Indeed, the discovery of new primes is no small task; every candidate prime must go through the time-consuming and rigorous process of being cut-up by any potential divisors.
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In 2004, Dr. Tao and Ben Green, a mathematician at the University of Oxford, cited Dr. Furstenberg and used ergodic theory arguments to prove a major result — that arbitrarily long progressions also exist among the prime numbers, the integers that have exactly two divisors: 1 and themselves.
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Thus after, at most, steps, one get a null remainder, say . As and have the same divisors, the set of the common divisors is not changed by Euclid's algorithm and thus all pairs have the same set of common divisors. The common divisors of and are thus the common divisors of and 0. Thus is a GCD of and .
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As a group, the Jacobian variety of a curve is isomorphic to the quotient of the group of divisors of degree zero by the subgroup of principal divisors, i.e., divisors of rational functions. This holds for fields that are not algebraically closed, provided one considers divisors and functions defined over that field.
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Euler's proof is short and depends on the fact that the sum of divisors function is multiplicative; that is, if and are any two relatively prime integers, then . For this formula to be valid, the sum of divisors of a number must include the number itself, not just the proper divisors. A number is perfect if and only if its sum of divisors is twice its value.
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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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New York: W. H. Freeman & Company (2000): 170 It is the nearest power of two from decimal 1000 and senary 10000 (decimal 1296). 1024 is the smallest number with exactly 11 divisors (but note that there are smaller numbers with more than 11 divisors; e.g., 60 has 12 divisors) .
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Twelve is a composite number, the smallest number with exactly six divisors, its divisors being 1, 2, 3, 4, 6 and 12. Twelve is also a highly composite number, the next one being twenty-four. Twelve is the smallest abundant number, since it is the smallest integer for which the sum of its proper divisors (1 + 2 + 3 + 4 + 6 = 16) is greater than itself. Twelve is a sublime number, a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number. Since there is a subset of 12's proper divisors that add up to 12 (all of them but with 4 excluded), 12 is a semiperfect number.
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In number theory, a weird number is a natural number that is abundant but not semiperfect. Section B2. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
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Additional registers can be added to provide additional integer divisors.
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As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles. On singular varieties, this good property can fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes.
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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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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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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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In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields. The background is that codimension-1 subvarieties are understood much better than higher- codimension subvarieties.
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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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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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When x is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) (). The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, ), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.
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Red auxiliary numbers selected divisors of denominators mp that best summed to numerator mn.
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Additionally, it follows that there are no Carmichael numbers with exactly two prime divisors.
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The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher- dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider.
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Illustration of the perfect number status of the number 6 In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, σ1(n) = 2n where σ1 is the sum-of- divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 × 28.
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In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
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In algebraic geometry, an algebraic generalization is given by the notion of a linear system of divisors.
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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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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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There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits.
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Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors., pp. 464–466.
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The distributive lattice of divisors of 120, and its representation as sets of prime powers. Consider the divisors of some composite number, such as (in the figure) 120, partially ordered by divisibility. Any two divisors of 120, such as 12 and 20, have a unique greatest common factor 12 ∧ 20 = 4, the largest number that divides both of them, and a unique least common multiple 12 ∨ 20 = 60; both of these numbers are also divisors of 120. These two operations ∨ and ∧ satisfy the distributive law, in either of two equivalent forms: (x ∧ y) ∨ z = (x ∨ z) ∧ (y ∨ z) and (x ∨ y) ∧ z = (x ∧ z) ∨ (y ∧ z), for all x, y, and z.
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34 is the ninth distinct semiprime and has four divisors including one and itself. Its neighbors, 33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers with the same number of divisors as it has. It is the ninth Fibonacci number and a companion Pell number. Since it is an odd-indexed Fibonacci number, 34 is a Markov number, appearing in solutions with other Fibonacci numbers, such as (1, 13, 34), (1, 34, 89), etc.
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No perfect number is untouchable, since, at the very least, it can be expressed as the sum of its own proper divisors (this situation happens at the case for 28). Similarly, none of the amicable numbers or sociable numbers are untouchable. Also, all Mersenne numbers are not untouchable, since Mn=2n-1 can be expressed as 2n's proper divisors' sum. No untouchable number is one more than a prime number, since if p is prime, then the sum of the proper divisors of p2 is p + 1\.
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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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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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16 (sixteen) is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 42 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and . In English speech, the numbers 16 and 60 are sometimes confused, as they sound very similar.
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It is the smallest number with exactly 12 divisors. It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. It is the sum of a pair of twin primes (29 + 31) and the sum of four consecutive primes (11 + 13 + 17 + 19). It is adjacent to two primes (59 and 61).
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Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.
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In number theory, the aliquot sum s(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. It can be used to characterize the prime numbers, perfect numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
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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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Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties.
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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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An example with period 4: :The sum of the proper divisors of 1264460 (=2^2\cdot5\cdot17\cdot3719) is ::1 + 2 + 4 + 5 + 10 + 17 + 20 + 34 + 68 + 85 + 170 + 340 + 3719 + 7438 + 14876 + 18595 + 37190 + 63223 + 74380 + 126446 + 252892 + 316115 + 632230 = 1547860, :the sum of the proper divisors of 1547860 (=2^2\cdot5\cdot193\cdot401) is ::1 + 2 + 4 + 5 + 10 + 20 + 193 + 386 + 401 + 772 + 802 + 965 + 1604 + 1930 + 2005 + 3860 + 4010 + 8020 + 77393 + 154786 + 309572 + 386965 + 773930 = 1727636, :the sum of the proper divisors of 1727636 (=2^2\cdot521\cdot829) is ::1 + 2 + 4 + 521 + 829 + 1042 + 1658 + 2084 + 3316 + 431909 + 863818 = 1305184, and :the sum of the proper divisors of 1305184 (=2^5\cdot40787) is ::1 + 2 + 4 + 8 + 16 + 32 + 40787 + 81574 + 163148 + 326296 + 652592 = 1264460.
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Sums of the divisors, in Cuisenaire rods, of the first six highly abundant numbers In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.
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104 is a primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and father 8 is one of those divisors, 104 is a refactorable number. The distinct prime factors of 104 add up to 15, and so do the ones of 105, hence the two numbers form a Ruth-Aaron pair under the first definition. In regular geometry, 104 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.
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The sequence of highly composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors . Highly composite numbers whose number of divisors is also a highly composite number are for n = 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 . It is extremely likely that this sequence is complete.
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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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In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve C.
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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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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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After his description of the greedy algorithm, Fibonacci suggests yet another method, expanding a fraction a/b by searching for a number c having many divisors, with b/2 < c < b, replacing a/b by ac/bc, and expanding ac as a sum of divisors of bc, similar to the method proposed by Hultsch and Bruins to explain some of the expansions in the Rhind papyrus.
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Since the greatest prime factor of 452 + 1 = 2026 is 1013, which is much more than 45 twice, 45 is a Størmer number. In base 10, it is a Kaprekar number and a Harshad number. 45 is the smallest odd number that has more divisors than n+1 and that has a larger sum of divisors than n+1 . 45 is conjectured R(5, 5) .
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7\. triangular number It is a composite number, its proper divisors being 1, 2, 4, 7, and 14. Twenty-eight is the second perfect number - it's the sum of its divisors: 1+2+4+7+14. As a perfect number, it is related to the Mersenne prime 7, since 2(3 − 1)(23 − 1) = 28. The next perfect number is 496, the previous being 6.
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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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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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A linear system of divisors algebraicizes the classic geometric notion of a family of curves, as in the Apollonian circles. In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a linear system of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisors D on a general scheme or even a ringed space (X, OX).
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Demonstration, with Cuisenaire rods, of the first four: 1, 2, 4, 6 __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The term was coined by Ramanujan (1915). However, Jean-Pierre Kahane has suggested that the concept might have been known to Plato, who set 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it.. Kahane cites Plato's Laws, 771c. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer.
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496 is the third perfect number, a number whose divisors add up to the actual number (1+2+4+8+16+31+62+124+248=496).
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Knowledge of the divisors of the group order can often be used to gain information about the order of the center or of the conjugacy classes.
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Colossally abundant numbers are one of several classes of integers that try to capture the notion of having many divisors. For a positive integer n, the sum-of-divisors function σ(n) gives the sum of all those numbers that divide n, including 1 and n itself. Paul Bachmann showed that on average, σ(n) is around πn / 6.G. Hardy, E. M. Wright, An Introduction to the Theory of Numbers.
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The prime factorization of 24601 is 73 and 337. Since these are the only nontrivial divisors of 24601, and , it follows that 24601 is a 60-hyperperfect number.
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If R possesses no non-zero zero divisors, it is called an integral domain (or domain). An element a satisfying for some positive integer n is called nilpotent.
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Since S in the construction contains no zero divisors, the natural map R \to Q(R) is injective, so the total quotient ring is an extension of R.
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284 (two hundred [and] eighty-four) is the natural number following 283 and preceding 285. Its divisors are 1, 2, 4, 71, and 142, adding up to 220, in turn, the divisors of 220 add up to 284, making the two a pair of amicable numbers. 284 equals the sum of the squares of the digits of its own square in base 15. This property is shared with 1, 159, 264, 306 and 387.
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Demonstration, with Cuisenaire rods, of the perfection of the number 6 In number theory, a semiperfect number or pseudoperfect number is a natural number n that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are :6, 12, 18, 20, 24, 28, 30, 36, 40, ...
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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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It is a composite number, with its divisors being 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, making it an amicable number with 284.Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 167 Every number up to 220 may be expressed as a sum of its divisors, making 220 a practical number. Also, being divisible by the sum of its digits, 220 is a Harshad number.
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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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Also, no untouchable number is three more than a prime number, except 5, since if p is an odd prime then the sum of the proper divisors of 2p is p + 3\.
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Having only 2 and 3 as its prime divisors, 162 is a 3-smooth number.. 162 is also an abundant number, since its sum of divisors 1+2+3+6+9+18+27+54+81 = 201 is greater than it. As the product 3\times 6\times 9=162 of numbers three units apart from each other, it is a triple factorial number. There are 162 ways of partitioning seven items into subsets of at least two items per subset.
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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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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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Demonstration, with Cuisenaire rods, that the number 8 is almost perfect, and deficient. In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number n such that the sum of all divisors of n (the sum-of-divisors function σ(n)) is equal to 2n − 1, the sum of all proper divisors of n, s(n) = σ(n) − n, then being equal to n − 1. The only known almost perfect numbers are powers of 2 with non- negative exponents . Therefore the only known odd almost perfect number is 20 = 1, and the only known even almost perfect numbers are those of the form 2k for some positive number k; however, it has not been shown that all almost perfect numbers are of this form.
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If all of the diagonals of a regular decagon are drawn, the resulting figure will have exactly 220 regions. It is the sum of the sums of the divisors of the first 16 positive integers.
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Ludwig Stickelberger (May 18, 1850 - April 11, 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields).
