Grassmannian of lines in P3. When C varies, by associating C to DC, we obtain a parameter space of degree d curves as a subset of the space of degree d divisors of the Grassmannian: Chow(d,P3).
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Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient.
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This can be written as an equation for x in modular arithmetic: : Let g be the greatest common divisor of a and b. Both terms in ax + by are divisible by g; therefore, c must also be divisible by g, or the equation has no solutions. By dividing both sides by c/g, the equation can be reduced to Bezout's identity : where s and t can be found by the extended Euclidean algorithm. This provides one solution to the Diophantine equation, x1 = s (c/g) and y1 = t (c/g).
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Polynomials in a single variable x can be added, multiplied and factored into irreducible polynomials, which are the analogs of the prime numbers for integers. The greatest common divisor polynomial of two polynomials and is defined as the product of their shared irreducible polynomials, which can be identified using the Euclidean algorithm. The basic procedure is similar to that for integers. At each step , a quotient polynomial and a remainder polynomial are identified to satisfy the recursive equation :r_{k-2}(x) = q_k(x)r_{k-1}(x) + r_k(x), where and .
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The Scottish elections are divided into two tiers. In the model of AMS used in the United Kingdom, the regional seats are divided using a D'Hondt method. However, the number of seats already won in the local constituencies is taken into account in the calculations for the list seats, and the first average taken in account for each party follows the number of FPTP seats won. For example, if a party won 5 constituency seats, then the first D'Hondt divisor taken for that party would be 6 (5 seats + 1), not 1.
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Suppose that R is a complete Noetherian local ring. If R has dimension greater than 0 and x is an element in the maximal ideal that is not a zero divisor then R is a complete intersection ring if and only if R/(x) is. (If the maximal ideal consists entirely of zero divisors then R is not a complete intersection ring.) If R has dimension 0, then showed that it is a complete intersection ring if and only if the Fitting ideal of its maximal ideal is non-zero.
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So, for commutative rings, the nilradical is contained in the Jacobson radical. The Jacobson radical is very similar to the nilradical in an intuitive sense. A weaker notion of being bad, weaker than being a zero divisor, is being a non-unit (not invertible under multiplication). The Jacobson radical of a ring consists of elements that satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not "act as a unit" in any module "internal to the ring".
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Cooper's own work has mainly been in elementary number theory, especially work related to digital representations of numbers. He collaborated extensively with Robert E. Kennedy. They have worked with Niven numbers, among other results, showing that no 21 consecutive integers can all be Niven numbers, and introduced the notion of tau numbers, numbers whose total number of divisors are itself a divisor of the number.. Independent of Kennedy, Cooper has also done work about generalizations of geometric series, and their application to probability.. Cooper is also the editor of the publication Fibonacci Quarterly.
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In this section, R is an integral domain. Given elements a and b of R, one says that a divides b, or that a is a divisor of b, or that b is a multiple of a, if there exists an element x in R such that ax = b. The units of R are the elements that divide 1; these are precisely the invertible elements in R. Units divide all other elements. If a divides b and b divides a, then a and b are associated elements or associates.
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Let n = p1 p2 … pk where the pi are distinct primes, then φ(n) = (p1 − 1)(p2 − 1)...(pk – 1). If no pi divides any (pj – 1), then n and φ(n) have no common (prime) divisor, and n is cyclic. The first cyclic numbers are 1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, ... .
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In algebra, the content of a polynomial with integer coefficients (or, more generally, with coefficients in a unique factorization domain) is the greatest common divisor of its coefficients. The primitive part of such a polynomial is the quotient of the polynomial by its content. Thus a polynomial is the product of its primitive part and its content, and this factorization is unique up to the multiplication of the content by a unit of the ring of the coefficients (and the multiplication of the primitive part by the inverse of the unit). A polynomial is primitive if its content equals 1.
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He recently spent a year in the Music Department at Harvard University doing research on musical similarity, a branch of music cognition. Since 2005 he has also been a researcher in the Centre for Interdisciplinary Research in Music Media and Technology in the Schulich School of Music at McGill University. He applies computational geometric and discrete mathematics methods to the analysis of symbolically represented music in general, and rhythm in particular. In 2004 he discovered that the Euclidean algorithm for computing the greatest common divisor of two numbers implicitly generates almost all the most important traditional rhythms of the world.
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Multiple crystals can be mixed in various combinations to produce various output frequencies. ;Phase locked loop (PLL): Using a varactor-controlled or voltage- controlled oscillator (VCO) (described above in varactor under analog VFO techniques) and a phase detector, a control-loop can be set up so that the VCO's output is frequency-locked to a crystal-controlled reference oscillator. The phase detector's comparison is made between the outputs of the two oscillators after frequency division by different divisors. Then by altering the frequency-division divisor(s) under computer control, a variety of actual (undivided) VCO output frequencies can be generated.
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In the 1970s, Szpiro's research in commutative algebra led to his proof of the Auslander zero divisor conjecture. Together with Christian Peskine, he developed the liaison theory of algebraic varieties. In the 1980s, Szpiro's research interests shifted to Diophantine geometry, first over function fields and then over number fields. The Institut des hautes études scientifiques described Szpiro as being "the first to realise the importance of a paper by Arakelov for questions of Diophantine geometry", which ultimately led to the development of Arakelov theory as a tool of modern Diophantine geometry exemplified by Gerd Faltings's proof of the Mordell conjecture.
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One application of log structures is the ability to define logarithmic forms on any log scheme. From this, one can for instance define corresponding notions of log-smoothness and log-étaleness which parallel the usual definitions for schemes. This then allows the study of deformation theory. In addition, log structures serve to define the mixed Hodge structure on any smooth variety X, by taking a compactification with boundary a normal crossings divisor D, and writing down the Hodge–De Rham complex associated to X with the standard log structure defined by D.Chris A.M. Peters; Joseph H.M. Steenbrink (2007).
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In principle, it suffices to test only prime divisors. This needs to have a table of prime numbers that may be generated for example with the sieve of Eratosthenes. As the method of factorization does essentially the same work as the sieve of Eratosthenes, it is generally more efficient to test for a divisor only those numbers for which it is not immediately clear whether they are prime or not. Typically, one may proceed by testing 2, 3, 5, and the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3.
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Dividing w_1,\,w_2,\,\ldots,\,w_n,\,W by their greatest common divisor is a way to improve the running time. Even if P≠NP, the O(nW) complexity does not contradict the fact that the knapsack problem is NP-complete, since W, unlike n, is not polynomial in the length of the input to the problem. The length of the W input to the problem is proportional to the number of bits in W, \log W, not to W itself. However, since this runtime is pseudopolynomial, this makes the (decision version of the) knapsack problem a weakly NP-complete problem.
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More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain , "rational numbers" must be replaced by "field of fractions of ". This implies that, if is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials.
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Basic arithmetic processes such as addition, subtraction, multiplication and division were present before the Han Dynasty. The Nine Chapters on the Mathematical Art take these basic operations for granted and simply instruct the reader to perform them. Han mathematicians calculated square and cubed roots in a similar manner as division, and problems on division and root extraction both occur in Chapter Four of The Nine Chapters on the Mathematical Art. Calculating the squared and cubed roots of numbers is done through successive approximation, the same as division, and often uses similar terms such as dividend (shi) and divisor (fa) throughout the process.
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With the help of this method, Vinogradov tackled questions such as the ternary Goldbach problem in 1937 (using Vinogradov's theorem), and the zero-free region for the Riemann zeta function. His own use of it was inimitable; in terms of later techniques, it is recognised as a prototype of the large sieve method in its application of bilinear forms, and also as an exploitation of combinatorial structure. In some cases his results resisted improvement for decades. He also used this technique on the Dirichlet divisor problem, allowing him to estimate the number of integer points under an arbitrary curve.
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Finding conditions for this equation to hold has been one of the major tasks of research in dynamical systems. Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition. If λ1, ..., λν are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others. As terms of the form λi - ∑ (multiples of other eigenvalues) occurs in the denominator of the terms for the function h, the non-resonant condition is also known as the small divisor problem.
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The algorithm is very similar to that provided above for computing the modular multiplicative inverse. There are two main differences: firstly the last but one line is not needed, because the Bézout coefficient that is provided always has a degree less than . Secondly, the greatest common divisor which is provided, when the input polynomials are coprime, may be any non zero elements of ; this Bézout coefficient (a polynomial generally of positive degree) has thus to be multiplied by the inverse of this element of . In the pseudocode which follows, is a polynomial of degree greater than one, and is a polynomial.
