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c_0pudiat4m1l5 | In number theory, specifically the study of Diophantine approximation, the lonely runner conjecture is a conjecture about the long-term behavior of runners on a circular track. It states that n {\displaystyle n} runners on a track of unit length, with constant speeds all distinct from one another, will each be lonely a... | Lonely runner conjecture |
c_xybdtnyrmy6w | In number theory, the Ankeny–Artin–Chowla congruence is a result published in 1953 by N. C. Ankeny, Emil Artin and S. Chowla. It concerns the class number h of a real quadratic field of discriminant d > 0. If the fundamental unit of the field is ε = t + u d 2 {\displaystyle \varepsilon ={\frac {t+u{\sqrt {d}}}{2}}} wit... | Ankeny–Artin–Chowla congruence |
c_kp5oog7w2ong | In number theory, the Baker–Heegner–Stark theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are unique factorization domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed clas... | Stark–Heegner theorem |
c_7x52dj6c6yep | The Baker–Heegner–Stark theorem can then be stated as follows: If d < 0, then the class number of Q(√d) is one if and only if d ∈ { − 1 , − 2 , − 3 , − 7 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.} These are known as the Heegner numbers. By replacing d with the... | Stark–Heegner theorem |
c_yj06et7noc06 | In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conj... | Bateman-Horn conjecture |
c_2vme9h4jdkwc | In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the fraction a/b has as its two children the numbers a/a + b and a + b/b. Every positive rational numb... | Calkin–Wilf tree |
c_6kgab8mx6pey | It is named after Neil Calkin and Herbert Wilf, but appears in other works including Kepler's Harmonices Mundi. The sequence of rational numbers in a breadth-first traversal of the Calkin–Wilf tree is known as the Calkin–Wilf sequence. Its sequence of numerators (or, offset by one, denominators) is Stern's diatomic ser... | Calkin–Wilf tree |
c_6ogmv16fqeud | In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by Ewald Warning (1935) and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by Chevalley (1935). Chevalley's theorem ... | Ax–Katz theorem |
c_uop22x45ghpx | In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n—in other words, that an integer n is prime if and only if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} . It is ... | Chinese hypothesis |
c_rxa5rpi2imfu | In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent.Let A = a 1 , a 2 , … {\displaystyle A=a_{1},a_{2},\dots } be a sequence of positive integers. Then the multiples of A {\displaystyle A} are another set M ( A ) {\displaystyl... | Davenport–Erdős theorem |
c_13mxuch6h9t8 | The sequential density, defined as the limit (as i {\displaystyle i} goes to infinity) of the densities of the sets M ( { a 1 , … a i } ) {\displaystyle M(\{a_{1},\dots a_{i}\})} of multiples of the first i {\displaystyle i} elements of A {\displaystyle A} . As these sets can be decomposed into finitely many disjoint a... | Davenport–Erdős theorem |
c_8f82ra0klk4v | In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by ψ ( n ) = n ∏ p | n ( 1 + 1 p ) , {\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right),} where the product is taken over all primes p {\displaystyle p} dividing n . {\displaystyle n.} (By convent... | Dedekind psi function |
c_dm7fxm4027ih | The value of ψ ( n ) {\displaystyle \psi (n)} for the first few integers n {\displaystyle n} is: 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the OEIS).The function ψ ( n ) {\displaystyle \psi (n)} is greater than n {\displaystyle n} for all n {\displaystyle n} greater than 1, and is even for all ... | Dedekind psi function |
c_yle3erw0w1u4 | This also leads to a proof of the generating function in terms of the Riemann zeta function, which is ∑ ψ ( n ) n s = ζ ( s ) ζ ( s − 1 ) ζ ( 2 s ) . {\displaystyle \sum {\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}.} This is also a consequence of the fact that we can write as a Dirichlet convolu... | Dedekind psi function |
c_afh3sonn6vi4 | There is an additive definition of the psi function as well. Quoting from Dickson, R. Dedekind proved that, if n {\displaystyle n} is decomposed in every way into a product a b {\displaystyle ab} and if e {\displaystyle e} is the g.c.d. of a , b {\displaystyle a,b} then ∑ a ( a / e ) φ ( e ) = n ∏ p | n ( 1 + 1 p ) {\d... | Dedekind psi function |