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Numbers do exist where the sum of all the divisors σ(n) is equal to 2n + 2: 20, 104, 464, 650, 1952, 130304, 522752 ... . Many of these numbers are of the form 2n−1(2n − 3) where 2n − 3 is prime (instead of 2n − 1 with perfect numbers). In addition, numbers exist where the sum of all the divisors σ(n) is equal to 2n − 1, such as the powers of 2. Betrothed numbers relate to quasiperfect numbers like amicable numbers relate to perfect numbers.
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203] Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself.Cohn (2003), [ Proposition 5.4.5, p. 150] Associative division algebras have no zero divisors.
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For example, the number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4\. The number 5 is untouchable as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2). The first few untouchable numbers are: :2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, ...
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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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In general linear systems became a basic tool of birational geometry as practised by the Italian school of algebraic geometry. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions -- the Riemann–Roch problem as it can be called -- can be better phrased in terms of homological algebra. The effect of working on varieties with singular points is to show up a difference between Weil divisors (in the free abelian group generated by codimension-one subvarieties), and Cartier divisors coming from sections of invertible sheaves.
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The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12\. The first few weird numbers are : 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... .
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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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The Erdős definition allows perfect numbers to be primitive abundant numbers too. For example, 20 is a primitive abundant number because: :#The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. :#The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are: :20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... The smallest odd primitive abundant number is 945.
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It is a composite number, with divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect number. It is the smallest number divisible by the numbers 1 to 6: there is no smaller number divisible by the numbers 1 to 5.
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144 is the twelfth Fibonacci number, and the largest one to also be a square,Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 165 as the square of 12 (which is also its index in the Fibonacci sequence), following 89 and preceding 233. 144 is the smallest number with exactly 15 divisors, but it is not highly composite since the smaller number 120 has 16 divisors. 144 is divisible by the value of its φ function, which returns 48 in this case.
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For the totient, see , p. 245. For the sum of divisors, see By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".
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Springer, Berlin, 2002. has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers. Katherine Stange K. Stange.
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For some special cases we even get explicit addition and doubling formulas which are very fast. For example, there are explicit formulas for hyperelliptic curves of genus 2Frank Leitenberger, About the Group Law for the Jacobi Variety of a Hyperelliptic Curve and genus 3. For hyperelliptic curves it is also fairly easy to visualize the adding of two reduced divisors. Suppose we have a hyperelliptic curve of genus 2 over the real numbers of the form :C: y^2 = f(x) and two reduced divisors :D_1 = [P] + [Q] - 2 [O] and :D_2 = [R] + [S] - 2 [O].
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Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that :\displaystyle ax + by = 1, from which it follows that :\displaystyle ax \equiv 1 \pmod y, or equivalently :a \equiv \frac1x \pmod y.
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A perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly (with remainder zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect. A Mersenne prime is a prime number of the form ; for a number of this form to be prime, itself must also be prime. The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form , where is a Mersenne prime..
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Brown earned a B.A. at Rice University in 1965.Alumni In The News: Ezra “Bud” Brown Rice Magazine He then studied mathematics at Louisiana State University (LSU), getting an M.S. in 1967 and a Ph.D. in 1969 with the dissertation "Representations of Discriminantal Divisors by Binary Quadratic Forms" under Gordon Pall.Representations of discriminantal divisors by binary quadratic forms Journal of Number Theory, Volume 3, Issue 2, May 1971, pp. 213-225 He joined Virginia Tech in 1969 becoming Assistant Professor (1969-73), Associate Professor (1973-81), Professor (1981-2005), and Alumni Distinguished Professor of Mathematicsand Distinguished Professor of Mathematics (2005-2017).
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If Grp were an additive category, then this set E of ten elements would be a ring. In any ring, the zero element is singled out by the property that 0x=x0=0 for all x in the ring, and so z would have to be the zero of E. However, there are no two nonzero elements of E whose product is z, so this finite ring would have no zero divisors. A finite ring with no zero divisors is a field, but there is no field with ten elements because every finite field has for its order, the power of a prime.
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Explicitly, a morphism from a variety X to projective space Pn over a field k determines a line bundle L on X, the pullback of the standard line bundle O(1) on Pn. Moreover, L comes with n+1 sections whose base locus (the intersection of their zero sets) is empty. Conversely, any line bundle L with n+1 global sections whose common base locus is empty determines a morphism X → Pn.Hartshorne (1977), Theorem II.7.1. These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.Lazarsfeld (2004), Chapter 1.
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For example, the proper divisors of 15 (that is, the positive divisors of 15 that are not equal to 15) are 1, 3 and 5, so the aliquot sum of 15 is 9 i.e. (1 + 3 + 5). The values of s(n) for n = 1, 2, 3, ... are: :0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ...
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Then there exists a reduced abelian p-group A of Ulm length τ whose Ulm factors are isomorphic to these p-groups, Uσ(A) ≅ Aσ. Ulm's original proof was based on an extension of the theory of elementary divisors to infinite matrices.
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Hasse diagram of the set P of divisors of 60, partially ordered by the relation "x divides y". The red subset S = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.
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This means that the non-real part of the "quotient" is arbitrary and division is therefore not defined for purely nonreal dual numbers. Indeed, they are (trivially) zero divisors and clearly form an ideal of the associative algebra (and thus ring) of the dual numbers.
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The ring was constructed from sequences of real numbers. Two sequences were considered equivalent if they differed only in a finite number of elements. Arithmetic operations were defined elementwise. However, the ring constructed in this way contains zero divisors and thus cannot be a field.
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More generally, a pencil is the special case of a linear system of divisors in which the parameter space is a projective line. Typical pencils of curves in the projective plane, for example, are written as : \lambda C + \mu C' = 0 \, where , are plane curves.
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Fifty-two is the 6th Bell number and a decagonal number. It is an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is never the answer to the equation x − φ(x).
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Betrothed numbers or quasi-amicable numbers are two positive integers such that the sum of the proper divisors of either number is one more than the value of the other number. In other words, (m, n) are a pair of betrothed numbers if s(m) = n + 1 and s(n) = m + 1, where s(n) is the aliquot sum of n: an equivalent condition is that σ(m) = σ(n) = m + n + 1, where σ denotes the sum-of-divisors function. The first few pairs of betrothed numbers are: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128). All known pairs of betrothed numbers have opposite parity.
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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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The integers and the polynomials over a field share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a unit, ±1 in the case of integers) and a product of irreducible elements (prime numbers, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. Integral domains which share this property are called unique factorization domains (UFD). Greatest common divisors exist in UFDs, and conversely, every integral domain in which greatest common divisors exist is an UFD. Every principal ideal domain is an UFD.
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Zero divisors have a topological interpretation, at least in the case of commutative rings: a ring R is an integral domain if and only if it is reduced and its spectrum Spec R is an irreducible topological space. The first property is often considered to encode some infinitesimal information, whereas the second one is more geometric. An example: the ring , where k is a field, is not a domain, since the images of x and y in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines and , is not irreducible.
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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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Detecting whether or not a given natural number n is a perfect power may be accomplished in many different ways, with varying levels of complexity. One of the simplest such methods is to consider all possible values for k across each of the divisors of n, up to k \leq \log_2 n. So if the divisors of n are n_1, n_2, \dots, n_j then one of the values n_1^2, n_2^2, \dots, n_j^2, n_1^3, n_2^3, \dots must be equal to n if n is indeed a perfect power. This method can immediately be simplified by instead considering only prime values of k.
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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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Then multiplying a global section of O(D) by a nonzero scalar in k does not change its zero locus. As a result, the projective space of lines in the k-vector space of global sections H0(X, O(D)) can be identified with the set of effective divisors linearly equivalent to D, called the complete linear system of D. A projective linear subspace of this projective space is called a linear system of divisors. One reason to study the space of global sections of a line bundle is to understand the possible maps from a given variety to projective space. This is essential for the classification of algebraic varieties.
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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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This chapter is focused on the function field case; the Riemann-Roch theorem is stated and proved in measure-theoretic language, with the canonical class defined as the class of divisors of non-trivial characters of the adele ring which are trivial on the embedded field.
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The usual proof involves another lemma called Bézout's identity. This states that if and are relatively prime integers (i.e. they share no common divisors other than 1 and -1) there exist integers and such that : rx+sy = 1. Let and be relatively prime, and assume that .
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Chinese astronomy was most interested in extracting the algebraic features of planetary motion (that is, the length of the cyclic periods) to establish astronomical theories. Thus astronomy was reduced to arithmetic operations, extracting common multiples and divisors from the observed cyclic motions of the heavenly bodies.
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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 nonnegative integer r is called the free rank or Betti number of the module M. The module is determined up to isomorphism by specifying its free rank , and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors. The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime (non-unit) factors, which will be powers of irreducible elements. This decomposition corresponds to maximally decomposing each submodule corresponding to an invariant factor by using the Chinese remainder theorem for R. Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one (which is a multiple of all others), as follows. For each irreducible element such that some power occurs in , take the highest such power, removing it from , and multiply these powers together for all (classes of associated) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
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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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Demonstration of the practicality of the number 12 In number theory, a practical number or panarithmic number cites and for the name "panarithmic numbers". is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers begins :1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66, 72, 78, 80, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 140, 144, 150.... Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions.
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A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideals domains, PID for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal.
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In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.Erdős adopts a wider definition that requires a primitive abundant number to be not deficient, but not necessarily abundant (Erdős, Surányi and Guiduli. Topics in the Theory of Numbers p214. Springer 2003.).
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720 (seven hundred [and] twenty) is the natural number following 719 and preceding 721. It is 6! (6 factorial), a composite number with thirty divisors, more than any number below, making it a highly composite number. It is a Harshad number in every base from binary to decimal.
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Typically output will be on the screen. It may also be saved as a text file (with res extension) and plotted in gnuplot. Direct plotting in gnuplot is also supported on Unix systems. This file shows centers of hyperbolic components of mandelbrot set for period 10 ( and its divisors).
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Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example, 1264460 \mapsto 1547860 \mapsto 1727636 \mapsto 1305184 \mapsto 1264460 \mapsto\dots are sociable numbers of order 4.
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The first few centered square numbers are: :1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4325, … . All centered square numbers are odd, and in base 10 one can notice the one's digits follows the pattern 1-5-3-5-1. All centered square numbers and their divisors have a remainder of one when divided by four. Hence all centered square numbers and their divisors end with digits 1 or 5 in base 6, 8 or 12.
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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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Demonstration, with Cuisenaire rods, that 1, 2, 8, 9, and 12 are refactorable A refactorable number or tau number is an integer n that is divisible by the count of its divisors, or to put it algebraically, n is such that \tau(n)\mid n. The first few refactorable numbers are listed in as :1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers.
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Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because (1,0)\cdot(0,1)=(0,0), but fields do not have zero divisors.
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In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is named after Francesco Severi and André Néron.
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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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Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing : J[n]\times J[n] \longrightarrow \mu_n for all n prime to the characteristic of k. As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.
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He received the Mathematical Association of America's first Chauvenet Prize, in 1925, for his article "Algebraic functions and their divisors," which culminated in his 1933 book Algebraic functions. Bliss once headed a government commission that devised rules for apportioning seats in the U.S. House of Representatives among the several states.
