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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 at some time—at least 1 / n {\displaystyle 1/n} units away from all others. The conjecture was first posed in 1967 by German mathematician Jörg M. Wills, in purely number-theoretic terms, and independently in 1974 by T. W. Cusick; its illustrative and now-popular formulation dates to 1998. The conjecture is known to be true for 7 runners or less, but the general case remains unsolved. Implications of the conjecture include solutions to view-obstruction problems and bounds on properties, related to chromatic numbers, of certain graphs. | https://en.wikipedia.org/wiki/Lonely_runner_conjecture |
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}}} with integers t and u, it expresses in another form h t u ( mod p ) {\displaystyle {\frac {ht}{u}}{\pmod {p}}\;} for any prime number p > 2 that divides d. In case p > 3 it states that − 2 m h t u ≡ ∑ 0 < k < d χ ( k ) k ⌊ k / p ⌋ ( mod p ) {\displaystyle -2{mht \over u}\equiv \sum _{0 | https://en.wikipedia.org/wiki/Ankeny–Artin–Chowla_congruence |
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 class number. Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q(√d) is a quadratic extension of Q. The class number of Q(√d) is one if and only if the ring of integers of Q(√d) is a principal ideal domain (or, equivalently, a unique factorization domain). | https://en.wikipedia.org/wiki/Stark–Heegner_theorem |
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 discriminant D of Q(√d) this list is often written as: D ∈ { − 3 , − 4 , − 7 , − 8 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle D\in \{-3,-4,-7,-8,-11,-19,-43,-67,-163\}.} | https://en.wikipedia.org/wiki/Stark–Heegner_theorem |
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 conjecture on the density of twin primes or their conjecture on primes of the form n2 + 1; it is also a strengthening of Schinzel's hypothesis H. | https://en.wikipedia.org/wiki/Bateman-Horn_conjecture |
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 number appears exactly once in the tree. | https://en.wikipedia.org/wiki/Calkin–Wilf_tree |
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 series, and can be computed by the fusc function. | https://en.wikipedia.org/wiki/Calkin–Wilf_tree |
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 implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields (Artin 1982, page x). | https://en.wikipedia.org/wiki/Ax–Katz_theorem |
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 true that if n is prime, then 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} (this is a special case of Fermat's little theorem), however the converse (if 2 n ≡ 2 mod n {\displaystyle 2^{n}\equiv 2{\bmod {n}}} then n is prime) is false, and therefore the hypothesis as a whole is false. The smallest counterexample is n = 341 = 11×31. Composite numbers n for which 2 n − 2 {\displaystyle 2^{n}-2} is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes. | https://en.wikipedia.org/wiki/Chinese_hypothesis |
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 ) {\displaystyle M(A)} that can be defined as the set M ( A ) = { k a ∣ k ∈ N , a ∈ A } {\displaystyle M(A)=\{ka\mid k\in \mathbb {N} ,a\in A\}} of numbers formed by multiplying members of A {\displaystyle A} by arbitrary positive integers.According to the Davenport–Erdős theorem, for a set M ( A ) {\displaystyle M(A)} , the following notions of density are equivalent, in the sense that they all produce the same number as each other for the density of M ( A ) {\displaystyle M(A)}: The lower natural density, the inferior limit as n {\displaystyle n} goes to infinity of the proportion of members of M ( A ) {\displaystyle M(A)} in the interval {\displaystyle } . The logarithmic density or multiplicative density, the weighted proportion of members of M ( A ) {\displaystyle M(A)} in the interval {\displaystyle } , again in the limit, where the weight of an element a {\displaystyle a} is 1 / a {\displaystyle 1/a} . | https://en.wikipedia.org/wiki/Davenport–Erdős_theorem |