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Members of the Council of Representatives are elected through the open list form of party-list proportional representation, using the 18 governorates of Iraq as the constituencies. The counting system uses the modified Sainte-Laguë method with a divisor of 1.7 which is considered as a disadvantage to smaller parties. Eight seats remain reserved for minority groups at the national level: five for Assyrians and one each for Mandaeans, Yazidis, and Shabaks.Iraq Amends Its Electoral Law and Is Ready for Parliamentary Elections in April 2014, historiae, 4 November 20132013 Report on Iraq, United States Commission on International Religious Freedom, p. 7.
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Integer triangles with a 120° angle can be generated bySelkirk, K., "Integer-sided triangles with an angle of 120°", Mathematical Gazette 67, December 1983, 251–255. :a = m^2 + mn + n^2, \, :b = 2mn+n^2, \, :c = m^2 - n^2 \, with coprime integers m, n with 0 < n < m (the angle of 120° is opposite to the side of length a). From here, all primitive solutions can be obtained by dividing a, b, and c by their greatest common divisor. The smallest solution, for m=2 and n=1, is the triangle with sides (3,5,7).
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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the natural numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple and the unique infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities.
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The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147\. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers.
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In other words, we can think of reflecting the rectangle rather than the path segments. Reducing to the coprime case: It is convenient to rescale the rectangle dividing a and b by their greatest common divisor, operation which does not alter the geometry of the path (e.g. the number of bouncing points). Reversing the time: The motion of the path is “time reversible”, meaning that if the path is currently traversing one particular unit square (in a particular direction), then there is absolutely no doubt from which unit square and from which direction it just came.
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Cruzeiro do Sul (, Southern Cross) is a municipality located on the Juruá river in the west of the Brazilian state of Acre. It is the second-largest city in Acre. It is bordered to the north by the state of Amazonas, to the south by Peru, to the east by the municipality of Porto Walter, to the west by the municipality of Rodrigues Alves, to the northwest by the municipality of Tarauacá, and to the northwest by the municipality of Mâncio Lima. The municipality contains 23% of the Serra do Divisor National Park, created in 1989.
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Several manufacturers show that their motors can easily maintain the 3% or 5% equality of step travel size as step size is reduced from full stepping down to 1/10 stepping. Then, as the microstepping divisor number grows, step size repeatability degrades. At large step size reductions it is possible to issue many microstep commands before any motion occurs at all and then the motion can be a "jump" to a new position. Some stepper controller ICs use increased current to minimise such missed steps, especially when the peak current pulses in one phase would otherwise be very brief.
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Because of this uniqueness, Euclidean division is often considered without referring to any method of computation, and without explicitly computing the quotient and the remainder. The methods of computation are called integer division algorithms, the best known of which being long division. Euclidean division, and algorithms to compute it, are fundamental for many questions concerning integers, such as the Euclidean algorithm for finding the greatest common divisor of two integers, and modular arithmetic, for which only remainders are considered. The operation consisting of computing only the remainder is called the modulo operation, and is used often in both mathematics and computer science.
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Proclus provides the only reference ascribing the Elements to Euclid. Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes (known as the Euclid–Euler theorem), the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers. The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible.
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However, the sequence cannot begin repeating itself any earlier than that. If it did, the length of the repeating section would have to be a divisor of p, so it would have to be 1 (since p is prime). But this contradicts our assumption that x0 is not a fixed point of Ta. In other words, the orbit contains exactly p distinct points. This holds for every orbit of S. Therefore, the set S, which contains ap − a points, can be broken up into orbits, each containing p points, so ap − a is divisible by p.
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In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM). A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian).
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An initial natural number is given, and players alternate choosing positive divisors of the initial number, but may not choose 1 or a multiple of a previously chosen divisor. This game models n-dimensional Chomp, where the initial natural number has n prime factors and the dimensions of the Chomp board are given by the exponents of the primes in its prime factorization. Ordinal Chomp is played on an infinite board with some of its dimensions ordinal numbers: for example a 2 × (ω + 4) bar. A move is to pick any block and remove all blocks with both indices greater than or equal the corresponding indices of the chosen block.
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This means that since 2012 only the player's 52 most recent tournaments (within the two-year period) are used to calculate his ranking average.Official World Ranking Board Approves Introduction of Maximum Divisor July 15, 2009 At first only the Championship Committee of the Royal and Ancient used the rankings for official purposes, but the PGA Tour recognized them in 1990, and in 1997 all five of the then principal men's golf tours did so. The rankings, which had previously been called the Sony Rankings, were renamed the Official World Golf Rankings at that time. They are run from offices in Virginia Water in Surrey, England.
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A set of integers S = {a1, a2, .... an} can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6, 10, 15 are coprime because 1 is the only positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime (or pairwise relatively prime, mutually coprime or mutually relatively prime). Pairwise coprimality is a stronger condition than setwise coprimality; every pairwise coprime finite set is also setwise coprime, but the reverse is not true.
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For wheel factorization, one starts from a small list of numbers, called the basis — generally the first few prime numbers; then one generates the list, called the wheel, of the integers that are coprime with all numbers of the basis. Then to find the smallest divisor of the number to be factorized, one divides it successively by the numbers in the basis, and in the wheel. With the basis {2, 3}, this method reduces the number of divisions to of the number necessary for trial division. Larger bases reduce this proportion even further; for example, with basis {2, 3, 5} to ; and with basis {2, 3, 5, 7} to .
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When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeatedly (computed).
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The emphasis on algebraic surfaces--algebraic varieties of dimension two--followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor). The classification of algebraic surfaces was a bold and successful attempt to repeat the division of algebraic curves by their genus g. The division of curves corresponds to the rough classification into the three types: g = 0 (projective line); g = 1 (elliptic curve); and g > 1 (Riemann surfaces with independent holomorphic differentials).
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More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.Lazarsfeld (2004), Definition 1.4.1. (The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.) A line bundle may also be called an invertible sheaf. The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" and "numerically effective", as well as for the phrase "numerically eventually free".Reid (1983), section 0.12f.
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Graphical demonstration that 1 = 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/(2×3×11×23×31). Each row of k squares of side length 1/k has total area 1/k, and all the squares together exactly cover a larger square with area 1. The bottom row of 47058 squares with side length 1/47058 is too small to see in the figure and is not shown. In number theory, Znám's problem asks which sets of k integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1.
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Roughly fifteen years later in England, Henry Goodwyn set out to create a much more ambitious version of Haros’ table. In particular, Goodwyn wanted to tabulate the decimal values for all irreducible fractions with denominators less than or equal to 1,024. There are 318,963 such fractions. As a warm up and a test of the commercial market for such a table in 1816 he published for private circulation The First Centenary of a Series of Concise and Useful Tables of all the Complete Decimal Quotients, which can arise from dividing a unit, or any whole Number less than each Divisor by all Integers from 1 to 1024.
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Mathematically, the number of strands is the greatest common divisor of the number of leads and the number of bights. The knot may be tied with a single strand if and only if the two numbers are coprime. For example, 3 lead × 5 bight (3×5), or 5 lead × 7 bight (5×7). Turk's head knots on netting There are three groupings of Turk's head knots: # Narrow, where the number of leads is two or more less than the number of bights (3×5, or 3×7). # Long or wide where the number of leads is two or more greater than the number of bights (5×3, or 16×7).
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The cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal.
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Berndtsson's first results concern zero sets of holomorphic functions, and in 1981 he showed that any divisor with finite area in the unit ball in the two-dimensional complex space is defined by a bounded holomorphic function (which is not true in higher dimensions). In the 1980s he also developed (together with Mats Andersson) a formalism to generate weighted integral representation formulas for holomorphic functions and solutions to the so-called dbar- equation,Berndtsson, B.; Andersson, M. Henkin-Ramirez formulas with weight factors. Annales de l'Institut Fourier 32 (1982), no. 3, v--vi, 91--110 which is the higher-dimensional generalization of the Cauchy–Riemann equations in the plane.
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If m and n are nonzero integers, and more generally, nonzero elements of an integral domain, it is said that m divides n, m is a divisor of n, or n is a multiple of m, and this is written as :m \mid n, if there exists an integer k, or an element k of the integral domain, such that mk = n.for instance, or This definition is sometimes extended to include zero. This does not add much to the theory, as 0 does not divide any other number, and every number divides 0. On the other hand, excluding zero from the definition simplifies many statements.
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Although the original problem asks for integer lattice points in a circle, there is no reason not to consider other shapes, for example conics; indeed Dirichlet's divisor problem is the equivalent problem where the circle is replaced by the rectangular hyperbola. Similarly one could extend the question from two dimensions to higher dimensions, and ask for integer points within a sphere or other objects. There is an extensive literature on these problems. If one ignores the geometry and merely considers the problem an algebraic one of Diophantine inequalities then there one could increase the exponents appearing in the problem from squares to cubes, or higher.