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Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number is that its square root is again an integer. For example, = 3, so 9 is a square number. A positive integer that has no perfect square divisors except 1 is called square-free.
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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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The condition that at least one of the derivatives , , or is nonzero and that , , and are coprime is used to show that is nonzero. For example, if then so divides (as and are coprime) so (as unless is constant). Step 3. is divisible by each of the greatest common divisors , , and .
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Matsumura, Exercise 10.4 Let L be a field extension of K. Then the integral closure of A in L is completely integrally closed.An exercise in Bourbaki. An integral domain is completely integrally closed if and only if the monoid of divisors of A is a group. See also: Krull domain.
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Moreover, each simple group (prime) or non-group irreducible semigroup (subsemigroup of the flip-flop monoid) that divides the transformation semigroup of A must divide the transition semigroup of some component of the cascade, and only the primes that must occur as divisors of the components are those that divide A's transition semigroup.
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Fibonacci brought the innovations from Islamic mathematics back to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler)., 8.
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Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and non-trivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the 2 × 2 real matrices. Split-octonions are non-associative and contain nilpotents.
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132 is the sixth Catalan number. It is a pronic number, the product of 11 and 12. As it has 12 divisors total, 132 is a refactorable number. If you take the sum of all 2-digit numbers you can make from 132, you get 132: 12 + 13 + 21 + 23 + 31 + 32 = 132.
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A Frobenioid consists of a category C together with a functor to an elementary Frobenioid, satisfying some complicated conditions related to the behavior of line bundles and divisors on models of global fields. One of Mochizuki's fundamental theorems states that under various conditions a Frobenioid can be reconstructed from the category C.
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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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Formally, α(D) is the infimum of the rational numbers r such that K_X + r D is in the closed real cone of effective divisors in the Néron–Severi group of X. If α is negative, then X is pseudo-canonical. It is expected that α(D) is always a rational number.
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If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero- divisors are exactly the fundamental ideal.Lam (2005) p. 282 If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal KP. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings Xk with the Harrison topology.
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A pencil is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the projective line. This means that in the case of a complex algebraic variety V, a Lefschetz pencil is something like a fibration over the Riemann sphere; but with two qualifications about singularity. The first point comes up if we assume that V is given as a projective variety, and the divisors on V are hyperplane sections. Suppose given hyperplanes H and H′, spanning the pencil -- in other words, H is given by L = 0 and H′ by L′= 0 for linear forms L and L′, and the general hyperplane section is V intersected with :\lambda L + \mu L^\prime = 0.
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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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Generally, if m divides n, then has n/m subgroups of type , and one subgroup ℤm. Therefore, the total number of subgroups of (n ≥ 1), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n. See list of small groups for the cases n ≤ 8\. The dihedral group of order 8 (D4) is the smallest example of a group that is not a T-group. Any of its two Klein four-group subgroups (which are normal in D4) has as normal subgroup order-2 subgroups generated by a reflection (flip) in D4, but these subgroups are not normal in D4.
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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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Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes. Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts. The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda: Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors,The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
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A highly composite number (HCN) is a positive integer with more divisors than any smaller positive integer. They are often used in geometry, grouping and time measurement. The first 20 highly composite numbers are: 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560.
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Possible (and useful) adequate equivalence relations include rational, algebraic, homological and numerical equivalence. They are called "adequate" because dividing out by the equivalence relation is functorial, i.e. push-forward (with change of co-dimension) and pull-back of cycles is well-defined. Codimension one cycles modulo rational equivalence form the classical group of divisors.
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A Fermat number cannot be a perfect number or part of a pair of amicable numbers. The series of reciprocals of all prime divisors of Fermat numbers is convergent. If nn + 1 is prime, there exists an integer m such that n = 22m. The equation nn + 1 = F(2m+m) holds in that case.
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124 is the sum of eight consecutive primes (5 + 7 + 11 + 13 + 17 + 19 + 23 + 29). It is a nontotient since there is no integer with 124 coprimes below it. It is an untouchable number since there is no integer whose proper divisors add up to 124. In base 5 it is a repdigit (4445).
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A Euclidean domain is an integral domain on which is defined a Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. In a Euclidean domain, Euclidean division allows defining a Euclidean algorithm for computing greatest common divisors. However this does not imply the existence of a factorization algorithm.
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Multiplicative number theory is a subfield of analytic number theory that deals with prime numbers and with factorization and divisors. The focus is usually on developing approximate formulas for counting these objects in various contexts. The prime number theorem is a key result in this subject. The Mathematics Subject Classification for multiplicative number theory is 11Nxx.
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133 is an n whose divisors (excluding n itself) added up divide φ(n). It is an octagonal number and a Harshad number. It is also a happy number. 133 is a repdigit in base 11 (111) and base 18 (77), whilst in base 20 it is a cyclic number formed from the reciprocal of the number three.
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100 is an 18-gonal number. It is divisible by 25, the number of primes below it. It can not be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. It can be expressed as a sum of some of its divisors, making it a semiperfect number.
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Small Eisenstein primes. If and are Eisenstein integers, we say that divides if there is some Eisenstein integer such that . A non-unit Eisenstein integer is said to be an Eisenstein prime if its only non-unit divisors are of the form , where is any of the six units. There are two types of Eisenstein primes.
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Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions (x,y,m,n) to the equations with prime divisors of x and y lying in a given finite set and that they may be effectively computed. showed that, for each fixed x and y, this equation has at most one solution.
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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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The division by a_n may not be well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing a_n by 1 in the first column of the Sylvester matrix—before computing the determinant. In any case, the discriminant is a polynomial in a_0, \ldots, a_n with integer coefficients.
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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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Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers.
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108 is a powerful number. Its prime factorization is 22 · 33, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as , where and are positive integers greater than 1, so 108 is an Achilles number. 360 is not an Achilles number because it is not powerful. One of its prime factors is 5 but 360 is not divisible by 52 = 25. Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power: :784=2^4 \cdot 7^2 = (2^2)^2 \cdot 7^2 = (2^2 \cdot 7)^2 = 28^2.
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The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
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In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i.e. an element a such that there exists a non-zero element b of the ring such that .
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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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128 is the seventh power of 2. It is the largest number which cannot be expressed as the sum of any number of distinct squares.OEIS:A001422. Similarly, the largest numbers that cannot be expressed as sums of distinct cubes and fourth powers, respectively, are 12758 and 5134240 . But it is divisible by the total number of its divisors, making it a refactorable number.OEIS:A033950.
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Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order of the number ω(n) of distinct prime divisors of an integer n. There is an exposition of Turán's proof in Hardy & Wright, §22.11. Tenenbaum gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.
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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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Applying his p-adic form of the Hardy- Littlewood-Ramanujan-Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Karatsuba obtained a new estimate of the well known Hardy function G(n) in the Waring's problem (for n \ge 400): : \\! G(n) < 2 n\log n + 2 n\log\log n + 12 n.
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Two independent conditions that are both strictly stronger than the BFD condition are the half-factorial domain condition (HFD: any two factorizations of any given x have the same length), and the finite factorization domain condition (FFD: any x has but a finite number of non-associate divisors). Every unique factorization domain obviously satisfies these two conditions, but neither implies unique factorization.
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The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
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In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
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The aliquot sequence can be represented as a directed graph, G_{n,s}, for a given integer n, where s(k) denotes the sum of the proper divisors of k. Cycles in G_{n,s} represent sociable numbers within the interval [1,n]. Two special cases are loops that represent perfect numbers and cycles of length two that represent amicable pairs.
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130 is a sphenic number. It is a noncototient since there is no answer to the equation x - φ(x) = 130. 130 is the only integer that is the sum of the squares of its first four divisors, including 1: 12 \+ 22 \+ 52 \+ 102 = 130. 130 is the largest number that cannot be written as the sum of four hexagonal numbers.
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In algebraic geometry, a Mori Dream Space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". showed that Mori dream spaces are quotients of affine varieties by torus actions. The notion is named so because it behaves nicely from the point of view of Mori's minimal model program.
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One of the simplest and most natural examples is the multiset of prime factors of a number . Here the underlying set of elements is the set of prime divisors of . For example, the number 120 has the prime factorization :120 = 2^3 3^1 5^1 which gives the multiset . A related example is the multiset of solutions of an algebraic equation.
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Along with his colleague Marie A. Vitulli, he developed a unified valuation theory for rings with zero divisors that generalized both Krull and Archimedean valuations. He was a Guggenheim Fellow for the academic year 1963–1964. He supervised 28 doctoral students including Joel Cunningham. Ann Hill Harrison endowed the Harrison Memory Award for outstanding mathematical students at the University of Oregon.
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120 is the factorial of 5 and one less than a square, making (5, 11) a Brown number pair. 120 is the sum of a twin prime pair (59 + 61) and the sum of four consecutive prime numbers (23 + 29 + 31 + 37), four consecutive powers of 2 (8 + 16 + 32 + 64), and four consecutive powers of 3 (3 + 9 + 27 + 81). It is highly composite, superabundant, and colossally abundant number, with its 16 divisors being more than any number lower than it has, and it is also the smallest number to have exactly that many divisors. It is also a sparsely totient number. 120 is the smallest number to appear six times in Pascal's triangle. 120 is also the smallest multiple of 6 with no adjacent prime number, being adjacent to 119 = 7 × 17 and 121 = 112.
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The existence of ultrasonic values is a consequence of the frequency-divider design; in order to have adequate resolution at audible frequencies it is necessary for the overall clock rate (and thus the output at small divisors) to be considerably higher than the audible range. Only divisors below 5 give entirely-ultrasonic output frequencies. Frequencies equivalent to the top octave of a piano keyboard can be defined with reasonable accuracy versus the accepted note values for the even-tempered scale, to nearly 1 Hz precision in the A440 range and even more finely at lower pitches. Despite the high maximum frequency, the ability to divide that figure by 4096 means the lowest directly definable output frequency is 30.6 Hz, roughly equal to B0, the third lowest note on a normal 88-key piano, and as good as subsonic with everyday speaker systems.
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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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The politeness of a positive number is defined as the number of ways it can be expressed as the sum of consecutive integers. For every x, the politeness of x equals the number of odd divisors of x that are greater than one. The politeness of the numbers 1, 2, 3, ... is :0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 3, 0, 1, 2, 1, 1, 3, ... . For instance, the politeness of 9 is 2 because it has two odd divisors, 3 and itself, and two polite representations :9 = 2 + 3 + 4 = 4 + 5; the politeness of 15 is 3 because it has three odd divisors, 3, 5, and 15, and (as is familiar to cribbage players). three polite representations :15 = 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 = 7 + 8. An easy way of calculating the politeness of a positive number is that of decomposing the number into its prime factors, taking the powers of all prime factors greater than 2, adding 1 to all of them, multiplying the numbers thus obtained with each other and subtracting 1. For instance 90 has politeness 5 because 90 = 2 \times 3^2 \times 5^1; the powers of 3 and 5 are respectively 2 and 1, and applying this method (2+1) \times (1+1)-1 = 5.