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 arithmetic progressions, their densities are well defined without resort to limits.However, there exist sequences A {\displaystyle A} and their sets of multiples M ( A ) {\displaystyle M(A)} for which the upper natural density (taken using the superior limit in place of the inferior limit) differs from the lower density, and for which the natural density itself (the limit of the same sequence of values) does not exist.The theorem is named after Harold Davenport and Paul Erdős, who published it in 1936. Their original proof used the Hardy–Littlewood tauberian theorem; later, they published another, elementary proof. | https://en.wikipedia.org/wiki/Davenport–Erdős_theorem |
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 convention, ψ ( 1 ) {\displaystyle \psi (1)} , which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions. | https://en.wikipedia.org/wiki/Dedekind_psi_function |
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 n {\displaystyle n} greater than 2. If n {\displaystyle n} is a square-free number then ψ ( n ) = σ ( n ) {\displaystyle \psi (n)=\sigma (n)} , where σ ( n ) {\displaystyle \sigma (n)} is the divisor function. The ψ {\displaystyle \psi } function can also be defined by setting ψ ( p n ) = ( p + 1 ) p n − 1 {\displaystyle \psi (p^{n})=(p+1)p^{n-1}} for powers of any prime p {\displaystyle p} , and then extending the definition to all integers by multiplicativity. | https://en.wikipedia.org/wiki/Dedekind_psi_function |
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 convolution of ψ = I d ∗ | μ | {\displaystyle \psi =\mathrm {Id} *|\mu |} . | https://en.wikipedia.org/wiki/Dedekind_psi_function |
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 ) {\displaystyle \sum _{a}(a/e)\varphi (e)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right)} where a {\displaystyle a} ranges over all divisors of n {\displaystyle n} and p {\displaystyle p} over the prime divisors of n {\displaystyle n} and φ {\displaystyle \varphi } is the totient function. | https://en.wikipedia.org/wiki/Dedekind_psi_function |
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 ∑ n ≤ x f ( n ) = ∑ n ≤ x ∑ a b = n g ( a ) h ( b ) = ∑ a ≤ x ∑ b ≤ x a g ( a ) h ( b ) + ∑ b ≤ x ∑ a ≤ x b g ( a ) h ( b ) − ∑ a ≤ x ∑ b ≤ x g ( a ) h ( b ) . {\displaystyle \sum _{n\leq x}f(n)=\sum _{n\leq x}\sum _{ab=n}g(a)h(b)=\sum _{a\leq {\sqrt {x}}}\sum _{b\leq {\frac {x}{a}}}g(a)h(b)+\sum _{b\leq {\sqrt {x}}}\sum _{a\leq {\frac {x}{b}}}g(a)h(b)-\sum _{a\leq {\sqrt {x}}}\sum _{b\leq {\sqrt {x}}}g(a)h(b).} | https://en.wikipedia.org/wiki/Dirichlet_hyperbola_method |
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 the Hecke operator Tp is congruent mod p to the sum of the Frobenius map Frob and its transpose Ver. In other words, Tp = Frob + Veras endomorphisms of the Jacobian J0(N)Fp of the modular curve X0N over the finite field Fp. The Eichler–Shimura congruence relation and its generalizations to Shimura varieties play a pivotal role in the Langlands program, by identifying a part of the Hasse–Weil zeta function of a modular curve or a more general modular variety, with the product of Mellin transforms of weight 2 modular forms or a product of analogous automorphic L-functions. | https://en.wikipedia.org/wiki/Eichler–Shimura_congruence_relation |
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 degree seven, and the other, C1344, gives Galois group AL(8), the semidirect product of a 2-elementary group of order eight acted on by PSL(2, 7), giving a transitive permutation subgroup of the symmetric group on eight roots of order 1344. The equation of the curve C168 is: y 2 = x ( 81 x 5 + 396 x 4 + 738 x 3 + 660 x 2 + 269 x + 48 ) {\displaystyle y^{2}=x(81x^{5}+396x^{4}+738x^{3}+660x^{2}+269x+48)} The curve is a plane algebraic curve model for a Galois resolvent for the trinomial polynomial equation x7 + bx + c = 0. If there exists a point (x, y) on the (projectivized) curve, there is a corresponding pair (b, c) of rational numbers, such that the trinomial polynomial either factors or has Galois group PSL(2,7), the finite simple group of order 168. | https://en.wikipedia.org/wiki/Elkies_trinomial_curves |