c_dfm6l2e62gqy | In number theory, the Dirichlet hyperbola method is a technique to evaluate the sum ∑ n ≤ x f ( n ) {\displaystyle \sum _{n\leq x}f(n)} where f , g , h {\displaystyle f,g,h} are multiplicative functions with f = g ∗ h {\displaystyle f=g*h} , where ∗ {\displaystyle *} is the Dirichlet convolution. It uses the fact that ... | Dirichlet hyperbola method |
c_4tuw01vuhqbn | In number theory, the Eichler–Shimura congruence relation expresses the local L-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators. It was introduced by Eichler (1954) and generalized by Shimura (1958). Roughly speaking, it says that the correspondence on the modular curve inducing ... | Eichler–Shimura congruence relation |
c_wva72lty7hp3 | In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particular Galois groups. One curve, C168, gives Galois group PSL(2,7) from a polynomial of deg... | Elkies trinomial curves |
c_onggjl4095a2 | The curve has genus two, and so by Faltings theorem there are only a finite number of rational points on it. These rational points were proven by Nils Bruin using the computer program Kash to be the only ones on C168, and they give only four distinct trinomial polynomials with Galois group PSL(2,7): x7-7x+3 (the Trinks... | Elkies trinomial curves |
c_vqemg2e2u11z | In number theory, the Elliott–Halberstam conjecture is a conjecture about the distribution of prime numbers in arithmetic progressions. It has many applications in sieve theory. It is named for Peter D. T. A. Elliott and Heini Halberstam, who stated the conjecture in 1968.Stating the conjecture requires some notation. | Elliott–Halberstam conjecture |
c_ooqv637fkamt | Let π ( x ) {\displaystyle \pi (x)} , the prime-counting function, denote the number of primes less than or equal to x {\displaystyle x} . If q {\displaystyle q} is a positive integer and a {\displaystyle a} is coprime to q {\displaystyle q} , we let π ( x ; q , a ) {\displaystyle \pi (x;q,a)} denote the number of prim... | Elliott–Halberstam conjecture |
c_l2t1url6h7y0 | If we then define the error function E ( x ; q ) = max gcd ( a , q ) = 1 | π ( x ; q , a ) − π ( x ) φ ( q ) | {\displaystyle E(x;q)=\max _{{\text{gcd}}(a,q)=1}\left|\pi (x;q,a)-{\frac {\pi (x)}{\varphi (q)}}\right|} where the max is taken over all a {\displaystyle a} coprime to q {\displaystyle q} , then the Elliott–H... | Elliott–Halberstam conjecture |
c_2h8umakftccl | One striking one is the result announced by Dan Goldston, János Pintz, and Cem Yıldırım, which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott–Halberstam conjecture, one can show the existence ... | Elliott–Halberstam conjecture |
c_1r0vha2vel3v | In number theory, the Erdős arcsine law, named after Paul Erdős in 1969, states that the prime divisors of a number have a distribution related to the arcsine distribution. Specifically, say that the jth prime factor p of a given number n (in the sorted sequence of distinct prime factors) is "small" when log log p < j.... | Erdős arcsine law |
c_zkg8nu4iktv6 | In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ω(n) is the number of distinct prime factors of n, then, loosely speaking, the probability distribution of ω ( n ) − log log n log log n {\displa... | Erdős–Kac theorem |
c_xhwqi60j2b4i | In number theory, the Erdős–Moser equation is 1 k + 2 k + ⋯ + m k = ( m + 1 ) k , {\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k},} where m {\displaystyle m} and k {\displaystyle k} are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist. | Erdős–Moser equation |
c_lwhulbrhjyc4 | In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. | Fermat pseudoprime |
c_5bac729vi81p | In number theory, the Fermat quotient of an integer a with respect to an odd prime p is defined as q p ( a ) = a p − 1 − 1 p , {\displaystyle q_{p}(a)={\frac {a^{p-1}-1}{p}},} or δ p ( a ) = a − a p p {\displaystyle \delta _{p}(a)={\frac {a-a^{p}}{p}}} .This article is about the former; for the latter see p-derivation.... | Fermat quotient |
c_kirvj3ansrt2 | In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation has only finitely many solutions (a,b,c,m,n,k) with distinct triplets of values (am, bn, ck) where a, b, c are positive coprime integers and m... | Fermat–Catalan conjecture |
c_wludzxddktvr | In number theory, the Gaussian moat problem asks whether it is possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded. More colorfully, if one imagines the Gaussian primes to be stepping stones in a sea of complex numbers,... | Gaussian moat |