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496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case. A triangular number and a hexagonal number, 496 is also a centered nonagonal number.
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In 1910, Mirimanoff expanded the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide . Granville and Monagan further proved that p2 must actually divide for every prime m ≤ 89. Suzuki extended the proof to all primes m ≤ 113. Let Hp be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, (x + y)p−1 ≡ 1 (mod p2), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p.
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A diagram of a long basket weave knot on a 2×5 grid A basket weave knot is made up of two sets of parallel lines drawn inside a rectangle such that the lines meet at the edges of the rectangle. For a true basket weave knot that can be tied with two strands, the number of intersections in each direction cannot have a common divisor. Within this constraint, there is no theoretical upper limit to the size of a basket weave knot. Thus, a knot that has two intersections in one direction can be lengthened with any odd number in the perpendicular direction.
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In the first stage, the fixed seats are distributed within each district according to the modified Sainte-Laguë method (jämkade uddatalsmetoden) with the first divisor adjusted to 1.2 (1.4 in elections before 2018). Only parties that have received at least 4 percent of the vote nationally or 12 percent of the vote within the district can participate in this distribution of seats. In the second stage, the 349 seats are distributed through a calculation based on the total number of votes summed up across the entire country. In this distribution only parties that have received more than 4 percent of the national vote are included.
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Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups. Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity.
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Baños de Agua Santa is an important tourist site The Andes is the watershed divisor between the Amazon watershed, which runs to the east, and the Pacific, including the north–south rivers Mataje, Santiago, Esmeraldas, Chone, Guayas, Jubones, and Puyango-Tumbes. Almost all of the rivers in Ecuador form in the Sierra region and flow east toward the Amazon River or west toward the Pacific Ocean. The rivers rise from snowmelt at the edges of the snowcapped peaks or from the abundant precipitation that falls at higher elevations. In the Sierra region, the streams and rivers are narrow and flow rapidly over precipitous slopes.
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When r is a divisor of n, the Turán graph is symmetric and strongly regular, although some authors consider Turán graphs to be a trivial case of strong regularity and therefore exclude them from the definition of a strongly regular graph. The Turán graph T(n,\lceil n/3\rceil) has 3a2b maximal cliques, where 3a + 2b = n and b ≤ 2; each maximal clique is formed by choosing one vertex from each partition subset. This is the largest number of maximal cliques possible among all n-vertex graphs regardless of the number of edges in the graph (Moon and Moser 1965); these graphs are sometimes called Moon–Moser graphs.
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Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India—or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words.
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Some properties of the GCD are in fact easier to see with this description, for instance the fact that any common divisor of a and b also divides the GCD (it divides both terms of ua + vb). The equivalence of this GCD definition with the other definitions is described below. The GCD of three or more numbers equals the product of the prime factors common to all the numbers, but it can also be calculated by repeatedly taking the GCDs of pairs of numbers. For example, : Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers.
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Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the Euclidean algorithm; Elements, Prop. VII.2) and the first known proof of the infinitude of primes (Elements, Prop. IX.20).
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Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials P with integer coefficients that always take on even number values are just those such that P/2 is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer coefficients of the binomial coefficients. In questions of prime number theory, such as Schinzel's hypothesis H and the Bateman–Horn conjecture, it is a matter of basic importance to understand the case when P has no fixed prime divisor (this has been called Bunyakovsky's property, after Viktor Bunyakovsky).
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Conway's prime generating algorithm above is essentially a quotient and remainder algorithm within two loops. Given input of the form 2^n 7^m where 0 ≤ m < n, the algorithm tries to divide n+1 by each number from n down to 1, until it finds the largest number k that is a divisor of n+1. It then returns 2n+1 7k-1 and repeats. The only times that the sequence of state numbers generated by the algorithm produces a power of 2 is when k is 1 (so that the exponent of 7 is 0), which only occurs if the exponent of 2 is a prime.
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In this context commensurability means that the ratio of the two planets' mean motions is very nearly equal to a ratio between a pair of small whole numbers. Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's. The corresponding difference between multiples of the mean motions, , corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period. As a result, the integrated perturbations with this period are disproportionately large, about 0.8° degrees of arc in orbital longitude for Saturn and about 0.3° for Jupiter.
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Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n2) operations in Fq using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in Fq using "fast" arithmetic. A Euclidean division (division with remainder) can be performed within the same time bounds. The cost of a polynomial greatest common divisor between two polynomials of degree at most n can be taken as O(n2) operations in Fq using classical methods, or as O(nlog2(n) log(log(n)) ) operations in Fq using fast methods.
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In arithmetic, short division is a division algorithm which breaks down a division problem into a series of easy steps. It is an abbreviated form of long division — whereby the products are omitted and the partial remainders are notated as superscripts. As a result, a short division tableau is always more notationally efficient than its long division counterpart — though sometimes at the expense of relying on mental arithmetic, which could limit the size of the divisor. For most people, small integer divisors up to 12 are handled using memorised multiplication tables, although the procedure could also be adapted to the larger divisors as well.
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The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by W. W. Rouse Ball in his Mathematical Recreations.. In A Mathematician's Apology, G. H. Hardy criticized Rouse Ball for including this problem, writing: :"These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to a mathematician. The proofs are neither difficult nor interesting—merely tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and proofs, which are not capable of any significant generalization.".
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Let N be a rational prime, and define :J0(N) = J as the Jacobian variety of the modular curve :X0(N) = X. There are endomorphisms Tl of J for each prime number l not dividing N. These come from the Hecke operator, considered first as an algebraic correspondence on X, and from there as acting on divisor classes, which gives the action on J. There is also a Fricke involution w (and Atkin–Lehner involutions if N is composite). The Eisenstein ideal, in the (unital) subring of End(J) generated as a ring by the Tl, is generated as an ideal by the elements : Tl − l - 1 for all l not dividing N, and by :w + 1.
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Suppose G is a finite group of order n, and d is a divisor of n. The number of order-d-elements in G is a multiple of φ(d) (possibly zero), where φ is Euler's totient function, giving the number of positive integers no larger than d and coprime to it. For example, in the case of S3, φ(3) = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility for composite d such as d=6, since φ(6)=2, and there are zero elements of order 6 in S3.
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Some Diophantine equations have no solution, some have one or a finite number, and others have infinitely many solutions. The monkey and the coconuts reduces algebraic- ally to a two variable linear Diophantine equation of the form :ax + by = c, or more generally, :(a/d)x + (b/d)y = c/d where d is the greatest common divisor of a and b.d can be found if necessary via Euclid's algorithm By Bézout's identity, the equation is solvable if and only if d divides c. If it does, the equation has infinitely many periodic solutions of the form :x = x0 \+ tb, :y = y0 \+ ta where (x0,y0) is a solution and t is a parameter than can be any integer.
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In algebra, a torsion-free module is a module over a ring such that zero is the only element annihilated by a regular element (non zero-divisor) of the ring. In integral domains the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the zero module.
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See, for example, Euclid's algorithm for finding the greatest common divisor of two numbers. By the High Middle Ages, the positional Hindu–Arabic numeral system had reached Europe, which allowed for systematic computation of numbers. During this period, the representation of a calculation on paper actually allowed calculation of mathematical expressions, and the tabulation of mathematical functions such as the square root and the common logarithm (for use in multiplication and division) and the trigonometric functions. By the time of Isaac Newton's research, paper or vellum was an important computing resource, and even in our present time, researchers like Enrico Fermi would cover random scraps of paper with calculation, to satisfy their curiosity about an equation.
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Therefore, the divisors form a finite distributive lattice. One may associate each divisor with the set of prime powers that divide it: thus, 12 is associated with the set {2,3,4}, while 20 is associated with the set {2,4,5}. Then 12 ∧ 20 = 4 is associated with the set {2,3,4} ∩ {2,4,5} = {2,4}, while 12 ∨ 20 = 60 is associated with the set {2,3,4} ∪ {2,4,5} = {2,3,4,5}, so the join and meet operations of the lattice correspond to union and intersection of sets. The prime powers 2, 3, 4, 5, and 8 appearing as elements in these sets may themselves be partially ordered by divisibility; in this smaller partial order, 2 ≤ 4 ≤ 8 and there are no order relations between other pairs.
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Flowchart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" or "true" (more accurately, the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
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In the case of , it may be stated as follows. Given two polynomials and of respective degrees and , if their monic greatest common divisor has the degree , then there is a unique pair of polynomials such that :ap+bq=g, and :\deg (a)\le n-d, \quad \deg(b) (For making this true in the limiting case where or , one has to define as negative the degree of the zero polynomial. Moreover, the equality \deg (a)= n-d can occur only if and are associated.) The uniqueness property is rather specific to . In the case of the integers the same property is true, if degrees are replaced by absolute values, but, for having uniqueness, one must require .