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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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If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) For example, 72 = 23 × 32, all the prime factors are repeated, so 72 is a powerful number. 42 = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree. Another way to classify composite numbers is by counting the number of divisors.
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In some non-English-speaking cultures, "a divided by b" is written . However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b"). With a knowledge of multiplication tables, two integers can be divided on paper using the method of long division. An abbreviated version of long division, short division, can be used for smaller divisors as well.
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To find the GCD of two polynomials using factoring, simply factor the two polynomials completely. Then, take the product of all common factors. At this stage, we do not necessarily have a monic polynomial, so finally multiply this by a constant to make it a monic polynomial. This will be the GCD of the two polynomials as it includes all common divisors and is monic.
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He is known for the introduction of the Cartier operator in algebraic geometry in characteristic p, and for work on duality of abelian varieties and on formal groups. He is the eponym of Cartier divisors and Cartier duality. From 1961 to 1971 he was a Professor at the University of Strasbourg. In 1970 he was an Invited Speaker at the International Congress of Mathematicians in Nice.
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146 is an octahedral number as well as a composite number. It is a nontotient since there is no integer with 146 coprimes below it, noncototient since there is no integer with 146 natural numbers below it which are not coprime to it, and an untouchable number since there is no integer whose proper divisors add up to 146. 146 is a repdigit in base 8 (222).
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For a cyclic group, the orders of the subgroups are just the divisors of the order of the group, so a cyclic group is a Leinster group if and only if its order is a perfect number. More strongly, as Leinster proved, an abelian group is a Leinster group if and only if it is a cyclic group whose order is a perfect number.
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Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example, :σ(9!) = σ(362880) = 1481040, but there is a smaller number with larger sum of divisors, :σ(360360) = 1572480, so 9! is not highly abundant. Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant.
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Some of the most interesting intersection numbers to compute are self-intersection numbers. This should not be taken in a naive sense. What is meant is that, in an equivalence class of divisors of some specific kind, two representatives are intersected that are in general position with respect to each other. In this way, self-intersection numbers can become well-defined, and even negative.
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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 positive composite integer n is a Lucas–Carmichael number if and only if n is square-free, and for all prime divisors p of n, it is true that p + 1 \mid n + 1. The first Lucas–Carmichael numbers are: :399, 935, 2015, 2915, 4991, 5719, 7055, 8855, 12719, 18095, 20705, 20999, 22847, 29315, 31535, 46079, 51359, 60059, 63503, 67199, 73535, 76751, 80189, 81719, 88559, 90287, ...
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An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable.
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Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them. By writing a number as the sum of each digit times a power of 10 each digit's power can be manipulated individually. Case where all digits are summed This method works for divisors that are factors of 10 − 1 = 9. Using 3 as an example, 3 divides 9 = 10 − 1\.
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Euclid proves that is perfect by observing that the geometric series with ratio 2 starting at , with the same number of terms, is proportional to the original series; therefore, since the original series sums to , the second series sums to , and both series together add to , two times the supposed perfect number. However, these two series are disjoint from each other and (by the primality of ) exhaust all the divisors of , so has divisors that sum to , showing that it is perfect.. Over a millennium after Euclid, Alhazen conjectured that even perfect number is of the form where is prime, but he was not able to prove this result. It was not until the 18th century that Leonhard Euler proved that the formula will yield all the even perfect numbers.. Originally read to the Berlin Academy on February 23, 1747, and published posthumously. See in particular section 8, p. 88.
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Then the factorization problem is reduced to factorize separately the content and the primitive part. Content and primitive part may be generalized to polynomials over the rational numbers, and, more generally, to polynomials over the field of fractions of a unique factorization domain. This makes essentially equivalent the problems of computing greatest common divisors and factorization of polynomials over the integers and of polynomials over the rational numbers.
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The depth of a local ring R is the number of elements in some (or, as can be shown, any) maximal regular sequence, i.e., a sequence a1, ..., an ∈ m such that all ai are non-zero divisors in :R / (a1, ..., ai−1). For any local Noetherian ring, the inequality :depth (R) ≤ dim (R) holds. A local ring in which equality takes place is called a Cohen–Macaulay ring.
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In mathematics, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties over algebraically closed fields, introduced by Eugenio Bertini. This is the simplest and broadest of the "Bertini theorems" applying to a linear system of divisors; simplest because there is no restriction on the characteristic of the underlying field, while the extensions require characteristic 0.Hartshorne, Ch. III.10.
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A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird. This leads to the definition of primitive weird numbers, i.e. weird numbers that are not multiple of other weird numbers . There are only 24 primitive weird numbers smaller than a million, compared to 1765 weird numbers up to that limit.
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A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes. The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms.
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Modulo a prime, there is only the subgroup of squares and a single coset. The fact that, e.g., modulo 15 the product of the nonresidues 3 and 5, or of the nonresidue 5 and the residue 9, or the two residues 9 and 10 are all zero comes from working in the full ring Z/nZ, which has zero divisors for composite n. For this reason some authorse.g.
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The Euler–Fokker genus may also be called a complete contracted chord. Euler coined the term complete chord, while Fokker coined the entire term. A complete chord has two pitches, the fundamental and a guide tone, the guide tone being a multiple of the fundamental. In between are other pitches which can be seen either as multiples of the fundamental or as divisors of the guide tone (otonality and utonality).
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In the Chinese calendar, a sexagenary cycle is commonly used, in which days or years are named by positions in a sequence of ten stems and in another sequence of 12 branches. The same stem and branch repeat every 60 steps through this cycle. Book VIII of Plato's Republic involves an allegory of marriage centered on the number 604 = and its divisors. This number has the particularly simple sexagesimal representation 1,0,0,0,0.
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For every positive integer n, the set of all positive divisors of n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free. A positive integer n is square-free if and only if μ(n) ≠ 0, where μ denotes the Möbius function.
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Consider the 2D isometry point group Dn. The conjugates of a rotation are the same and the inverse rotation. The conjugates of a reflection are the reflections rotated by any multiple of the full rotation unit. For odd n these are all reflections, for even n half of them. This group, and more generally, abstract group Dihn, has the normal subgroup Zm for all divisors m of n, including n itself.
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Artin conjectured that every proper scheme over the integers has finite Brauer group.Milne (1980), Question IV.2.19. This is far from known even in the special case of a smooth projective variety X over a finite field. Indeed, the finiteness of the Brauer group for surfaces in that case is equivalent to the Tate conjecture for divisors on X, one of the main problems in the theory of algebraic cycles.
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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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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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Even and odd numbers: An integer is even if it is a multiple of two, and is odd otherwise. Prime number: An integer with exactly two positive divisors: itself and 1. The primes form an infinite sequence 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Composite number: A number that can be factored into a product of smaller integers. Every integer greater than one is either prime or composite.
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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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The algebra is not a division algebra or field since the null elements are not invertible. All of the nonzero null elements are zero divisors. Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring. The algebra of split-complex numbers forms a composition algebra since :\lVert zw \rVert = \lVert z \rVert \lVert w \rVert for any numbers z and w.
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In mathematics, a Hessian pair or Hessian duad, named for Otto Hesse, is a pair of points of the projective line canonically associated with a set of 3 points of the projective line. More generally, one can define the Hessian pair of any triple of elements from a set that can be identified with a projective line, such as a rational curve, a pencil of divisors, a pencil of lines, and so on.
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The Babylonians were able to make great advances in mathematics for two reasons. First, the number 60 has many divisors (2, 3, 4, 5, 6, 10, 12, 15, 20, and 30), making calculations easier. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values (much as in our base-ten system: 734 = 7×100 + 3×10 + 4×1).
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The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes (67 + 71 + 73 + 79). The sum of the squares of the divisors of 17 is 290. Not only is it a nontotient and a noncototient, it is also an untouchable number. 290 is the 16th member of the Mian–Chowla sequence; it can not be obtained as the sum of any two previous terms in the sequence.
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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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This situation signals a breakdown of perturbation theory: it stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is a asymptotic series: a useful approximation for a few terms, but ultimately inexact. The breakthrough from chaos theory was an explanation of why this happened: the small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other.
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Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36, hence 36 is a semiperfect number. This number is the sum of a twin prime pair (17 + 19), the sum of the cubes of the first three positive integers, and also the product of the squares of the first three positive integers. 36 is the number of degrees in the interior angle of each tip of a regular pentagram.
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In number theory, reversing the digits of a number sometimes produces another number that is divisible by . This happens trivially when is a palindromic number; the nontrivial reverse divisors are :1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... . For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178.... As cited by .. The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples..
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Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces. The elements of a free abelian group with basis B may be described in several equivalent ways.
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152 is the sum of four consecutive primes (31 + 37 + 41 + 43). It is a nontotient since there is no integer with 152 coprimes below it. 152 is a refactorable number since it is divisible by the total number of divisors it has, and in base 10 it is divisible by the sum of its digits, making it a Harshad number. Recently, the smallest repunit probable prime in base 152 was found, it has 589570 digits.
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The top left block is "poisoned" and the player who eats this loses. The chocolate-bar formulation of Chomp is due to David Gale, but an equivalent game expressed in terms of choosing divisors of a fixed integer was published earlier by Frederik Schuh. Chomp is a special case of a poset game where the partially ordered set on which the game is played is a product of total orders with the minimal element (poisonous block) removed.
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Chilkat Range from south of Haines The Chilkat Range is a mountain range in Haines Borough and the Hoonah-Angoon Census Area in the U.S. state of Alaska, west of the city of Juneau. The Chilkat Range is one of the principal divisors between Haines Borough and Glacier Bay National Park and Preserve. It also separates Chilkat Inlet and Lynn Canal from Muir Inlet in Glacier Bay. The northern boundary is generally considered to be the Klehini River.
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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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The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. logarithmic. A prime number is a positive integer with no divisors other than 1 and itself, excluding 1. Euclid recorded a proof that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.
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Another consequence of 8128 being a perfect number is that it has the same prime factors as the sum of its divisors, its cototient is a power of two, and it is a harmonic seed number (though there are deficient and abundant numbers that share these properties). 8128 is the 127th triangular number, the 64th hexagonal number, a happy number, the eighth 292-gonal number, and the fourth 1356-gonal number, as well as the 43rd centered nonagonal number.
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If X is a scheme of finite type over a field there is a natural map from divisors to line bundles. If X is either projective or reduced then this map is surjective. Kleiman found an example of a non-reduced and non-projective X for which this map is not surjective as follows. Take Hironaka's example of a variety with two rational curves A and B such that A+B is numerically equivalent to 0.
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There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the Picard group Pic(X) of divisors modulo linear equivalence, its quotient the Néron–Severi group NS(X) with rank the Picard number ρ, topological invariants such as the fundamental group π1 and the integral homology and cohomology groups, and invariants of the underlying smooth 4-manifold such as the Seiberg–Witten invariants and Donaldson invariants.