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 polynomial), (1/11)x7-14x+32 (the Erbach-Fisher-McKay polynomial) and two new polynomials with Galois group PSL(2,7), 37 2 x 7 − 28 x + 3 2 {\displaystyle 37^{2}x^{7}-28x+3^{2}} and ( 499 2 / 113 ) x 7 − 212 x + 3 4 {\displaystyle (499^{2}/113)x^{7}-212x+3^{4}} .On the other hand, the equation of curve C1344 is: y 2 = 2 x 6 + 4 x 5 + 36 x 4 + 16 x 3 − 45 x 2 + 190 x + 1241 {\displaystyle y^{2}=2x^{6}+4x^{5}+36x^{4}+16x^{3}-45x^{2}+190x+1241} Once again the genus is two, and by Faltings theorem the list of rational points is finite. It is thought the only rational points on it correspond to polynomials x8+16x+28, x8+576x+1008, 19453x8+19x+2 which have Galois group AL(8), and x8+324x+567, which comes from two different rational points and has Galois group PSL(2, 7) again, this time as the Galois group of a polynomial of degree eight. | https://en.wikipedia.org/wiki/Elkies_trinomial_curves |
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. | https://en.wikipedia.org/wiki/Elliott–Halberstam_conjecture |
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 primes less than or equal to x {\displaystyle x} which are equal to a {\displaystyle a} modulo q {\displaystyle q} . Dirichlet's theorem on primes in arithmetic progressions then tells us that π ( x ; q , a ) ≈ π ( x ) φ ( q ) {\displaystyle \pi (x;q,a)\approx {\frac {\pi (x)}{\varphi (q)}}} where φ {\displaystyle \varphi } is Euler's totient function. | https://en.wikipedia.org/wiki/Elliott–Halberstam_conjecture |
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–Halberstam conjecture is the assertion that for every θ < 1 {\displaystyle \theta <1} and A > 0 {\displaystyle A>0} there exists a constant C > 0 {\displaystyle C>0} such that ∑ 1 ≤ q ≤ x θ E ( x ; q ) ≤ C x log A x {\displaystyle \sum _{1\leq q\leq x^{\theta }}E(x;q)\leq {\frac {Cx}{\log ^{A}x}}} for all x > 2 {\displaystyle x>2} . This conjecture was proven for all θ < 1 / 2 {\displaystyle \theta <1/2} by Enrico Bombieri and A. I. Vinogradov (the Bombieri–Vinogradov theorem, sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the generalized Riemann hypothesis. It is known that the conjecture fails at the endpoint θ = 1 {\displaystyle \theta =1} .The Elliott–Halberstam conjecture has several consequences. | https://en.wikipedia.org/wiki/Elliott–Halberstam_conjecture |
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 of infinitely many pairs of consecutive primes that differ by at most 12. In August 2014, Polymath group showed that subject to the generalized Elliott–Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 6. Without assuming any form of the conjecture, the lowest proven bound is 246. | https://en.wikipedia.org/wiki/Elliott–Halberstam_conjecture |
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. Then, for any fixed parameter u, in the limit as x goes to infinity, the proportion of the integers n less than x that have fewer than u log log n small prime factors converges to 2 π arcsin u . {\displaystyle {\frac {2}{\pi }}\arcsin {\sqrt {u}}.} | https://en.wikipedia.org/wiki/Erdős_arcsine_law |
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 {\displaystyle {\frac {\omega (n)-\log \log n}{\sqrt {\log \log n}}}} is the standard normal distribution. ( ω ( n ) {\displaystyle \omega (n)} is sequence A001221 in the OEIS.) This is an extension of the Hardy–Ramanujan theorem, which states that the normal order of ω(n) is log log n with a typical error of size log log n {\displaystyle {\sqrt {\log \log n}}} . | https://en.wikipedia.org/wiki/Erdős–Kac_theorem |
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. | https://en.wikipedia.org/wiki/Erdős–Moser_equation |
In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem. | https://en.wikipedia.org/wiki/Fermat_pseudoprime |
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. The quotient is named after Pierre de Fermat. If the base a is coprime to the exponent p then Fermat's little theorem says that qp(a) will be an integer. If the base a is also a generator of the multiplicative group of integers modulo p, then qp(a) will be a cyclic number, and p will be a full reptend prime. | https://en.wikipedia.org/wiki/Fermat_quotient |