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Another definition of the GCD is helpful in advanced mathematics, particularly ring theory. The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua + vb where u and v are integers. The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer). In modern mathematical language, the ideal generated by a and b is the ideal generated by g alone (an ideal generated by a single element is called a principal ideal, and all ideals of the integers are principal ideals).
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Bézout's identity is essential to many applications of Euclid's algorithm, such as demonstrating the unique factorization of numbers into prime factors. To illustrate this, suppose that a number L can be written as a product of two factors u and v, that is, L = uv. If another number w also divides L but is coprime with u, then w must divide v, by the following argument: If the greatest common divisor of u and w is 1, then integers s and t can be found such that : by Bézout's identity. Multiplying both sides by v gives the relation : Since w divides both terms on the right-hand side, it must also divide the left-hand side, v.
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A Lehmer sieve, which is a primitive digital computer once used for finding primes and solving simple Diophantine equations.While the word algorithm goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period. An interesting early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in Elements, together with a proof of correctness.
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Most root- finding algorithms behave badly when there are multiple roots or very close roots. However, for polynomials whose coefficients are exactly given as integers or rational numbers, there is an efficient method to factorize them into factors that have only simple roots and whose coefficients are also exactly given. This method, called square-free factorization, is based on the multiple roots of a polynomial being the roots of the greatest common divisor of the polynomial and its derivative. The square-free factorization of a polynomial p is a factorization p=p_1p_2^2\cdots p_k^k where each p_i is either 1 or a polynomial without multiple roots, and two different p_i do not have any common root.
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If d is a divisor of n, then the number of elements in Z/nZ which have order d is φ(d), and the number of elements whose order divides d is exactly d. If G is a finite group in which, for each , G contains at most n elements of order dividing n, then G must be cyclic. The order of an element m in Z/nZ is n/gcd(n,m). If n and m are coprime, then the direct product of two cyclic groups Z/nZ and Z/mZ is isomorphic to the cyclic group Z/nmZ, and the converse also holds: this is one form of the Chinese remainder theorem.
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Division can only be performed if the result has no remainder (i.e., the divisor is a factor of the numerator). Fractions are not allowed, and only positive integers may be obtained as a result at any stage of the calculation. As in the letters rounds, any contestant who does not write down their calculations in time must go first if both declare the same result, and both contestants must show their work to each other if their results and calculations are identical. Only the contestant whose result is closer to the target number scores points: 10 for reaching it exactly, 7 for being 1–5 away, 5 for being 6–10 away.
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No algorithm is known for computing any of these square-free factors which is faster than computing the complete prime factorization. In particular, there is no known polynomial-time algorithm for computing the square-free part of an integer, or even for determining whether an integer is square-free. In contrast, polynomial-time algorithms are known for primality testing. This is a major difference between the arithmetic of the integers, and the arithmetic of the univariate polynomials, as polynomial-time algorithms are known for square-free factorization of polynomials (in short, the largest square-free factor of a polynomial is its quotient by the greatest common divisor of the polynomial and its formal derivative).
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The Frobenius number exists as long as the set of coin denominations has no common divisor greater than 1. There is an explicit formula for the Frobenius number when there are only two different coin denominations, x and y: xy − x − y. If the number of coin denominations is three or more, no explicit formula is known; but, for any fixed number of coin denominations, there is an algorithm computing the Frobenius number in polynomial time (in the logarithms of the coin denominations forming an input). No known algorithm is polynomial time in the number of coin denominations, and the general problem, where the number of coin denominations may be as large as desired, is NP-hard.
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However, the space used to represent 2^{2^n} is proportional to 2^n, and thus exponential rather than polynomial in the space used to represent the input. Hence, it is not possible to carry out this computation in polynomial time on a Turing machine, but it is possible to compute it by polynomially many arithmetic operations. Conversely, there are algorithms that run in a number of Turing machine steps bounded by a polynomial in the length of binary-encoded input, but do not take a number of arithmetic operations bounded by a polynomial in the number of input numbers. The Euclidean algorithm for computing the greatest common divisor of two integers is one example.
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Typically an n-bit CRC applied to a data block of arbitrary length will detect any single error burst not longer than n bits, and the fraction of all longer error bursts that it will detect is . Specification of a CRC code requires definition of a so-called generator polynomial. This polynomial becomes the divisor in a polynomial long division, which takes the message as the dividend and in which the quotient is discarded and the remainder becomes the result. The important caveat is that the polynomial coefficients are calculated according to the arithmetic of a finite field, so the addition operation can always be performed bitwise- parallel (there is no carry between digits).
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If one of those values is 0, then we have found a root (and so a factor). If none are 0, then each one has a finite number of divisors. Now, 2 can only factor as :1×2, 2×1, (−1)×(−2), or (−2)×(−1). Therefore, if a second degree integer polynomial factor exists, it must take one of the values :1, 2, −1, or −2 at x=0, and likewise at x=-1. There are eight different ways to factor 6 (one for each divisor of 6), so there are :4×4×8 = 128 possible combinations, of which half can be discarded as the negatives of the other half, corresponding to 64 possible second degree integer polynomials that must be checked.
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The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra. In the case of univariate polynomials over a field, any multiple factor of a polynomial introduces a nontrivial common factor of f and its formal derivative f ′, so a sufficient condition for f to be square-free is that the greatest common divisor of f and f ′ is 1\.
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The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
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180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a refactorable number. 180 is the sum of two square numbers: 122 \+ 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37.
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Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on K and occur only to exponent one. The modulus m is a product of a non- archimedean (finite) part mf and an archimedean (infinite) part m∞. The non- archimedean part mf is a nonzero ideal in the ring of integers OK of K and the archimedean part m∞ is simply a set of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals.
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For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski-Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a branch locus, namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possible "open subsets of failure" to be an étale morphism.
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An informal definition could be "a set of rules that precisely defines a sequence of operations",Stone 1973:4 which would include all computer programs (including programs that do not perform numeric calculations), and (for example) any prescribed bureaucratic procedure or cook-book recipe. In general, a program is only an algorithm if it stops eventuallyStone simply requires that "it must terminate in a finite number of steps" (Stone 1973:7–8). \- even though infinite loops may sometimes prove desirable. A prototypical example of an algorithm is the Euclidean algorithm, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the flowchart above and as an example in a later section.
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The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form . Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains.
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Twenty-eight is the sum of the totient function for the first nine integers. Since the greatest prime factor of 282 \+ 1 = 785 is 157, which is more than 28 twice, 28 is a Størmer number. Twenty-eight is a harmonic divisor number, a happy number, a triangular number, a hexagonal number, and a centered nonagonal number. It appears in the Padovan sequence, preceded by the terms 12, 16, 21 (it is the sum of the first two of these). It is also a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 2, 8, 10, 18, 28... Twenty-eight is the third positive integer with a prime factorization of the form 2q where q is an odd prime.
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An ECL procedure for computing the greatest common divisor of two integers according to the Euclidean algorithm could be defined as follows: gcd <\- EXPR(m:INT BYVAL, n: INT BYVAL; INT) BEGIN DECL r:INT; REPEAT r <\- rem(m, n); r = 0 => n; m <\- n; n <\- r; END; END This is an assignment of a procedure constant to the variable `gcd`. The line EXPR(m:INT BYVAL, n: INT BYVAL; INT) indicates that the procedure takes two parameters, of type `INT`, named `m` and `n`, and returns a result of type `INT`. (Data types are called modes in ECL.) The bind-class `BYVAL` in each parameter declaration indicates that that parameter is passed by value. The computational components of an ECL program are called forms.
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Next, :(98+2) \cdot a + b and again expanding :98 \cdot a + 2 \cdot a + b, and after eliminating the known multiple of 7, the result is :2 \cdot a + b, which is the rule "double the number formed by all but the last two digits, then add the last two digits". Case where the last digit(s) is multiplied by a factor The representation of the number may also be multiplied by any number relatively prime to the divisor without changing its divisibility. After observing that 7 divides 21, we can perform the following: :10 \cdot a + b, after multiplying by 2, this becomes :20 \cdot a + 2 \cdot b, and then :(21 - 1) \cdot a + 2 \cdot b.
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The Nevanlinna invariant has similar formal properties to the abscissa of convergence of the height zeta function and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let X be a projective variety over a number field K with ample divisor D giving rise to an embedding and height function H, and let U denote a Xariski open subset of X. Let α = α(D) be the Nevanlinna invariant of D and β the abscissa of convergence of Z(U, H; s). Then for every ε > 0 there is a U such that β < α + ε: in the opposite direction, if α > 0 then α = β for all sufficiently large fields K and sufficiently small U.