c_2gb46cfzob25 | + 3, ..., n! + n are all composite.The problem of finding a path between two Gaussian primes that minimizes the maximum hop size is an instance of the minimax path problem, and the hop size of an optimal path is equal to the width of the widest moat between the two primes, where a moat may be defined by a partition of ... | Gaussian moat |
c_8macmms6coap | It is known that, for any positive number k, there exist Gaussian primes whose nearest neighbor is at distance k or larger. In fact, these numbers may be constrained to be on the real axis. For instance, the number 20785207 is surrounded by a moat of width 17. Thus, there definitely exist moats of arbitrarily large wid... | Gaussian moat |
c_cnevby1woyh1 | In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemeréd... | Green–Tao theorem |
c_lzhdbdzqhsnw | In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm. Here to be a global norm means to be an element k of K such that there is an element l of L with N L / K ( l ) = k {\displaystyle \math... | Hasse norm theorem |
c_p2788bjgb4on | Serre and Tate showed that another counterexample is given by the field Q ( 13 , 17 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {13}},{\sqrt {17}})/{\mathbf {Q} }} where every rational square is a local norm everywhere but 5 2 {\displaystyle 5^{2}} is not a global norm. This is an example of a theorem stating a local-gl... | Hasse norm theorem |
c_rzmp68ur0yov | The full theorem is due to Hasse (1931). The special case when the degree n of the extension is 2 was proved by Hilbert (1897), and the special case when n is prime was proved by Furtwangler in 1902.The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K) is trivial ... | Hasse norm theorem |
c_hu9l63ajjz3t | In number theory, the Katz–Lang finiteness theorem, proved by Nick Katz and Serge Lang (1981), states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely generated over the prime field, and Ker(X/K) is the kernel of the maps between their abelianized fundamental groups, t... | Katz–Lang finiteness theorem |
c_zqwiq9k5ayg8 | In number theory, the Kempner function S ( n ) {\displaystyle S(n)} is defined for a given positive integer n {\displaystyle n} to be the smallest number s {\displaystyle s} such that n {\displaystyle n} divides the factorial s ! {\displaystyle s!} . For example, the number 8 {\displaystyle 8} does not divide 1 ! | Kempner function |
c_tpnwuljgcqz5 | {\displaystyle 1!} , 2 ! {\displaystyle 2!} | Kempner function |
c_060o637bazh3 | , or 3 ! {\displaystyle 3!} , but does divide 4 ! | Kempner function |
c_e82ak5soxqjb | {\displaystyle 4!} , so S ( 8 ) = 4 {\displaystyle S(8)=4} . This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers. | Kempner function |
c_u8hauktl162t | In number theory, the Kronecker symbol, written as ( a n ) {\displaystyle \left({\frac {a}{n}}\right)} or ( a | n ) {\displaystyle (a|n)} , is a generalization of the Jacobi symbol to all integers n {\displaystyle n} . It was introduced by Leopold Kronecker (1885, page 770). | Kronecker symbol |
c_kip2999m3tpn | In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. There are many versions of "arithmetic derivatives", including the one disc... | Arithmetic derivative |
c_w1nj0wosf84z | In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0. The Legendre symbol was introduced by A... | Legendre symbol |
c_39s69bjc6jxe | In number theory, the Mertens function is defined for all positive integers n as M ( n ) = ∑ k = 1 n μ ( k ) , {\displaystyle M(n)=\sum _{k=1}^{n}\mu (k),} where μ ( k ) {\displaystyle \mu (k)} is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive real num... | Mertens function |
c_s1jjzftfuggw | {\displaystyle M(x)=M(\lfloor x\rfloor ).} Less formally, M ( x ) {\displaystyle M(x)} is the count of square-free integers up to x that have an even number of prime factors, minus the count of those that have an odd number. The first 143 M(n) values are (sequence A002321 in the OEIS) The Mertens function slowly grows ... | Mertens function |
c_cf0hfr4xx7kt | This implies, for θ = 0 {\displaystyle \theta =0} that M ( x ) = O ( x log h x ) . {\displaystyle M(x)=O\left({\frac {x}{\log ^{h}x}}\right)\ .} The Mertens conjecture went further, stating that there would be no x where the absolute value of the Mertens function exceeds the square root of x. The Mertens conjecture w... | Mertens function |
c_50gnxkdmbhn5 | However, the Riemann hypothesis is equivalent to a weaker conjecture on the growth of M(x), namely M(x) = O(x1/2 + ε). Since high values for M(x) grow at least as fast as x {\displaystyle {\sqrt {x}}} , this puts a rather tight bound on its rate of growth. Here, O refers to big O notation. | Mertens function |