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Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17-gonal number. 48 is the smallest number with exactly ten divisors. There are 11 solutions to the equation φ(x) = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number.
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After this extension, again, x/y is the middle term. By this construction, the polite representations of a number and its odd divisors greater than one may be placed into a one-to-one correspondence, giving a bijective proof of the characterization of polite numbers and politeness.. More generally, the same idea gives a two-to-one correspondence between, on the one hand, representations as a sum of consecutive integers (allowing zero, negative numbers, and single-term representations) and on the other hand odd divisors (including 1). Another generalization of this result states that, for any n, the number of partitions of n into odd numbers having k distinct values equals the number of partitions of n into distinct numbers having k maximal runs of consecutive numbers... Here a run is one or more consecutive values such that the next larger and the next smaller consecutive values are not part of the partition; for instance the partition 10 = 1 + 4 + 5 has two runs, 1 and 4 + 5\.
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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 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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His results on 0-cycles on algebraic varieties with isolated singularities effectively reduces their study to the corresponding study on the desingularization, together with information about multiples of the exceptional divisors. This allows the complete calculation of the Chow group of 0-cycles on an algebraic variety in many cases, like the case of rational varieties or cones. Working initially with Levine, and later with Park, Krishna built up the original constructions of Bloch-Esnault on additive Chow groups into a full theory.
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The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.
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A prime number is a natural number that has exactly two distinct natural number divisors: the number 1 and itself. To find all the prime numbers less than or equal to a given integer by Eratosthenes' method: # Create a list of consecutive integers from 2 through : . # Initially, let equal 2, the smallest prime number. # Enumerate the multiples of by counting in increments of from to , and mark them in the list (these will be ; the itself should not be marked).
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For a more sophisticated approach, the linear system of divisors defining the embedding of V can be related to the line bundle or invertible sheaf defining the embedding by its space of sections. The tautological line bundle on Pn pulls back to V. The degree determines the first Chern class. The degree can also be computed in the cohomology ring of Pn, or Chow ring, with the class of a hyperplane intersecting the class of V an appropriate number of times.
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More generally a Prüfer ring is a commutative ring in which every non-zero finitely generated ideal consisting only of non-zero-divisors is invertible (that is, projective). A commutative ring is said to be arithmetical if for every maximal ideal m in R, the localization Rm of R at m is a chain ring. With this definition, an arithmetical domain is a Prüfer domain. Noncommutative right or left semihereditary domains could also be considered as generalizations of Prüfer domains.
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Schützenberger, Marcel-Paul, "On finite monoids having only trivial subgroups," Information and Control, Vol 8 No. 2, pp. 190–194, 1965. A consequence of the Krohn–Rhodes theorem is that every finite aperiodic monoid divides a wreath product of copies of the three- element flip-flop monoid, consisting of an identity element and two right zeros. The two-sided Krohn–Rhodes theorem alternatively characterizes finite aperiodic monoids as divisors of iterated block products of copies of the two- element semilattice.
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Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux. Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind.
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A hash function can be designed to exploit existing entropy in the keys. If the keys have leading or trailing zeros, or particular fields that are unused, always zero or some other constant, or generally vary little, then masking out only the volatile bits and hashing on those will provide a better and possibly faster hash function. Selected divisors or multipliers in the division and multiplicative schemes may make more uniform hash functions if the keys are cyclic or have other redundancies.
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Highly composite numbers higher than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first HCN that is not a Harshad number is 245,044,800, which has a digit sum of 27, but 27 does not divide evenly into 245,044,800. 10 of the first 38 highly composite numbers are superior highly composite numbers.
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The nef R-divisors form a closed convex cone in N^1(X), the nef cone Nef(X). The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space N_1(X) of 1-cycles modulo numerical equivalence. The vector spaces N^1(X) and N_1(X) are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.Lazarsfeld (2004), Definition 1.4.25.
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However, there is no integer which has 120 as the sum of its proper divisors, making 120 an untouchable number. The sum of Euler's totient function φ(x) over the first nineteen integers is 120. 120 figures in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that multiplied with any of the other numbers in the sequence yields a number that is one less than a square.
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In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only one analytic branch at a point. proved that if a local ring of an algebraic variety is a normal ring, then it is analytically irreducible. There are many examples of reduced and irreducible local rings that are analytically reducible, such as the local ring of a node of an irreducible curve, but it is hard to find examples that are also normal.
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During the same period Maier investigated as well the size of the coefficients of cyclotomic polynomials and later collaborated with Sergei Konyagin and E. Wirsing on this topic. He also collaborated with Hugh Lowell Montgomery on the size of the sum of the Möbius function under the assumption of the Riemann Hypothesis. Maier and Gérald Tenenbaum in joint work investigated the sequence of divisors of integers, solving the famous propinquity problem of Paul Erdős. Since 1993 Maier is a Professor at the University of Ulm, Germany.
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Originally from Champaign, Illinois, Krieger holds an undergraduate degree from the University of Illinois at Urbana–Champaign and master's and Ph.D. degrees from the University of Illinois at Chicago. She completed her Ph.D. in 2013; her dissertation, Primitive prime divisors in polynomial dynamics, was jointly supervised by Laura DeMarco and Ramin Takloo-Bighash. Subsequently she held a National Science Foundation postdoctoral fellowship at Massachusetts Institute of Technology supervised by Bjorn Poonen, and in 2016, Krieger was hired as a lecturer at the University of Cambridge.
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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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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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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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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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Thus (x + y) and (x − y) each contain factors of n, but those factors can be trivial. In this case we need to find another x and y. Computing the greatest common divisors of (x + y, n) and of (x - y, n) will give us these factors; this can be done quickly using the Euclidean algorithm. Congruences of squares are extremely useful in integer factorization algorithms and are extensively used in, for example, the quadratic sieve, general number field sieve, continued fraction factorization, and Dixon's factorization.
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In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety. The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors.
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Paul Erdős asked whether for any arbitrarily large N there exists an incongruent covering system the minimum of whose moduli is at least N. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 (Erdős gave an example where the moduli are in the set of the divisors of 120; a suitable cover is 0(3), 0(4), 0(5), 1(6), 1(8), 2(10), 11(12), 1(15), 14(20), 5(24), 8(30), 6(40), 58(60), 26(120) ) D. Swift gave an example where the minimum of the moduli is 4 (and the moduli are in the set of the divisors of 2880). S. L. G. Choi proved that it is possible to give an example for N = 20, and Pace P Nielsen demonstrates the existence of an example with N = 40, consisting of more than 10^{50} congruences. Erdős's question was resolved in the negative by Bob Hough. Hough used the Lovász local lemma to show that there is some maximum N<1016 which can be the minimum modulus on a covering system.
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Thus the primitive part of a polynomial is a primitive polynomial. Gauss's lemma for polynomials states that the product of primitive polynomials (with coefficients in the same unique factorization domain) also is primitive. This implies that the content and the primitive part of the product of two polynomials are, respectively, the product of the contents and the product of the primitive parts. As the computation of greatest common divisors is generally much easier than polynomial factorization, the first step of a polynomial factorization algorithm is generally the computation of its primitive part–content factorization (see ).
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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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These numbers are to be used in schools, and in practical life > millions of times as multipliers and divisors, and every unnecessary > additional figure is justly objectionable. > In a popular sense of the word, however, the numbers in the schedule may be > said to be exact. The length of the meter, for example, is given as 39.37 > inches. The mean of the best English and the best American determinations > differs from this only by about the amount by which the standard bar changes > its length by a change of one degree of temperature.
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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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It is mentioned by Pomponius Mela (DC III 1, 7), Pliny (HN IV 35, 116), Ptolemy (GH: II 5, 2), and Marcianus of Heracleia (PME: II, 13). Mints bronze asses and its lead divisors (semis, quadrans, triens, sextans) about mid 1st century BCE, in Latin alphabet with marine motives (tunas, dolphins, several kinds of boats). The name BALSA, recorded in these coins is the oldest attestation of the toponym. According to Mela (DC III 1, 7) Balsa was situated in the Cuneus Ager, a Roman geographical region corresponding to modern Central and Eastern Algarve.
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In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple. Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers". A number that is not part of any friendly pair is called solitary.
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The well-known article of Kleiman, listed in the bibliography, gives such an example. The correct solution is to proceed as follows: :For each open set U, let SU be the set of all elements in Γ(U, OX) that are not zero divisors in any stalk OX,x. Let KXpre be the presheaf whose sections on U are localizations SU−1Γ(U, OX) and whose restriction maps are induced from the restriction maps of OX by the universal property of localization. Then KX is the sheaf associated to the presheaf KXpre.
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Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. In order to avoid these issues, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.
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For univariate polynomials over a field, one can additionally require the GCD to be monic (that is to have 1 as its coefficient of the highest degree), but in more general cases there is no general convention. Therefore, equalities like or are common abuses of notation which should be read " is a GCD of and " and " and have the same set of GCDs as and ". In particular, means that the invertible constants are the only common divisors. In this case, by analogy with the integer case, one says that and are '.
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In 1934, Turán used the Turán sieve to give a new and very simple proof of a 1917 result of G. H. Hardy and Ramanujan on the normal order of the number of distinct prime divisors of a number n, namely that it is very close to \ln \ln n. In probabilistic terms he estimated the variance from \ln \ln n. Halász says "Its true significance lies in the fact that it was the starting point of probabilistic number theory". The Turán–Kubilius inequality is a generalization of this work.
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In a lattice, an element x is join-irreducible if x is not the join of a finite set of other elements. Equivalently, x is join-irreducible if it is neither the bottom element of the lattice (the join of zero elements) nor the join of any two smaller elements. For instance, in the lattice of divisors of 120, there is no pair of elements whose join is 4, so 4 is join-irreducible. An element x is join-prime if, whenever x ≤ y ∨ z, either x ≤ y or x ≤ z.
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As used in Belgium's national parliamentary elections from 1919 to 2003 the system could be said to have existed in a two-tier form, until it was replaced by a single-tier PR system. First, the Hare rather than Hagenbach-Bischoff quota was applied in the constituencies of provinces, and second, any seats remaining after quota allocation were aggregated, along with parties' provincial vote totals, at the provincial level where the D'Hondt method was then applied, including in the divisors for each party the number of seats it had won in the constituencies.
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Vitulli's research is in commutative algebra and applications to algebraic geometry. More specific topics in her research include deformations of monomial curves, seminormal rings, the weak normality of commutative rings and algebraic varieties, weak subintegrality, and the theory of valuations for commutative rings. Along with her colleague David K. Harrison, she developed a unified valuation theory for rings with zero divisors that generalized both Krull and Archimedean valuations. She was an undergraduate at the University of Rochester and obtained her Ph.D. in 1976 at the University of Pennsylvania under the supervision of Dock-Sang Rim.
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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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If R and S are rings and there exists a ring homomorphism R → S, then the characteristic of S divides the characteristic of R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring with characteristic 1 is the trivial ring, which has only a single element . If a nontrivial ring R does not have any nontrivial zero divisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, and to all division rings.