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The divisor is chosen as necessary so that the resulting quotients, disregarding any fractional remainders, sum to the required total; in other words, pick a number so that there is no need to examine the remainders. Any number in one range of quotas will accomplish this, with the highest number in the range always being the same as the lowest number used by the D'Hondt method to award a seat (if it is used rather than the Jefferson method), and the lowest number in the range being the smallest number larger than the next number which would award a seat in the D'Hondt calculations. Applied to the above example of party lists, this range extends as integers from 20,001 to 25,000. More precisely, any number n for which 20,000 < n ≤ 25,000 can be used.
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An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings. A module M over a ring R is called a torsion module if all its elements are torsion elements, and torsion-free if zero is the only torsion element.
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These numbers (or at least one of them) is often chosen among primes to create an even contact between every cog of both wheels, and thereby avoid unnecessary wear and damage. An even uniform gear wear is achieved by ensuring the tooth counts of the two gears meshing together are relatively prime to each other; this occurs when the greatest common divisor (GCD) of each gear tooth count equals 1, e.g. GCD(16,25)=1; if a 1:1 gear ratio is desired a relatively prime gear may be inserted in between the two gears; this maintains the 1:1 ratio but reverses the gear direction; a second relatively prime gear could also be inserted to restore the original rotational direction while maintaining uniform wear with all 4 gears in this case.
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The analogous conjecture with the integers replaced by the one- variable polynomial ring over a finite field is false. For example, Swan noted in 1962 (for reasons unrelated to Hypothesis H) that the polynomial ::x^8 + u^3\, over the ring F2[u] is irreducible and has no fixed prime polynomial divisor (after all, its values at x = 0 and x = 1 are relatively prime polynomials) but all of its values as x runs over F2[u] are composite. Similar examples can be found with F2 replaced by any finite field; the obstructions in a proper formulation of Hypothesis H over F[u], where F is a finite field, are no longer just local but a new global obstruction occurs with no classical parallel, assuming hypothesis H is in fact correct.
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Unlike many similar mathematical games, sylver coinage has not been completely solved, mainly because many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions, due to R. L. Hutchings, guarantees that such a position has a winning strategy but does not identify the strategy. Hutchings's Theorem states that any of the prime numbers 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known. When the greatest common divisor of the moves that have been made so far is 1, the remaining set of numbers that can be played will be a finite set, and can be described mathematically as the set of gaps of a numerical semigroup.
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Consider the simple set of simultaneous congruences : x ≡ 3 (mod 4) : x ≡ 5 (mod 6) Now, for x ≡ 3 (mod 4) to be true, x = 3 + 4j for some integer j. Substitute this in the second equation : 3+4j ≡ 5 (mod 6) since we are looking for a solution to both equations. Subtract 3 from both sides (this is permitted in modular arithmetic) : 4j ≡ 2 (mod 6) We simplify by dividing by the greatest common divisor of 4,2 and 6. Division by 2 yields: : 2j ≡ 1 (mod 3) The Euclidean modular multiplicative inverse of 2 mod 3 is 2. After multiplying both sides with the inverse, we obtain: : j ≡ 2 × 1 (mod 3) or : j ≡ 2 (mod 3) For the above to be true: j = 2 + 3k for some integer k.
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For, if not, it would be congruent to 1 mod 3 and 2p + 1 would be congruent to 3 mod 3, impossible for a prime number.. Similar restrictions hold for larger prime moduli, and are the basis for the choice of the "correction factor" 2C in the Hardy–Littlewood estimate on the density of the Sophie Germain primes. If a Sophie Germain prime p is congruent to 3 (mod 4) (, Lucasian primes), then its matching safe prime 2p + 1 will be a divisor of the Mersenne number 2p − 1\. Historically, this result of Leonhard Euler was the first known criterion for a Mersenne number with a prime index to be composite.. It can be used to generate the largest Mersenne numbers (with prime indices) that are known to be composite..
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In algebra, Gauss's lemma,Article 42 of Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801) named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers.
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The text of the Suan shu shu is however much less systematic than the Nine Chapters, and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Book of Computations contains many perquisites to problems that would be expanded upon in The Nine Chapters on the Mathematical Art. An example of the elementary mathematics in the Suàn shù shū, the square root is approximated by using false position method which says to "combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend." Furthermore, The Book of Computations solves systems of two equations and two unknowns using the same false position method.
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Mauritia flexuosa, or moriche palm, is an economically important species dominant in some parts of the ecoregion. The southwest Amazon moist forest region covers an extensive area of the Upper Amazon Basin comprising four sub-basins: (1) both the Pastaza-Marañon and (2) Ucayali River sub-basins drain into the Upper Amazon River in Peru; (3) the Acre and (4) Madre de Dios- Beni sub-basins drain to the east into the Juruá, Purus and Madeira Rivers; which, in turn, feed into the Amazon River lower down in Brazil. The region is bisected north to south between Peru and Brazil by the small mountain range Serra do Divisor. It extends east to the edge of the Purus Arch, or ancient zone of uplift, in the southwestern area of the Brazilian State of Amazonas.
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For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing :A[n] \times A^\vee[n] \longrightarrow \mu_n for all n prime to the characteristic of K.James Milne, Abelian Varieties, available at www.jmilne.org/math/ Here A^\vee denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation :\lambda: A \longrightarrow A^\vee, then composition gives a (possibly degenerate) pairing :A[n] \times A[n] \longrightarrow \mu_n. If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians).
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In general, for any finite group G of order n, it is straightforward to determine the signature of the permutation πg made by left-multiplication by the element g of G. The permutation πg will be even, unless there are an odd number of orbits of even size. Assuming n even, therefore, the condition for πg to be an odd permutation, when g has order k, is that n/k should be odd, or that the subgroup generated by g should have odd index. We will apply this to the group of nonzero numbers mod p, which is a cyclic group of order p − 1\. The jth power of a primitive root modulo p will by index calculus have index the greatest common divisor :i = (j, p − 1).
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At any step, it may perform one of three actions: it may copy any pointer it has to another object in memory, it may apply and replace any of its pointers by a pointer to the next object in the sequence, or it may apply a subroutine for determining whether two of its pointers represent equal values in the sequence. The equality test action may involve some nontrivial computation: for instance, in Pollard's rho algorithm, it is implemented by testing whether the difference between two stored values has a nontrivial greatest common divisor with the number to be factored. In this context, by analogy to the pointer machine model of computation, an algorithm that only uses pointer copying, advancement within the sequence, and equality tests may be called a pointer algorithm.
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For m=0 the generalized Jacobian Jm is just the usual Jacobian J, an abelian variety of dimension g, the genus of C. For m a nonzero effective divisor the generalized Jacobian is an extension of J by a connected commutative affine algebraic group Lm of dimension deg(m)−1. So we have an exact sequence :0 → Lm → Jm → J → 0 The group Lm is a quotient :0 → Gm → ΠRi → Lm → 0 of a product of groups Ri by the multiplicative group Gm of the underlying field. The product runs over the points Pi in the support of m, and the group Ri is the group of invertible elements of the local ring modulo those that are 1 mod m. The group Ri has dimension ni, the number of times Pi occurs in m.
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The Euclidean rhythm in music was discovered by Godfried Toussaint in 2004 and is described in a 2005 paper "The Euclidean Algorithm Generates Traditional Musical Rhythms".The Euclidean algorithm generates traditional musical rhythms by G. T. Toussaint, Proceedings of BRIDGES: Mathematical Connections in Art, Music, and Science, Banff, Alberta, Canada, July 31 to August 3, 2005, pp. 47–56. The greatest common divisor of two numbers is used rhythmically giving the number of beats and silences, generating almost all of the most important world music rhythms,Comparative Musicology – Musical Rhythm and Mathematics (except Indian).The Euclidean Algorithm Generates Traditional Musical Rhythms, by Godfried Toussaint, Extended version of the paper that appeared in the Proceedings of BRIDGES: Mathematical Connections in Art, Music and Science’’, Banff, Alberta, Canada, July 31–August 3, 2005, pp. 47–56.
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These problems already trivially lie in co-NP. This was the first strong evidence that these problems are not NP-complete, since if they were, it would imply that NP is subset of co-NP, a result widely believed to be false; in fact, this was the first demonstration of a problem in NP intersect co-NP not known, at the time, to be in P. Producing certificates for the complement problem, to establish that a number is composite, is straightforward: it suffices to give a nontrivial divisor. Standard probabilistic primality tests such as the Baillie–PSW primality test, the Fermat primality test, and the Miller–Rabin primality test also produce compositeness certificates in the event where the input is composite, but do not produce certificates for prime inputs.