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, n, k are positive integers satisfying The inequality on m, n, and k is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with k = 1 (for any a, b, m, and n and with c = am + bn) or with m, n, and k all equal to two (for the infinitely many known Pythagorean triples). | https://en.wikipedia.org/wiki/Fermat–Catalan_conjecture |
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, the question is whether one can walk from the origin to infinity with steps of bounded size, without getting wet. The problem was first posed in 1962 by Basil Gordon (although it has sometimes been erroneously attributed to Paul Erdős) and it remains unsolved.With the usual prime numbers, such a sequence is impossible: the prime number theorem implies that there are arbitrarily large gaps in the sequence of prime numbers, and there is also an elementary direct proof: for any n, the n − 1 consecutive numbers n! + 2, n! | https://en.wikipedia.org/wiki/Gaussian_moat |
+ 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 the primes into two subsets and its width is the distance between the closest pair that has one element in each subset. Thus, the Gaussian moat problem may be phrased in a different but equivalent form: is there a finite bound on the widths of the moats that have finitely many primes on the side of the origin?Computational searches have shown that the origin is separated from infinity by a moat of width 6. | https://en.wikipedia.org/wiki/Gaussian_moat |
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 width, but these moats do not necessarily separate the origin from infinity. | https://en.wikipedia.org/wiki/Gaussian_moat |
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 Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770. | https://en.wikipedia.org/wiki/Green–Tao_theorem |
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 \mathbf {N} _{L/K}(l)=k} ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean. The theorem is no longer true in general if the extension is abelian but not cyclic. Hasse gave the counterexample that 3 is a local norm everywhere for the extension Q ( − 3 , 13 ) / Q {\displaystyle {\mathbf {Q} }({\sqrt {-3}},{\sqrt {13}})/{\mathbf {Q} }} but is not a global norm. | https://en.wikipedia.org/wiki/Hasse_norm_theorem |
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-global principle. | https://en.wikipedia.org/wiki/Hasse_norm_theorem |
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 if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to the Tate cohomology group H0(L/K) which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere. | https://en.wikipedia.org/wiki/Hasse_norm_theorem |
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, then Ker(X/K) is finite if K has characteristic 0, and the part of the kernel coprime to p is finite if K has characteristic p > 0. | https://en.wikipedia.org/wiki/Katz–Lang_finiteness_theorem |
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 ! | https://en.wikipedia.org/wiki/Kempner_function |
{\displaystyle 1!} , 2 ! {\displaystyle 2!} | https://en.wikipedia.org/wiki/Kempner_function |
, or 3 ! {\displaystyle 3!} , but does divide 4 ! | https://en.wikipedia.org/wiki/Kempner_function |
{\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. | https://en.wikipedia.org/wiki/Kempner_function |
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). | https://en.wikipedia.org/wiki/Kronecker_symbol |
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 discussed in this article (the Lagarias arithmetic derivative), such as Ihara's arithmetic derivative and Buium's arithmetic derivatives. | https://en.wikipedia.org/wiki/Arithmetic_derivative |
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 Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol. | https://en.wikipedia.org/wiki/Legendre_symbol |
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 numbers as follows: M ( x ) = M ( ⌊ x ⌋ ) . | https://en.wikipedia.org/wiki/Mertens_function |