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In number theory, Gillies' conjecture is a conjecture about the distribution of prime divisors of Mersenne numbers and was made by Donald B. Gillies in a 1964 paper in which he also announced the discovery of three new Mersenne primes. The conjecture is a specialization of the prime number theorem and is a refinement of conjectures due to I. J. Good and Daniel Shanks. The conjecture remains an open problem: several papers give empirical support, but it disagrees with the widely accepted (but also open) Lenstra–Pomerance–Wagstaff conjecture.
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A "preferred" stoichiometric vector is one for which all of its elements can be converted to integers with no common divisors by multiplication by a suitable constant. Generally, the composition matrix is degenerate: That is to say, not all of its rows will be linearly independent. In other words, the rank (JR) of the composition matrix is generally less than its number of columns (J). By the rank-nullity theorem, the null space of aij will have J-JR dimensions and this number is called the nullity (JN) of aij.
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It is unknown if there are infinitely many pairs of amicable numbers. A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers. The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992). .
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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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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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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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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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International standard ISO 2848 (Building construction – Modular coordination – Principles and rules, International Organization for Standardization, 1984) is an ISO standard used by the construction industry. It is based on multiples of 300 mm and 600 mm The numbers 300 and 600 were chosen because they are preferred numbers due to their large number of divisors – any multiple can be evenly divided into 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, etc., making them easy to use in mental arithmetic. This system is known as "modular coordination".
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Many positive results were proven by Peter Linnell. For example, if the group acting is a free group, then the l^2-Betti numbers are integers. The most general question open as of late 2011 is whether l^2-Betti numbers are rational if there is a bound on the orders of finite subgroups of the group which acts. In fact, a precise relationship between possible denominators and the orders in question is conjectured; in the case of torsion-free groups, this statement generalizes the zero-divisors conjecture.
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In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by . Hodge diamond: Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors in an Abelian variety. The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties.
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In the Euclidean plane there are 3 regular forms equilateral triangle, squares, and regular hexagons; and 8 semiregular forms; and 4-demiregular forms which can tile the plane with other planigons. All angles of a planigon are whole divisors of 360°. Tilings are made by edge-to-edge connections by perpendicular bisectors of the edges of the original uniform lattice, or centroids along common edges (they coincide). Tilings made from planigons can be seen as dual tilings to the regular, semiregular, and demiregular tilings of the plane by regular polygons.
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Due to their general nature and versatility, sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology.
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Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in D; if the coefficient in D at z is negative, then we require that h has a zero of at least that multiplicity at z – if the coefficient in D is positive, h can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar).
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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 the early 19th century, Carl Friedrich Gauss observed that non- zero integer solutions to homogeneous polynomial equations with rational coefficients exist if non-zero rational solutions exist. In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, and made numerous other connections between number theory and algebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.
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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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The positive (equally non-negative) integers form a cancellative semigroup under addition. The non-negative integers form a cancellative monoid under addition. In fact, any free semigroup or monoid obeys the cancellative law, and in general, any semigroup or monoid embedding into a group (as the above examples clearly do) will obey the cancellative law. In a different vein, (a subsemigroup of) the multiplicative semigroup of elements of a ring that are not zero divisors (which is just the set of all nonzero elements if the ring in question is a domain, like the integers) has the cancellation property.
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By using a theorem by Carl Ludwig Siegel providing an upper bound for the real zeros (see Siegel zero) of Dirichlet L-functions formed with real non-principal characters, Walfisz obtained the Siegel-Walfisz theorem, from which the prime number theorem for arithmetic progressions can be deduced. By using estimates on exponential sums due to I. M. Vinogradov and , Walfisz obtained the currently best O-estimates for the remainder terms of the summatory functions of both the sum-of-divisors function \sigma and the Euler function \phi (in: "Weylsche Exponentialsummen in der neueren Zahlentheorie", see below).
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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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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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Although the book may be readable by some undergraduate mathematics students, reviewer David Perkinson suggests that its main audience should be graduate students in mathematics, for whom it could be used as the basis of a graduate course or seminar. He calls it "a thorough introduction to an exciting and growing subject", with "clear and concise exposition". Reviewer Paul Dreyer calls it a "deep dive" into "incredibly deep mathematics". Another book on the same general topic, published at approximately the same time, is Divisors and Sandpiles: An Introduction to Chip-Firing by Corry and Perkinson (American Mathematical Society, 2018).
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At the beginning of computer algebra, circa 1970, when the long-known algorithms were first put on computers, they turned out to be highly inefficient. Therefore, a large part of the work of the researchers in the field consisted in revisiting classical algebra in order to make it effective and to discover efficient algorithms to implement this effectiveness. A typical example of this kind of work is the computation of polynomial greatest common divisors, which is required to simplify fractions. Surprisingly, the classical Euclid's algorithm turned out to be inefficient for polynomials over infinite fields, and thus new algorithms needed to be developed.
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Different choices of α give different pseudo-remainder sequences, which are described in the next subsections. As the common divisors of two polynomials are not changed if the polynomials are multiplied by invertible constants (in Q), the last nonzero term in a pseudo-remainder sequence is a GCD (in Q[X]) of the input polynomials. Therefore, pseudo-remainder sequences allows computing GCD's in Q[X] without introducing fractions in Q. In some contexts, it is essential to control the sign of the leading coefficient of the pseudo- remainder. This is typically the case when computing resultants and subresultants, or for using Sturm's theorem.
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The Hagenbach-Bischoff system is a variant of the D'Hondt method, used for allocating seats in party-list proportional representation. It usually uses the Hagenbach-Bischoff quota for allocating seats, and for any seats remaining the D'Hondt method is then applied so that the first and subsequent divisors (number of seats won plus 1) for each party list's vote total includes the number of seats that have been allocated by the quota. The system gives results identical to the D'Hondt method and it is often referred to as such in countries using the system e.g. Switzerland and Belgium.
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Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of primes. Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers n that evenly divide (n-1)!+1. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it. Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by testing only the divisors up to the square root of the largest number to be tested.
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Babylonian cuneiform numerals The Babylonian cuneiform numerals had a base of 60, inherited from the Sumerian and Akkadian civilizations, and possibly motivated by the large number of divisors that 60 has. The sexagesimal measurement of time and of geometric angles is a legacy of the Babylonian system. The number system in the Mali Empire was based on 60, reflected in the counting system of the Maasina Fulfulde, a variant of the Fula language spoken in contemporary Mali. The Ekagi of Western New Guinea used base 60, and the sexagenary cycle plays a role in Chinese calendar and numerology.
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Rational equivalence of divisors (known as linear equivalence) was studied in various forms during the 19th century, leading to the ideal class group in number theory and the Jacobian variety in the theory of algebraic curves. For higher-codimension cycles, rational equivalence was introduced by Francesco Severi in the 1930s. In 1956, Wei-Liang Chow gave an influential proof that the intersection product is well-defined on cycles modulo rational equivalence for a smooth quasi-projective variety, using Chow's moving lemma. Starting in the 1970s, Fulton and MacPherson gave the current standard foundation for Chow groups, working with singular varieties wherever possible.
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To draw attention to this new computer design in the field of mathematics, he wrote an implementation of the Lucas–Lehmer primality test and found three new Mersenne primes, and published them in a paper, "Three new Mersenne primes and a statistical theory." The new Mersenne primes were reported in the Guinness Book of World Records, and the largest one was immortalized on all mail sent from the Post Office (Annex) at the Math department of the University of Illinois. In the same paper, Gillies made a conjecture about the distribution of prime divisors of Mersenne numbers.
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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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For example, if D has negative degree, then this vector space is zero (because a meromorphic function cannot have more zeros than poles). If D has positive degree, then the dimension of H0(X, O(mD)) grows linearly in m for m sufficiently large. The Riemann-Roch theorem is a more precise statement along these lines. On the other hand, the precise dimension of H0(X, O(D)) for divisors D of low degree is subtle, and not completely determined by the degree of D. The distinctive features of a compact Riemann surface are reflected in these dimensions.
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If p and q are two prime divisors of n, then implies the same equation also and These two smaller elliptic curves with the \boxplus-addition are now genuine groups. If these groups have Np and Nq elements, respectively, then for any point P on the original curve, by Lagrange's theorem, is minimal such that kP=\infty on the curve modulo p implies that k divides Np; moreover, N_p P=\infty. The analogous statement holds for the curve modulo q. When the elliptic curve is chosen randomly, then Np and Nq are random numbers close to and respectively (see below).
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For a prime number p, the following are equivalent: # The modular curve X0+(p) = X0(p) / wp, where wp is the Fricke involution of X0(p), has genus zero. # Every supersingular elliptic curve in characteristic p can be defined over the prime subfield Fp. # The order of the Monster group is divisible by p. The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then conjectural) existence of a sporadic simple group having exactly these primes as prime divisors.
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This strange coincidence was the beginning of the theory of monstrous moonshine. Three non-supersingular primes occur in the orders of two other sporadic simple groups: 37 and 67 divide the order of the Lyons group, and 37 and 43 divide the order of the fourth Janko group. It immediately follows that these two are not subquotients of the Monster group (they are two of the six pariah groups). The rest of the sporadic groups (including the other four pariahs, and also the Tits group, if that is counted among the sporadics) have orders with only supersingular prime divisors.
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In fact, other than the Baby Monster group, they all have orders divisible only by primes less than or equal to 31, although no single sporadic group, other than the Monster itself, has all of them as prime divisors. The supersingular prime 47 also divides the order of the Baby Monster group, and the other three supersingular primes (41, 59, and 71) do not divide the order of any sporadic group other than the Monster itself. All supersingular primes are Chen primes, but 37, 53, and 67 are also Chen primes, and there are infinitely many Chen primes greater than 73.
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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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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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The Drigganita system propounded by Parameshvara was a revision of the Parahita system introduced by Haridatta in the year 683 CE. No new methodology was introduced as part of the Drigganita system. Instead, new multipliers and divisors were given for the computation of the Kali days and for the calculation of the mean positions of the planets. Revised values are given for the positions of planets at zero Kali. Also the values of the sines of arc of anomaly (manda-jya) and of commutation (sighra-jya) are revised and are given for intervals of 6 degrees.
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36 is both the square of six and a triangular number, making it a square triangular number. It is the smallest square triangular number other than one, and it is also the only triangular number other than one whose square root is also a triangular number. It is also a circular number – a square number that ends with the same integer by itself (6×6=36). It is the smallest number n with exactly eight solutions to the equation φ(x) = n. Being the smallest number with exactly nine divisors, 36 is a highly composite number.
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This example is due to Selberg and is given as an exercise with hints by Cojocaru & Murty. The problem is to estimate separately the number of numbers ≤ x with no prime divisors ≤ x1/2, that have an even (or an odd) number of prime factors. It can be shown that, no matter what the choice of weights in a Brun- or Selberg-type sieve, the upper bound obtained will be at least (2 + o(1)) x / ln x for both problems. But in fact the set with an even number of factors is empty and so has size 0.