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A systematic code is one in which the message appears verbatim somewhere within the codeword. Therefore, systematic BCH encoding involves first embedding the message polynomial within the codeword polynomial, and then adjusting the coefficients of the remaining (non-message) terms to ensure that s(x) is divisible by g(x). This encoding method leverages the fact that subtracting the remainder from a dividend results in a multiple of the divisor. Hence, if we take our message polynomial p(x) as before and multiply it by x^{n-k} (to "shift" the message out of the way of the remainder), we can then use Euclidean division of polynomials to yield: :p(x)x^{n-k} = q(x)g(x) + r(x) Here, we see that q(x)g(x) is a valid codeword.
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The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor.. This applies notably to rational expressions over a field. The irreducible fraction for a given element is unique up to multiplication of denominator and numerator by the same invertible element. In the case of the rational numbers this means that any number has two irreducible fractions, related by a change of sign of both numerator and denominator; this ambiguity can be removed by requiring the denominator to be positive. In the case of rational functions the denominator could similarly be required to be a monic polynomial..
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All tournaments recognised by the world's professional tours, and some leading invitational events, were graded into categories ranging from major championship (whose winners would receive 50 points) to "other tournaments" (whose winners would receive a minimum of 8). In all events, other finishers received points on a diminishing scale that began with runners-up receiving 60% of the winners' points, and the number of players in the field receiving points would be the same as the points awarded to the winner. In a major, for example, all players finishing 30th to 40th would receive 2 points, and all players finishing 50th or higher, 1 point. Beginning in April 1989, the rankings were changed to be based on the average points per event played instead of simply total points earned, subject to a minimum divisor of 60 (20 events per year).
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Each digit of the product depends only on a neighborhood of two digits in the given number: the digit in the same position and the digit one position to the right. More generally, multiplication or division of doubly infinite digit sequences in any radix , by a multiplier or divisor all of whose prime factors are also prime factors of , is an operation that forms a cellular automaton because it depends only on a bounded number of nearby digits, and is reversible because of the existence of multiplicative inverses., p. 1093. Multiplication by other values (for instance, multiplication of decimal numbers by three) remains reversible, but does not define a cellular automaton, because there is no fixed bound on the number of digits in the initial value that are needed to determine a single digit in the result.
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Starting state of the standard puzzle Water pouring puzzles (also called water jug problems, decanting problems or measuring puzzles) are a class of puzzle involving a finite collection of water jugs of known integer capacities (in terms of a liquid measure such as liters or gallons). Initially each jug contains a known integer volume of liquid, not necessarily equal to its capacity. Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs. By Bézout's identity, such puzzles have solution if and only if the desired volume is a multiple of the greatest common divisor of all the integer volume capacities of jugs.
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Hence, at the very least, our varieties must have nK_{X'} to be a Cartier divisor for some positive integer n.) The first key result is the cone theorem of Shigefumi Mori, describing the structure of the cone of curves of X. Briefly, the theorem shows that starting with X, one can inductively construct a sequence of varieties X_i, each of which is "closer" than the previous one to having K_{X_i} nef. However, the process may encounter difficulties: at some point the variety X_i may become "too singular". The conjectural solution to this problem is the flip, a kind of codimension-2 surgery operation on X_i. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model X' in finitely many steps.) showed that flips exist in the 3-dimensional case.
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For any field F, if two polynomials p(x),q(x) ∈ F[x] are relatively prime then they do not have a common root, for if a ∈ F was a common root, then p(x) and q(x) would both be multiples of x − a and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields. If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor. Then, since r(x) is not constant, it will have some root a, which will be then a common root of p(x) and q(x).
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Demonstration, with rods, of the amicability of the pair of numbers (220,284) Amicable numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number. The smallest pair of amicable numbers is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2.
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If G has genus g ≥ 1 then the ΣnC are closely related to the Jacobian variety J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors. For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil's technique of constructing J as an abstract variety from 'birational data'.
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In mathematical terms the problem can be stated: :Given positive integers a1, a2, ..., an such that gcd(a1, a2, ..., an) = 1, find the largest integer that cannot be expressed as an integer conical combination of these numbers, i.e., as a sum :: k1a1 + k2a2 + ··· + knan, :where k1, k2, ..., kn are non-negative integers. This largest integer is called the Frobenius number of the set { a1, a2, ..., an }, and is usually denoted by g(a1, a2, ..., an). The requirement that the greatest common divisor (GCD) equal 1 is necessary in order for the Frobenius number to exist. If the GCD were not 1, then starting at some number m the only sums possible are multiples of the GCD; every number past m that is not divisible by the GCD cannot be represented by any linear combination of numbers from the set.
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Hare was said to have been 'conspicuous for great industry – to have wide interests in life and clearness of intellectual vision'. He was a member of the London-based Political Economy Club and the British Dictionary of National Biography says of him: :Hare's energies were concentrated in an attempt to devise a system which would secure proportional representation of all classes in the United Kingdom, including minorities in the House of Commons and other electoral assemblies. His original electoral system ideas included making The United Kingdom one huge electorate (Hare, in 1854, set the divisor as 654 the number of seats in the UK Parliament) (later he changed this to seven or eight hundred electorates) and that each voter would sign and check his vote. By 1873, however, he had adapted his ideas to take account of the secret vote.
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In mathematics and its applications, the mean square is defined as the arithmetic mean of the squares of a set of numbers or of a random variable, or as the arithmetic mean of the squares of the differences between a set of numbers and a given "origin" that may not be zero (e.g. may be a mean or an assumed mean of the data). When the mean square is calculated relative to a given "target" or "correct" value, or as the mean square of differences from a sequence of correct values, it is known as mean squared error. A typical estimate for the variance from a set of sample values x_i uses a divisor of one less then the number n of values, rather than a simple arithmetic average, and this is still called the mean square (e.g.
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This theorem may be viewed as a generalization of Bézout's identity, which provides a condition under which an integer or a univariate polynomial h may be expressed as an element of the ideal generated by two other integers or univariate polynomials f and g: such a representation exists exactly when h is a multiple of the greatest common divisor of f and g. The AF+BG condition expresses, in terms of divisors (sets of points, with multiplicities), a similar condition under which a homogeneous polynomial H in three variables can be written as an element of the ideal generated by two other polynomials F and G. This theorem is also a refinement, for this particular case, of Hilbert's Nullstellensatz, which provides a condition expressing that some power of a polynomial h (in any number of variables) belongs to the ideal generated by a finite set of polynomials.
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Many commentators saw the latter change as directed at Michelle Wie, who at the time was ranked second in the world despite having competed in only 16 women's professional events in the two-year period. However, the chairman of the Rolex Rankings Technical Committee defended the change as one designed to make the women's rankings more comparable to the Official World Golf Ranking for men, which use a minimum divisor of 40 events. On 16 April 2007, another modification in the formula was introduced. Instead of points being awarded on an accumulated 104-week rolling period, with the points awarded in the most recent 13-week period carrying a stronger value, points began to be reduced in 91 equal decrements following week 13 for the remaining 91 weeks of the two- year Rolex Ranking period rather than the seven equal 13 week decrements previously used.
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For a scheme X of finite type over k, the group of i-cycles rationally equivalent to zero is the subgroup of Z_i(X) generated by the cycles (f) for all (i+1)-dimensional subvarieties W of X and all nonzero rational functions f on W. The Chow group CH_i(X) of i-dimensional cycles on X is the quotient group of Z_i(X) by the subgroup of cycles rationally equivalent to zero. Sometimes one writes [Z] for the class of a subvariety Z in the Chow group, and if two subvarieties Z and W have [Z] = [W], then Z and W are said to be rationally equivalent. For example, when X is a variety of dimension n, the Chow group CH_{n-1}(X) is the divisor class group of X. When X is smooth over k, this is isomorphic to the Picard group of line bundles on X.
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All subgroups and quotient groups of cyclic groups are cyclic. Specifically, all subgroups of Z are of the form ⟨m⟩ = mZ, with m a positive integer. All of these subgroups are distinct from each other, and apart from the trivial group {0} = 0Z, they all are isomorphic to Z. The lattice of subgroups of Z is isomorphic to the dual of the lattice of natural numbers ordered by divisibility.. Thus, since a prime number p has no nontrivial divisors, pZ is a maximal proper subgroup, and the quotient group Z/pZ is simple; in fact, a cyclic group is simple if and only if its order is prime.. All quotient groups Z/nZ are finite, with the exception For every positive divisor d of n, the quotient group Z/nZ has precisely one subgroup of order d, generated by the residue class of n/d. There are no other subgroups.
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For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a regular divisor, and is measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertible sheaf). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of intersection number requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with superabundant divisors, as discussed in the Riemann–Roch theorem for surfaces. Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any k distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the cross ratio.