c_9gghakryguip | The true rate of growth of M(x) is not known. An unpublished conjecture of Steve Gonek states that 0 < lim sup x → ∞ | M ( x ) | x ( log log log x ) 5 / 4 < ∞ . {\displaystyle 0<\limsup _{x\to \infty }{\frac {|M(x)|}{{\sqrt {x}}(\log \log \log x)^{5/4}}}<\infty .} | Mertens function |
c_7loheb7hysz5 | Probabilistic evidence towards this conjecture is given by Nathan Ng. In particular, Ng gives a conditional proof that the function e − y / 2 M ( e y ) {\displaystyle e^{-y/2}M(e^{y})} has a limiting distribution ν {\displaystyle \nu } on R {\displaystyle \mathbb {R} } . That is, for all bounded Lipschitz continuous fu... | Mertens function |
c_6d5vpqf7a8ho | In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4. Equivalently, they are the numbers whose binary representations are nonzero only in even positions. The Moser–de Bruijn numbers in this sequence grow... | Moser–de Bruijn sequence |
c_j1owxcdzet8c | The difference of two Moser–de Bruijn numbers, multiplied by two, is never square. Every natural number can be formed in a unique way as the sum of a Moser–de Bruijn number and twice a Moser–de Bruijn number. This representation as a sum defines a one-to-one correspondence between integers and pairs of integers, listed... | Moser–de Bruijn sequence |
c_y6gd16482mi7 | In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate. | Néron–Tate height |
c_2dojkfk6e9e2 | In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle P(0)=P(1)=P(2)=1,} and the recurrence relation P ( n ) = P ( n − 2 ) + P ( n − 3 ) . {\displaystyle P(n)=P(n-2)+P(n-3).} The first few values of P(n) are 1, 1, 1, 2, 2, ... | Padovan sequence |
c_2ufgk3onx4s9 | Hans van der Laan: Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. | Padovan sequence |
c_f54sk3d8nkl0 | He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics". The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets. | Padovan sequence |
c_nuuayad3nxt4 | In number theory, the Poussin proof is the proof of an identity related to the fractional part of a ratio. In 1838, Peter Gustav Lejeune Dirichlet proved an approximate formula for the average number of divisors of all the numbers from 1 to n: ∑ k = 1 n d ( k ) n ≈ ln n + 2 γ − 1 , {\displaystyle {\frac {\sum _{k=1}^... | Poussin proof |
c_jx24k6fyr9nh | In number theory, the Pólya conjecture (or Pólya's conjecture) stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors. The conjecture was set forth by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. ... | Pólya conjecture |
c_cswve3vy56vk | In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. | Selberg sieve |
c_ahz72irs40rh | In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information about the coefficient of the leading term in the Taylor expansion of an Artin L-function associated with a Galois extension K/k of algebraic number fields. The conjecture... | Stark conjectures |
c_0j7gtcgxibdj | In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree. The Stern–Brocot tree was introduced independently by Moritz Stern (1858) and Achille Brocot... | Stern–Brocot tree |
c_pglk2vzf485g | In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where q = p {\displaystyle q=p} if p {\displaystyle p} is odd and q = 4 {\displaystyle q=4} if p = 2 {\displaystyle p=2} , taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identify... | Teichmüller character |
c_mw5edbdd1dyk | In number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934. | Turán sieve |
c_xwjzlvay2y2i | 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. That is, s ( n ) = ∑ d | n , d ≠ n d . {\displaystyle s(n)=\sum \nolimits _{d|n,\ d\neq n}d.} It can be used to characterize the prime numbers, perfect numbers, sociable... | Restricted divisor function |
c_me93cw8frr5t | In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. | Class number formula |
c_qete72nkijg7 | In number theory, the classical modular curve is an irreducible plane algebraic curve given by an equation Φn(x, y) = 0,such that (x, y) = (j(nτ), j(τ)) is a point on the curve. Here j(τ) denotes the j-invariant. The curve is sometimes called X0(n), though often that notation is used for the abstract algebraic curve fo... | Classical modular curve |
c_fkxgp6sh8mzt | A related object is the classical modular polynomial, a polynomial in one variable defined as Φn(x, x). It is important to note that the classical modular curves are part of the larger theory of modular curves. In particular it has another expression as a compactified quotient of the complex upper half-plane H. | Classical modular curve |