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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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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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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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For the large primes used in cryptography, Provable primes can be generated using variants of Pocklington primality test or Probable primes using standard probabilistic primality tests such as the Baillie–PSW primality test or the Miller–Rabin primality test. Both the provable and probable primality tests use modular exponentiation, a comparatively expensive computation. To reduce the computational cost, the integers are first checked for any small prime divisors using either sieves similar to the Sieve of Eratosthenes or Trial division. Integers with special forms, such as Mersenne prime or Fermat primes, can be efficiently tested for primality if the prime factorization of p − 1 or p + 1 is known.
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Similarly, tangency to a given line L (tangency is intersection with multiplicity two) is one quadratic condition, so determined a quadric in P5. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact each such quadric contains the Veronese surface, which parametrizes the conics :(aX + bY + cZ)2 = 0 called 'double lines'. This is because a double line intersects every line in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is tangent to the line.
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Monitor in the station building Situated in a central location, the station and its railway yard have served as divisors of the city's districts almost in the same way as Tammerkoski: Tamperean places are often located by saying which side of the station they are on. For decades, there has been discussion of a unified travel centre in Tampere: currently, the long-distance bus terminal is quite far away from the railway station. There are currently three platforms in the Tampere railway station, two of which have a roof. There are five tracks in total, but there are plans to add a fourth platform, making seven tracks in total.
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Highest average systems involve dividing the votes received by each party by a series of divisors, producing figures that determine seat allocation; for example the D'Hondt method (of which there are variants including Hagenbach-Bischoff) and the Webster/Sainte-Laguë method. Under largest remainder systems, parties' vote shares are divided by the quota (obtained by dividing the total number of votes by the number of seats available). This usually leaves some seats unallocated, which are awarded to parties based on the largest fractions of seats that they have remaining. Examples of largest remainder systems include the Hare quota, Droop quota, the Imperiali quota and the Hagenbach-Bischoff quota.
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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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Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n, :\sigma(n) > \sigma(m) where σ denotes the sum-of-divisors function. The first few highly abundant numbers are :1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3.
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Every uniruled variety over a field of characteristic zero has Kodaira dimension −∞. The converse is a conjecture which is known in dimension at most 3: a variety of Kodaira dimension −∞ over a field of characteristic zero should be uniruled. A related statement is known in all dimensions: Boucksom, Demailly, Păun and Peternell showed that a smooth projective variety X over a field of characteristic zero is uniruled if and only if the canonical bundle of X is not pseudo-effective (that is, not in the closed convex cone spanned by effective divisors in the Néron-Severi group tensored with the real numbers).Boucksom, Demailly, Păun and Peternell.
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An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion. :Theorem (A. Korselt 1899): A positive composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of n, it is true that p - 1 \mid n - 1. It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor, and thus p - 1 \mid n - 1 results in an even dividing an odd, a contradiction.
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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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Free modular lattice generated by three elements {x,y,z} The definition of modularity is due to Richard Dedekind, who published most of the relevant papers after his retirement. In a paper published in 1894 he studied lattices, which he called dual groups () as part of his "algebra of modules" and observed that ideals satisfy what we now call the modular law. He also observed that for lattices in general, the modular law is equivalent to its dual. In another paper in 1897, Dedekind studied the lattice of divisors with gcd and lcm as operations, so that the lattice order is given by divisibility.
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136 is itself a factor of the Eddington number. With a total of 8 divisors, 8 among them, 136 is a refactorable number. It is a composite number. 136 is a triangular number, a centered triangular number and a centered nonagonal number. The sum of the ninth row of Lozanić's triangle is 136. 136 is a self-descriptive number in base 4, and a repdigit in base 16. In base 10, the sum of the cubes of its digits is 1^3 + 3^3 + 6^3 = 244. The sum of the cubes of the digits of 244 is 2^3 + 4^3 + 4^3 = 136.
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Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime ideal, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction. The irrelevant ideal J generated by all the Xi corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space.
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Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that Benjamin Jowett, in the introduction to his translation of Laws, wrote, "Plato, writing under Pythagorean influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice and moderation."Laws, by Plato, translated By Benjamin Jowett, at Project Gutenberg; retrieved 7 July 2009. Jean-Pierre Kahane has suggested that Plato's use of the number 5040 marks the first appearance of the concept of a highly composite number, a number with more divisors than any smaller number..
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The number of signed permutation matrices of size n \times n can be described by the sequence y(n) which is determined by the recurrence equation4 (n+1)^2 \, y(n) + 2 \, y(n+1) + (-1) \, y(n+2) = 0over \Q. Taking a(n) = n+1,b(n)=1 as monic divisors of p_0 (n) = 4(n+1)^2, p_2(n) = -1 respectively, one gets z = \pm 2. For z=2 the corresponding recurrence equation which is solved in Petkovšek's algorithm is4(n+1)^2 \, c(n) + 4(n+1)\, c(n+1) - 4(n+1)(n+2) \, c(n+2) = 0.This recurrence equation has the polynomial solution c(n) = c_0 for an arbitrary c_0 \in \Q.
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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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Adding 19, however, gives 510529, which is prime. Hence 19 is a Fortunate number. The Fortunate number for pn# is always above pn and all its divisors are larger than pn. This is because pn#, and thus pn# + m, is divisible by the prime factors of m not larger than pn. The Fortunate numbers for the first primorials are: :3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, etc. . The Fortunate numbers sorted in numerical order with duplicates removed: :3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, ... .
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An (imaginary) hyperelliptic curve of genus g over a field K is given by the equation C : y^2 + h(x) y = f(x) \in K[x,y] where h(x) \in K[x] is a polynomial of degree not larger than g and f(x) \in K[x] is a monic polynomial of degree 2g + 1. From this definition it follows that elliptic curves are hyperelliptic curves of genus 1. In hyperelliptic curve cryptography K is often a finite field. The Jacobian of C, denoted J(C), is a quotient group, thus the elements of the Jacobian are not points, they are equivalence classes of divisors of degree 0 under the relation of linear equivalence.
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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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The minimal degree of a faithful complex representation is 196,883, which is the product of the three largest prime divisors of the order of M. The smallest faithful linear representation over any field has dimension 196,882 over the field with two elements, only one less than the dimension of the smallest faithful complex representation. The smallest faithful permutation representation of the monster is on 24 · 37 · 53 · 74 · 11 · 132 · 29 · 41 · 59 · 71 (about 1020) points. The monster can be realized as a Galois group over the rational numbers , and as a Hurwitz group. The monster is unusual among simple groups in that there is no known easy way to represent its elements.
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As the computation of a resultant may be reduced to computing determinants and polynomial greatest common divisors, there are algorithms for computing resultants in a finite number of steps. However, the generic resultant is a polynomial of very high degree (exponential in ) depending on a huge number of indeterminates. It follows that, except for very small and very small degrees of input polynomials, the generic resultant is, in practice, impossible to compute, even with modern computers. Moreover, the number of monomials of the generic resultant is so high, that, if it would be computable, the result could not be stored on available memory devices, even for rather small values of and of the degrees of the input polynomials.
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In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product :C × C × ... × C or Cn by the group action of the symmetric group on n letters permuting the factors. It exists as a smooth algebraic variety ΣnC; if C is a compact Riemann surface it is therefore a complex manifold. Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients. For C the projective line (say the Riemann sphere) ΣnC can be identified with projective space of dimension n.
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Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors. , it is still the best algorithm for divisors not exceeding 50 to 60 digits, as its running time is dominated by the size of the smallest factor p rather than by the size of the number n to be factored. Frequently, ECM is used to remove small factors from a very large integer with many factors; if the remaining integer is still composite, then it has only large factors and is factored using general- purpose techniques. The largest factor found using ECM so far has 83 decimal digits and was discovered on 7 September 2013 by R. Propper.
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10 is square-free, as its divisors greater than 1 are 2, 5, and 10, none of which is a perfect square (the first few perfect squares being 1, 4, 9, and 16) In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-free, but is not, because 18 is divisible by . The smallest positive square-free numbers are :1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, ...
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Using his improved Hardy-Littlewood method, I. M. Vinogradov published numerous refinements leading to :G(k)\le k(3\log k +11) in 1947 and, ultimately, :G(k)\le k(2\log k +2\log\log k + C\log\log\log k) for an unspecified constant C and sufficiently large k in 1959. Applying his p-adic form of the Hardy-Littlewood-Ramanujan- Vinogradov method to estimating trigonometric sums, in which the summation is taken over numbers with small prime divisors, Anatolii Alexeevitch Karatsuba obtained (1985) a new estimate of the Hardy function G(k) (for k \ge 400): : \\! G(k) < 2 k\log k + 2 k\log\log k + 12 k. Further refinements were obtained by Vaughan [1989].
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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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After attending a lecture on mathematics by French mathematician François Cluzet, Dimitris learns about "eteros ego" (έτερος εγώ English: the other me) a formula known by the pythagoreans regarding friend numbers (Amicable numbers), numbers so related that the sum of the proper divisors of each is equal to the other number. Such a pair is 220 and 284. This information proved to be very useful as it is later on reported that Kleio rapti was wearing an amulet with the number 284 carved on it. At the end, Dimitris successfully solves the case as he finds out that the murderer is Kleio's best friend Danae, who owns another amulet similar to Kleio's with the number 220 on it.
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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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Emergence holes The nymphs emerge in large numbers about the same time, sometimes more than 1.5 million individuals per acre (>370/m²). Their mass emergence is a survival trait called predator satiation: for the first week after emergence, the periodical cicadas are an easy prey for reptiles, birds, squirrels, cats, and other small and large mammals. Early ideas maintained that the cicadas' overall survival mechanism was simply to overwhelm predators by their sheer numbers, ensuring the survival of most of the individuals. The emergence period of large prime numbers (13 and 17 years) was hypothesized to be a predator avoidance strategy adopted to eliminate the possibility of potential predators receiving periodic population boosts by synchronizing their own generations to divisors of the cicada emergence period.
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In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring over a field . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems.
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It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra. So, given an integral domain , it is often very useful to know that has a Euclidean function: in particular, this implies that is a PID.
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The main disadvantage of the brute-force method is that, for many real-world problems, the number of natural candidates is prohibitively large. For instance, if we look for the divisors of a number as described above, the number of candidates tested will be the given number n. So if n has sixteen decimal digits, say, the search will require executing at least 1015 computer instructions, which will take several days on a typical PC. If n is a random 64-bit natural number, which has about 19 decimal digits on the average, the search will take about 10 years. This steep growth in the number of candidates, as the size of the data increases, occurs in all sorts of problems.
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The variety V in its projective embedding is projectively normal if R is integrally closed. This condition implies that V is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions.Robin Hartshorne, Algebraic Geometry (1977), p. 23. Another equivalent condition is in terms of the linear system of divisors on V cut out by the dual of the tautological line bundle on projective space, and its d-th powers for d = 1, 2, 3, ... ; when V is non- singular, it is projectively normal if and only if each such linear system is a complete linear system.Hartshorne, p. 159.