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Let X be a variety of nonnegative Kodaira dimension over a field of characteristic zero, and let B be the canonical model of X, B = Proj R(X, KX); the dimension of B is equal to the Kodaira dimension of X. There is a natural rational map X – → B; any morphism obtained from it by blowing up X and B is called the Iitaka fibration. The minimal model and abundance conjectures would imply that the general fiber of the Iitaka fibration can be arranged to be a Calabi–Yau variety, which in particular has Kodaira dimension zero. Moreover, there is an effective Q-divisor Δ on B (not unique) such that the pair (B, Δ) is klt, KB \+ Δ is ample, and the canonical ring of X is the same as the canonical ring of (B, Δ) in degrees a multiple of some d > 0.O. Fujino and S. Mori, J. Diff. Geom.
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Therefore, assume that lines and weights are constructed to obey the rule using a common measure (or unit) N, and at a ratio of four to three (as per the sketch). Now, double the length of ED by duplicating the longer arm on the left, and the shorter arm on the right. 270px For demonstration's sake, reorder the lines so that CD is adjacent to LE (the two red lines together), and juxtapose with the original (as below): 271px Weights and levers at a ratio of eight to six. It is clear then, that both lines are double the length of the original line ED, that LH has its centre at E (see adjacent red lines), and HK its centre at D. Note, additionally, that EH (which is equal to CD) carries the common divisor (or unit) N, an exact number of times, as does EC, and therefore, by inference, CH too.
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1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan's number and the Hardy–Ramanujan number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital. He related their conversation:Quotations by Hardy The anecdote about 1729 occurs on pages lvii and lviii The two different ways are: : 1729 = 13 \+ 123 = 93 \+ 103 The quotation is sometimes expressed using the term "positive cubes", since allowing negative perfect cubes (the cube of a negative integer) gives the smallest solution as 91 (which is a divisor of 1729): :91 = 63 \+ (−5)3 = 43 \+ 33 Numbers that are the smallest number that can be expressed as the sum of two cubes in n distinct ways have been dubbed "taxicab numbers". The number was also found in one of Ramanujan's notebooks dated years before the incident, and was noted by Frénicle de Bessy in 1657.
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Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R: # rM = M for all nonzero r in R. (It is sometimes required that r is not a zero-divisor, and some authors require that R is a domain.) # For every principal left ideal Ra, any homomorphism from Ra into M extends to a homomorphism from R into M. (This type of divisible module is also called principally injective module.) # For every finitely generated left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M. The last two conditions are "restricted versions" of the Baer's criterion for injective modules. Since injective left modules extend homomorphisms from all left ideals to R, injective modules are clearly divisible in sense 2 and 3. If R is additionally a domain then all three definitions coincide.
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However, since division almost immediately introduces infinitely repeating sequences of digits (such as 4/7 in decimal, or 1/10 in binary), should this possibility arise then either the representation would be truncated at some satisfactory size or else rational numbers would be used: a large integer for the numerator and for the denominator. But even with the greatest common divisor divided out, arithmetic with rational numbers can become unwieldy very quickly: 1/99 − 1/100 = 1/9900, and if 1/101 is then added, the result is 10001/999900. The size of arbitrary- precision numbers is limited in practice by the total storage available, the variables used to index the digit strings, and computation time. A 32-bit operating system may limit available storage to less than 4 GB. A programming language using 32-bit integers can only index 4 GB. If multiplication is done with a algorithm, it would take on the order of steps to multiply two one- million-word numbers.
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Notation: n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The outer automorphism group is isomorphic to the semidirect product D \rtimes (F \times G) where all these groups D, F, G are cyclic of the respective orders d, f, g, except for type D_n(q), q odd, where the group of order d=4 is C_2 \times C_2, and (only when n=4) G = S_3, the symmetric group on three elements. The notation (a,b) represents the greatest common divisor of the integers a and b.
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Knuth (1968, 1973) has given a list of five properties that are widely accepted as requirements for an algorithm: # Finiteness: "An algorithm must always terminate after a finite number of steps ... a very finite number, a reasonable number" # Definiteness: "Each step of an algorithm must be precisely defined; the actions to be carried out must be rigorously and unambiguously specified for each case" # Input: "...quantities which are given to it initially before the algorithm begins. These inputs are taken from specified sets of objects" # Output: "...quantities which have a specified relation to the inputs" # Effectiveness: "... all of the operations to be performed in the algorithm must be sufficiently basic that they can in principle be done exactly and in a finite length of time by a man using paper and pencil" Knuth offers as an example the Euclidean algorithm for determining the greatest common divisor of two natural numbers (cf. Knuth Vol. 1 p. 2).
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Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. More generally, any two principal ideals in a commutative ring have a greatest common divisor in the sense of ideal multiplication. In principal ideal domains, this allows us to calculate greatest common divisors of elements of the ring, up to multiplication by a unit; we define \gcd(a, b) to be any generator of the ideal \langle a, b \rangle. For a Dedekind domain R, we may also ask, given a non-principal ideal I of R, whether there is some extension S of R such that the ideal of S generated by I is principal (said more loosely, I becomes principal in S). This question arose in connection with the study of rings of algebraic integers (which are examples of Dedekind domains) in number theory, and led to the development of class field theory by Teiji Takagi, Emil Artin, David Hilbert, and many others.
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The origin of the class of such problems has been attributed to the Indian mathematician Mahāvīra in chapter VI, §131, 132 of his Ganita-sara-sangraha (“Compendium of the Essence of Mathematics”), circa 850CE, which dealt with serial division of fruit and flowers with specified remainders.CHRONOLOGY OF RECREATIONAL MATHEMATICS by David Singmaster That would make progenitor problems over 1000 years old before their resurgence in the modern era. Problems involving division which invoke the Chinese remainder theorem appeared in Chinese literature as early as the first century CE. Sun Tzu asked: Find a number which leaves the remainders 2,3,2 when divided by 3,5,7 respectively. Diophantus of Alexandria first studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such problems was discovered by the Greek geometer Euclid and published in his Elements in 300CE. Singmaster traces a series of less plausibly related problems through the middle ages, with a few references as far back as the Babylonian empire circa 1700BC.
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In a sociable chain, or aliquot cycle, a sequence of divisor-sums returns to the initial number. These are the two chains Poulet described in 1918: 12496 → 14288 → 15472 → 14536 → 14264 → 12496 (5 links) 14316 → 19116 → 31704 → 47616 → 83328 → 177792 → 295488 → 629072 → 589786 → 294896 → 358336 → 418904 → 366556 → 274924 → 275444 → 243760 → 376736 → 381028 → 285778 → 152990 → 122410 → 97946 → 48976 → 45946 → 22976 → 22744 → 19916 → 17716 → 14316 (28 links) The second chain remains by far the longest known, despite the exhaustive computer searches begun by the French mathematician Henri Cohen in 1969. Poulet introduced sociable chains in a paper in the journal L'Intermédiaire des Mathématiciens #25 (1918). The paper ran like this: :If one considers a whole number a, the sum b of its proper divisors, the sum c of the proper divisors of b, the sum d of the proper divisors of c, and so on, one creates a sequence that, continued indefinitely, can develop in three ways: :The most frequent is to arrive at a prime number, then at unity [i.e.
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The 16 sets that are associated with divisors of 120 are the lower sets of this smaller partial order, subsets of elements such that if x ≤ y and y belongs to the subset, then x must also belong to the subset. From any lower set L, one can recover the associated divisor by computing the least common multiple of the prime powers in L. Thus, the partial order on the five prime powers 2, 3, 4, 5, and 8 carries enough information to recover the entire original 16-element divisibility lattice. Birkhoff's theorem states that this relation between the operations ∧ and ∨ of the lattice of divisors and the operations ∩ and ∪ of the associated sets of prime powers is not coincidental, and not dependent on the specific properties of prime numbers and divisibility: the elements of any finite distributive lattice may be associated with lower sets of a partial order in the same way. As another example, the application of Birkhoff's theorem to the family of subsets of an n-element set, partially ordered by inclusion, produces the free distributive lattice with n generators.
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This variant is identical to the original, as a third container capable of holding the contents of the first two is mathematically equivalent to a tap or drain capable of filling or emptying both containers. An optimal solution can be easily obtained using a biliard-shape barycentric plot (or a mathematical billiard). Another variant is when one of the jugs has a known volume of water to begin with; In that case, the achievable volumes are either a mutiple of the greatest common divisor between the two containers away from the existing known volume, or from zero. For example, if one jug that holds 8 liters is empty and the other jug that hold 12 liters has 9 liters of water in it to begin with, then with a source (tap) and a drain (sink), these two jugs can measure volumes of 9 liters, 5 liters, 1 liter, as well as 12 liters, 8 liters, 4 liters and 0 liters. The simplest solution for 5 liters is [9,0] → [9,8] → [12,5]; The simplest solution for 4 liters is [9,0] → [12,0] → [4,8].