c_2upkdk4et9a1 | In number theory, the continued fraction factorization method (CFRAC) is an integer factorization algorithm. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer n, not depending on special form or properties. It was described by D. H. Lehmer and R. E. Powers in 1931, and developed a... | Continued fraction factorization |
c_q9qo7rhn3689 | In number theory, the crank of a partition is a certain integer associated with the partition in number theory. Dyson first introduced the term without a definition in a 1944 paper in a journal published by the Mathematics Society of Cambridge University. He then gave a list of properties this yet-to-be-defined quantit... | Freeman Dyson |
c_b894thdxrcn1 | In number theory, the crank of a partition of an integer is a certain integer associated with the partition. The term was first introduced without a definition by Freeman Dyson in a 1944 paper published in Eureka, a journal published by the Mathematics Society of Cambridge University. Dyson then gave a list of properti... | Crank of a partition |
c_01xvgt1y5ffv | In number theory, the diamond operators 〈d〉 are operators acting on the space of modular forms for the group Γ1(N), given by the action of a matrix (a bc δ) in Γ0(N) where δ ≈ d mod N. The diamond operators form an abelian group and commute with the Hecke operators. | Diamond operator |
c_aw6tulhhomit | In number theory, the distribution of zeros of the Riemann zeta function (and other L-functions) is modeled by the distribution of eigenvalues of certain random matrices. The connection was first discovered by Hugh Montgomery and Freeman Dyson. It is connected to the Hilbert–Pólya conjecture. | Gaussian unitary ensemble |
c_c7insf4uk45r | In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems. | Dirichlet's divisor problem |
c_zpt6wuhfpose | In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert: 92–93 write: A curious feature of sieve literature is that while there is frequent use of Brun's method there are only a few attempts to ... | Fundamental lemma of sieve theory |
c_h0fuo9vggrfi | In number theory, the fundamental theorem of ideal theory in number fields states that every nonzero proper ideal in the ring of integers of a number field admits unique factorization into a product of nonzero prime ideals. In other words, every ring of integers of a number field is a Dedekind domain. | Fundamental theorem of ideal theory in number fields |
c_f86ch0flnsje | In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every n {\displaystyle n} by P ( n ) = ∑ k = 1 n gcd ( k , n ) {\displaystyle P(n)=\sum _{k=1}^{n}\gcd(k,n)} or equivalently P ( n ) = ∑ d ∣ n d φ ( n / d ) {\displaystyle P(n)=\sum _{d\mid n}d\varphi (n/d)} where d {\dis... | Pillai's arithmetical function |
c_nzgi8bm3t3ur | In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log2 n⌋ + 1 bits) is of the form exp ( ( ( 64 / 9 ) 1 / 3 + o ( 1 ) ) ( log n ) 1 / 3 ( log log... | Number Field Sieve |
c_b7ev6hyfzs49 | The principle of the number field sieve (both special and general) can be understood as an improvement to the simpler rational sieve or quadratic sieve. When using such algorithms to factor a large number n, it is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n1/2. The size of ... | Number Field Sieve |
c_xw9oiz6tb137 | The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve. In order to... | Number Field Sieve |
c_wh9yxan6csjv | In number theory, the home prime HP(n) of an integer n greater than 1 is the prime number obtained by repeatedly factoring the increasing concatenation of prime factors including repetitions. The mth intermediate stage in the process of determining HP(n) is designated HPn(m). For instance, HP(10) = 773, as 10 factors a... | Home prime |
c_d15odypbuwnb | Investigations into home primes make up a minor side issue in number theory. Its questions have served as test fields for the implementation of efficient algorithms for factoring composite numbers, but the subject is really one in recreational mathematics. The outstanding computational problem as of 2016 is whether HP(... | Home prime |
c_m696ty6lp3jv | As each iteration is greater than the previous up until a prime is reached, factorizations generally grow more difficult so long as an end is not reached. As of August 2016 the pursuit of HP(49) concerns the factorization of a 251-digit composite factor of HP49(119) after a break was achieved on 3 December 2014 with th... | Home prime |