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888 is a repdigit (a number all of whose digits are equal), and a strobogrammatic number (one that reads the same upside-down on a seven-segment calculator display). 8883 = 700227072 is the smallest cube in which each digit occurs exactly three times, and the only cube in which three distinct digits each occur three times.. 888 the smallest multiple of 24 whose digit sum is 24, and as well as being divisible by its digit sum it is divisible by all of its digits. 888 is a practical number, meaning that every positive integer up to 888 itself may be represented as a sum of distinct divisors of 888.Nombres pratiques (in French), Jeux et Mathématiques, Jean-Paul Davalan, retrieved 2013-01-31.
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A variable or value of that type is usually represented as a fraction m/n where m and n are two integer numbers, either with a fixed or arbitrary precision. Depending on the language, the denominator n may be constrained to be non-zero, and the two numbers may be kept in reduced form (without any common divisors except 1). Languages that support a rational data type usually provide special syntax for building such values, and also extend the basic arithmetic operations ('+', '−', '×', '/', integer powers) and comparisons ('=', '<', '>', '≤') to act on them — either natively or through operator overloading facilities provided by the language. These operations may be translated by the compiler into a sequence of integer machine instructions, or into library calls.
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Therefore, divisibility by 2, 5, and 10 only depend on whether the last 1 digit is divisible by those divisors. The factors of 102 include 4 and 25, and divisibility by those only depend on the last 2 digits. Case where only the last digit(s) are removed Most numbers do not divide 9 or 10 evenly, but do divide a higher power of 10n or 10n − 1\. In this case the number is still written in powers of 10, but not fully expanded. For example, 7 does not divide 9 or 10, but does divide 98, which is close to 100. Thus, proceed from :100 \cdot a + b where in this case a is any integer, and b can range from 0 to 99.
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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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It is still used, in varying degrees, in everyday life in the United Kingdom, Canada, New Zealand, Australia, and some other former British colonies, despite their official adoption of the metric system. The avoirdupois weight systems general attributes were originally developed for the international wool trade in the Late Middle Ages, when trade was in recovery. It was historically based on a physical standardized pound or "prototype weight" that could be divided into 16 ounces. There were a number of competing measures of mass, and the fact that the avoirdupois pound had three even numbers as divisors (half and half and half again) may have been a cause of much of its popularity, so that the system won out over systems with 12 or 10 or 15 subdivisions.
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In 2012, amassed numerical evidence of an extension of Mathieu moonshine, where families of mock modular forms were attached to divisors of 24. After some group-theoretic discussion with Glauberman, found that this earlier extension was a special case (the A-series) of a more natural encoding by Niemeier lattices. For each Niemeier root system X, with corresponding lattice LX, they defined an umbral group GX, given by the quotient of the automorphism group of LX by the subgroup of reflections- these are also known as the stabilizers of deep holes in the Leech lattice. They conjectured that for each X, there is an infinite dimensional graded representation KX of GX, such that the characters of elements are given by a list of vector-valued mock modular forms that they computed.
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Suppose that H is a subgroup of a finite group G, K is an invariant subset of H such that if two elements in K are conjugate in G, then they are conjugate in H, and π a set of primes containing all prime divisors of the orders of elements of K. The Dade lifting is a linear map f → fσ from class functions f of H with support on K to class functions fσ of G, which is defined as follows: fσ(x) is f(k) if there is an element k ∈ K conjugate to the π-part of x, and 0 otherwise. The Dade lifting is an isometry if for each k ∈ K, the centralizer CG(k) is the semidirect product of a normal Hall π' subgroup I(K) with CH(k).
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In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point.
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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 Ore condition, which (if true) allows a ring of fractions to be defined, and the Ore extension, a non- commutative analogue of rings of polynomials, are part of this work. In more elementary number theory, Ore's harmonic numbers are the numbers whose divisors have an integer harmonic mean. As a teacher, Ore is notable for teaching mathematics to two doctoral students who would make contributions to science and mathematics, Grace Hopper, who would eventually become a United States rear admiral and computer scientist, who was a pioneer in the development of the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics. In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether.
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For example, the direct sum of the Ri form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime. An element x in R is a unit if and only if all of its components are units, i.e., if and only if is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of Ri. A product of two or more non-zero rings always has nonzero zero divisors: if x is an element of the product whose coordinates are all zero except , and y is an element of the product with all coordinates zero except where , then in the product ring.
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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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If the coefficients do not belong to Fp, the p-th root of a polynomial with zero derivative is obtained by the same substitution on x, completed by applying the inverse of the Frobenius automorphism to the coefficients. This algorithm works also over a field of characteristic zero, with the only difference that it never enters in the blocks of instructions where pth roots are computed. However, in this case, Yun's algorithm is much more efficient because it computes the greatest common divisors of polynomials of lower degrees. A consequence is that, when factoring a polynomial over the integers, the algorithm which follows is not used: one compute first the square-free factorization over the integers, and to factor the resulting polynomials, one chooses a p such that they remain square-free modulo p.
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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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It follows from the Chinese remainder theorem that there are at least four distinct square roots of 1 modulo N (since there are two roots for each modular equation). The aim of the algorithm is to find a square root b of 1 modulo N that is different from 1 and \- 1 , because then : b^2 - 1 = (b+1)(b-1) = mN for a non-zero integer m which gives us the non- trivial divisors \gcd(N, b+1) and \gcd(N, b-1) of N . This idea is similar to other factoring algorithms, like the quadratic sieve. In turn, finding such a b is reduced to finding an element a of even period with a certain additional property (as explained below, it is required that the condition of Step 6 of the classical part does not hold).
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If the definition of a prime number were changed to call 1 a prime, many statements involving prime numbers would need to be reworded in a more awkward way. For example, the fundamental theorem of arithmetic would need to be rephrased in terms of factorizations into primes greater than 1, because every number would have multiple factorizations with different numbers of copies of 1. Similarly, the sieve of Eratosthenes would not work correctly if it handled 1 as a prime, because it would eliminate all multiples of 1 (that is, all other numbers) and output only the single number 1. Some other more technical properties of prime numbers also do not hold for the number 1: for instance, the formulas for Euler's totient function or for the sum of divisors function are different for prime numbers than they are for 1.
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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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The goal is to find an inverse to 10 modulo the prime under consideration (does not work for 2 or 5) and use that as a multiplier to make the divisibility of the original number by that prime depend on the divisibility of the new (usually smaller) number by the same prime. Using 31 as an example, since 10 × (−3) = −30 = 1 mod 31, we get the rule for using y − 3x in the table above. Likewise, since 10 × (28) = 280 = 1 mod 31 also, we obtain a complementary rule y + 28x of the same kind - our choice of addition or subtraction being dictated by arithmetic convenience of the smaller value. In fact, this rule for prime divisors besides 2 and 5 is really a rule for divisibility by any integer relatively prime to 10 (including 33 and 39; see the table below).
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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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Sun-tzu's original formulation: In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime. The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the Sun-tzu Suan- ching in the 3rd century AD. The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem (expressed in terms of congruences) is true over every principal ideal domain. It has been generalized to any commutative ring, with a formulation involving ideals.
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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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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 games and puzzles of Lewis Carroll were the subject of Martin Gardner's March 1960 Mathematical Games column in Scientific American. Other items include a rule for finding the day of the week for any date; a means for justifying right margins on a typewriter; a steering device for a velociam (a type of tricycle); fairer elimination rules for tennis tournaments; a new sort of postal money order; rules for reckoning postage; rules for a win in betting; rules for dividing a number by various divisors; a cardboard scale for the Senior Common Room at Christ Church which, held next to a glass, ensured the right amount of liqueur for the price paid; a double- sided adhesive strip to fasten envelopes or mount things in books; a device for helping a bedridden invalid to read from a book placed sideways; and at least two ciphers for cryptography. He also proposed alternative systems of parliamentary representation. He proposed the so-called Dodgson's method, using the Condorcet method.
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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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However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient. The subresultant pseudo- remainder sequences were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism on the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many prime numbers and then reconstructs the result with the Chinese remainder theorem. The use of fast multiplication of integers and polynomials allows algorithms for resultants and greatest common divisors that have a better time complexity, which is of the order of the complexity of the multiplication, multiplied by the logarithm of the size of the input (\log(s(d+e)), where is an upper bound of the number of digits of the input polynomials).
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In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the solution and checking whether each candidate satisfies the problem's statement. A brute-force algorithm to find the divisors of a natural number n would enumerate all integers from 1 to n, and check whether each of them divides n without remainder. A brute-force approach for the eight queens puzzle would examine all possible arrangements of 8 pieces on the 64-square chessboard, and, for each arrangement, check whether each (queen) piece can attack any other. While a brute-force search is simple to implement, and will always find a solution if it exists, its cost is proportional to the number of candidate solutionswhich in many practical problems tends to grow very quickly as the size of the problem increases (§Combinatorial explosion).
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The number 5 is believed to be the only odd untouchable number, but this has not been proven: it would follow from a slightly stronger version of the Goldbach conjecture, since the sum of the proper divisors of pq (with p, q distinct primes) is 1+p+q. Thus, if a number n can be written as a sum of two distinct primes, then n+1 is not an untouchable number. It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 7 is an untouchable number, and 1=\sigma(2)-2, 3=\sigma(4)-4, 7=\sigma(8)-8, so only 5 can be an odd untouchable number.The stronger version is obtained by adding to the Goldbach conjecture the further requirement that the two primes be distinct - see Thus it appears that besides 2 and 5, all untouchable numbers are composite numbers (since except 2, all even numbers are composite).
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A finite- dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion. The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F. One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).
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It follows that the algebraic integers in F form a ring denoted OF called the ring of integers of F. It is a subring of (that is, a ring contained in) F. A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of F is an integral domain. The field F is the field of fractions of the integral domain OF. This way one can get back and forth between the algebraic number field F and its ring of integers OF. Rings of algebraic integers have three distinctive properties: firstly, OF is an integral domain that is integrally closed in its field of fractions F. Secondly, OF is a Noetherian ring. Finally, every nonzero prime ideal of OF is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
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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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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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He held the No. 1 position until August 3, when McIlroy regained the top spot by following his Open Championship victory with another at the WGC- Bridgestone Invitational. On August 16, 2015, following Jordan Spieth's second-place finish at the 2015 PGA Championship (that followed earlier wins at the Masters and the U.S. Open), Spieth became the 18th world No. 1. Over the following three weeks, the No. 1 spot passed back and forth between McIlroy and Spieth, due to the way each player's average points (which were almost identical) fluctuated (as their point weightings and events played divisors changed), until, on September 20, both were overtaken by Jason Day, the 2015 PGA Championship winner, who became the 19th world No. 1 with victory in the BMW Championship, his fifth of the season. A week later, Spieth regained the No. 1 spot from Day after winning the Tour Championship (and with it, the FedEx Cup), and concluded 2015 as world No. 1, but Day's continued good form took him back to number one after winning the WGC Matchplay in March 2016. On February 19, 2017, Dustin Johnson became the 20th player to reach number one in the rankings following his victory at the Genesis Open.
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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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