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An ideal does not have any zero (the system of equations is inconsistent) if and only if 1 belongs to the ideal (this is Hilbert's Nullstellensatz), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if the corresponding reduced Gröbner basis is [1]. Given the Gröbner basis G of an ideal I, it has only a finite number of zeros, if and only if, for each variable x, G contains a polynomial with a leading monomial that is a power of x (without any other variable appearing in the leading term). If it is the case the number of zeros, counted with multiplicity, is equal to the number of monomials that are not multiple of any leading monomial of G. This number is called the degree of the ideal. When the number of zeros is finite, the Gröbner basis for a lexicographical monomial ordering provides, theoretically, a solution: the first coordinates of a solution is a root of the greatest common divisor of polynomials of the basis that depends only of the first variable.
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The period of a lag-r MWC generator is the order of b in the multiplicative group of numbers modulo p = abr − 1\. While it is theoretically possible to choose a non-prime modulus, a prime modulus eliminates the possibility of the initial seed sharing a common divisor with the modulus, which would reduce the generator's period. Because 2 is a quadratic residue of numbers of the form 8k±1, b = 2k cannot be a primitive root of p = abr − 1\. Therefore, MWC generators with base 2k have their parameters chosen so their period is (abr−1)/2. This is one of the difficulties that use of b = 2k − 1 overcomes. The basic form of an MWC generator has parameters a, b and r, and r+1 words of state. The state consists of r residues modulo b : 0 ≤ x0, x1, x2,..., xr−1 < b, and a carry cr−1 < a. The initial state ("seed") values are arbitrary, except that they must not be all zero, nor all at the maximum permitted values (xi = b−1 and cr−1 = a−1).
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Classical results of Bernhard Riemann describe Θ in another way, in the case that A is the Jacobian variety J of an algebraic curve (compact Riemann surface) C. There is, for a choice of base point P on C, a standard mapping of C to J, by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is an algebraic group, C may be added to itself k times on J, giving rise to subvarieties Wk. If g is the genus of C, Riemann proved that Θ is a translate on J of Wg − 1. He also described which points on Wg − 1 are non-singular: they correspond to the effective divisors D of degree g − 1 with no associated meromorphic functions other than constants. In more classical language, these D do not move in a linear system of divisors on C, in the sense that they do not dominate the polar divisor of a non constant function.
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In applications that require only one solution, rather than all solutions, the expected running time of a brute force search will often depend on the order in which the candidates are tested. As a general rule, one should test the most promising candidates first. For example, when searching for a proper divisor of a random number n, it is better to enumerate the candidate divisors in increasing order, from 2 to , than the other way aroundbecause the probability that n is divisible by c is 1/c. Moreover, the probability of a candidate being valid is often affected by the previous failed trials. For example, consider the problem of finding a 1 bit in a given 1000-bit string P. In this case, the candidate solutions are the indices 1 to 1000, and a candidate c is valid if P[c] = 1. Now, suppose that the first bit of P is equally likely to be 0 or 1, but each bit thereafter is equal to the previous one with 90% probability.
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Euclidean division of polynomials is very similar to Euclidean division of integers and leads to polynomial remainders. Its existence is based on the following theorem: Given two univariate polynomials a(x) and b(x) (where b(x) is a non-zero polynomial) defined over a field (in particular, the reals or complex numbers), there exist two polynomials q(x) (the quotient) and r(x) (the remainder) which satisfy: :a(x) = b(x)q(x) + r(x) where :\deg(r(x)) < \deg(b(x)), where "deg(...)" denotes the degree of the polynomial (the degree of the constant polynomial whose value is always 0 can be defined to be negative, so that this degree condition will always be valid when this is the remainder). Moreover, q(x) and r(x) are uniquely determined by these relations. This differs from the Euclidean division of integers in that, for the integers, the degree condition is replaced by the bounds on the remainder r (non-negative and less than the divisor, which insures that r is unique.) The similarity between Euclidean division for integers and that for polynomials motivates the search for the most general algebraic setting in which Euclidean division is valid.
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Likewise the first procedure should return Λ if there are no candidates at all for the instance P. The brute-force method is then expressed by the algorithm c ← first(P) while c ≠ Λ do if valid(P,c) then output(P, c) c ← next(P, c) end while For example, when looking for the divisors of an integer n, the instance data P is the number n. The call first(n) should return the integer 1 if n ≥ 1, or Λ otherwise; the call next(n,c) should return c + 1 if c < n, and Λ otherwise; and valid(n,c) should return true if and only if c is a divisor of n. (In fact, if we choose Λ to be n + 1, the tests n ≥ 1 and c < n are unnecessary.)The brute-force search algorithm above will call output for every candidate that is a solution to the given instance P. The algorithm is easily modified to stop after finding the first solution, or a specified number of solutions; or after testing a specified number of candidates, or after spending a given amount of CPU time.
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Let F' be the union of the lines corresponding to g'(S). The threefold F' is isomorphic to F. Thus knowing a Fano surface S, we can recover the threefold F. By the Tangent Bundle Theorem, we can also understand geometrically the invariants of S: a) Recall that the second Chern number of a rank 2 vector bundle on a surface is the number of zeroes of a generic section. For a Fano surface S, a 1-form w defines also a hyperplane section {w=0} into P4 of the cubic F. The zeros of the generic w on S corresponds bijectively to the numbers of lines into the smooth cubic surface intersection of {w=0} and F, therefore we recover that the second Chern class of S equals 27. b) Let w1, w2 be two 1-forms on S. The canonical divisor K on S associated to the canonical form w1 ∧ w2 parametrizes the lines on F that cut the plane P={w1=w2=0} into P4. Using w1 and w2 such that the intersection of P and F is the union of 3 lines, one can recover the fact that K2=45.
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Let F, G, and H be homogeneous polynomials in three variables, with H having higher degree than F and G; let a = deg H − deg F and b = deg H − deg G (both positive integers) be the differences of the degrees of the polynomials. Suppose that the greatest common divisor of F and G is a constant, which means that the projective curves that they define in the projective plane P2 have an intersection consisting in a finite number of points. For each point P of this intersection, the polynomials F and G generate an ideal (F, G)P of the local ring of P2 at P (this local ring is the ring of the fractions n/d, where n and d are polynomials in three variables and d(P) ≠ 0). The theorem asserts that, if H lies in (F, G)P for every intersection point P, then H lies in the ideal (F, G); that is, there are homogeneous polynomials A and B of degrees a and b, respectively, such that H = AF + BG. Furthermore, any two choices of A differ by a multiple of G, and similarly any two choices of B differ by a multiple of F.
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Vieta's formulas are frequently used with polynomials with coefficients in any integral domain . Then, the quotients a_i/a_n belong to the ring of fractions of (and possibly are in itself if a_n happens to be invertible in ) and the roots r_i are taken in an algebraically closed extension. Typically, is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers. Vieta's formulas are then useful because they provide relations between the roots without having to compute them. For polynomials over a commutative ring which is not an integral domain, Vieta's formulas are only valid when a_n is a non zero-divisor and P(x) factors as a_n(x-r_1)(x-r_2)\dots(x-r_n). For example, in the ring of the integers modulo 8, the polynomial P(x)=x^2-1 has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, r_1=1 and r_2=3, because P(x) eq (x-1)(x-3). However, P(x) does factor as (x-1)(x-7) and as (x-3)(x-5), and Vieta's formulas hold if we set either r_1=1 and r_2=7 or r_1=3 and r_2=5.
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Early examples of these algorithms are primarily decrease and conquer – the original problem is successively broken down into single subproblems, and indeed can be solved iteratively. Binary search, a decrease-and-conquer algorithm where the subproblems are of roughly half the original size, has a long history. While a clear description of the algorithm on computers appeared in 1946 in an article by John Mauchly, the idea of using a sorted list of items to facilitate searching dates back at least as far as Babylonia in 200 BC. Another ancient decrease-and-conquer algorithm is the Euclidean algorithm to compute the greatest common divisor of two numbers by reducing the numbers to smaller and smaller equivalent subproblems, which dates to several centuries BC. An early example of a divide-and-conquer algorithm with multiple subproblems is Gauss's 1805 description of what is now called the Cooley–Tukey fast Fourier transform (FFT) algorithm,Heideman, M. T., D. H. Johnson, and C. S. Burrus, "Gauss and the history of the fast Fourier transform", IEEE ASSP Magazine, 1, (4), 14–21 (1984). although he did not analyze its operation count quantitatively, and FFTs did not become widespread until they were rediscovered over a century later.
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Demonstration, with Cuisenaire rods, of the divisors of the composite number 10 Comparison of prime and composite numbers A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit. For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7\. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself. The composite numbers up to 150 are :4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150.
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