c_kbd8okiuhs09 | Details of the history of this search, as well as the sequences leading to home primes for all other numbers through 100, are maintained at Patrick De Geest's worldofnumbers website. A wiki primarily associated with the Great Internet Mersenne Prime Search maintains the complete known data through 1000 in base 10 and a... | Home prime |
c_irqrr7mdaltr | In number theory, the ideal class group (or class group) of an algebraic number field K is the quotient group JK /PK where JK is the group of fractional ideals of the ring of integers of K, and PK is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ... | Ideal class group |
c_trx26grkba63 | In number theory, the integer complexity of an integer is the smallest number of ones that can be used to represent it using ones and any number of additions, multiplications, and parentheses. It is always within a constant factor of the logarithm of the given integer. | Integer complexity |
c_rgoitopkzjo7 | In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n, For example, isqrt ( 27 ) = ⌊ 27 ⌋ = ⌊ 5.19615242270663... ⌋ = 5. {\displaystyle \operatorname {isqrt} (27)=\lfloor {\sqrt {27}}\rfloor ... | Integer square root |
c_plbvsori7aoa | In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of... | Larger sieve |
c_lpomwr30okjs | In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: This law, together with its supplements, allows the easy calc... | Quadratic Reciprocity |
c_hwoo2bjjgir8 | {\displaystyle \left(\pm a^{\frac {p+1}{4}}\right)^{2}=a^{\frac {p+1}{2}}=a\cdot a^{\frac {p-1}{2}}\equiv a\left({\frac {a}{p}}\right)=a{\bmod {p}}.} This formula only works if it is known in advance that a {\displaystyle a} is a quadratic residue, which can be checked using the law of quadratic reciprocity. The quadra... | Quadratic Reciprocity |
c_r6ojeotwo773 | (Art. 151)Privately, Gauss referred to it as the "golden theorem". | Quadratic Reciprocity |
c_5yx0u0rgysn6 | He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs. The shortest known proof is included below, together with short proofs of the law's supplements (the Legendre symbols of −1 and 2). Generalizing the reciprocity law to higher powers has been a lead... | Quadratic Reciprocity |
c_no6ayjytmgss | In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published. | Proofs of quadratic reciprocity |
c_z6lf1bu6ydjx | In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) = exp ( ∑ m = 1 ∞ N m m ( q − s ) m ) {\displaystyle Z(V,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)} where V is a non-singular n-dime... | Local zeta-function |
c_p51gkbg2tc0l | In number theory, the multiplicative digital root of a natural number n {\displaystyle n} in a given number base b {\displaystyle b} is found by multiplying the digits of n {\displaystyle n} together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of n {... | Multiplicative persistence |
c_uu2bz7mqtoz9 | In number theory, the nth Pisano period, written as π(n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774. | Pisano period |
c_jeazltmvnx7j | In number theory, the numbers of the form x2 + xy + y2 for integer x, y are called the Löschian numbers (or Loeschian numbers). These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes c... | Löschian number |
c_djb5mj8gs6p7 | In number theory, the odd greedy expansion problem asks whether a greedy algorithm for finding Egyptian fractions with odd denominators always succeeds. As of 2021, it remains unsolved. | Odd greedy expansion |
c_uwhydmf9nhh8 | In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers a and b to equal the reciprocal of a third positive integer c: 1 a + 1 b = 1 c . {\displaystyle {\frac {1}{a}}+{\frac {1}{b}}={\frac {1}{c}}.} Multiplying both sides by abc shows that the optic equation... | Optic equation |
c_uzuq7j3rqoas | In number theory, the p-adic valuation or p-adic order of an integer n is the exponent of the highest power of the prime number p that divides n. It is denoted ν p ( n ) {\displaystyle \nu _{p}(n)} . Equivalently, ν p ( n ) {\displaystyle \nu _{p}(n)} is the exponent to which p {\displaystyle p} appears in the prime fa... | P-adic order |
c_z0zxenx3bzax | In number theory, the parity problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was identified and named by Atle Selberg in 1949. Beginning around 1996, John Friedlander and Henryk Iwaniec developed some parity-sensitive s... | Parity problem (sieve theory) |
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