{\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 in positive and negative directions both on average and in peak value, oscillating in an apparently chaotic manner passing through zero when n has the values 2, 39, 40, 58, 65, 93, 101, 145, 149, 150, 159, 160, 163, 164, 166, 214, 231, 232, 235, 236, 238, 254, 329, 331, 332, 333, 353, 355, 356, 358, 362, 363, 364, 366, 393, 401, 403, 404, 405, 407, 408, 413, 414, 419, 420, 422, 423, 424, 425, 427, 428, ... (sequence A028442 in the OEIS).Because the Möbius function only takes the values −1, 0, and +1, the Mertens function moves slowly, and there is no x such that |M(x)| > x. H. Davenport demonstrated that, for any fixed h, ∑ n = 1 x μ ( n ) exp ( i 2 π n θ ) = O ( x log h x ) {\displaystyle \sum _{n=1}^{x}\mu (n)\exp(i2\pi n\theta )=O\left({\frac {x}{\log ^{h}x}}\right)} uniformly in θ {\displaystyle \theta } . | https://en.wikipedia.org/wiki/Mertens_function |
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 was proven false in 1985 by Andrew Odlyzko and Herman te Riele. | https://en.wikipedia.org/wiki/Mertens_function |
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. | https://en.wikipedia.org/wiki/Mertens_function |
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 .} | https://en.wikipedia.org/wiki/Mertens_function |
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 functions f {\displaystyle f} on the reals we have that lim Y → ∞ 1 Y ∫ 0 Y f ( e − y / 2 M ( e y ) ) d y = ∫ − ∞ ∞ f ( x ) d ν ( x ) , {\displaystyle \lim _{Y\to \infty }{\frac {1}{Y}}\int _{0}^{Y}f{\big (}e^{-y/2}M(e^{y}){\big )}\,dy=\int _{-\infty }^{\infty }f(x)\,d\nu (x),} if one assumes various conjectures about the Riemann zeta function. | https://en.wikipedia.org/wiki/Mertens_function |
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 in proportion to the square numbers. They are the squares for a modified form of arithmetic without carrying. | https://en.wikipedia.org/wiki/Moser–de_Bruijn_sequence |
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 in order of their positions on a Z-order curve. The Moser–de Bruijn sequence can be used to construct pairs of transcendental numbers that are multiplicative inverses of each other and both have simple decimal representations. A simple recurrence relation allows values of the Moser–de Bruijn sequence to be calculated from earlier values, and can be used to prove that the Moser–de Bruijn sequence is a 2-regular sequence. | https://en.wikipedia.org/wiki/Moser–de_Bruijn_sequence |
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. | https://en.wikipedia.org/wiki/Néron–Tate_height |
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, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence A000931 in the OEIS)A Padovan prime is a Padovan number that is prime. The first Padovan primes are: 2, 3, 5, 7, 37, 151, 3329, 23833, 13091204281, 3093215881333057, 1363005552434666078217421284621279933627102780881053358473, 1558877695141608507751098941899265975115403618621811951868598809164180630185566719, ... (sequence A100891 in the OEIS).The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. | https://en.wikipedia.org/wiki/Padovan_sequence |
Hans van der Laan: Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996. | https://en.wikipedia.org/wiki/Padovan_sequence |
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. | https://en.wikipedia.org/wiki/Padovan_sequence |
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}^{n}d(k)}{n}}\approx \ln n+2\gamma -1,} where d represents the divisor function, and γ represents the Euler-Mascheroni constant. In 1898, Charles Jean de la Vallée-Poussin proved that if a large number n is divided by all the primes up to n, then the average fraction by which the quotient falls short of the next whole number is γ: ∑ p ≤ n { n p } π ( n ) ≈ 1 − γ , {\displaystyle {\frac {\sum _{p\leq n}\left\{{\frac {n}{p}}\right\}}{\pi (n)}}\approx 1-\gamma ,} where {x} represents the fractional part of x, and π represents the prime-counting function. For example, if we divide 29 by 2, we get 14.5, which falls short of 15 by 0.5. | https://en.wikipedia.org/wiki/Poussin_proof |
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. Though mathematicians typically refer to this statement as the Pólya conjecture, Pólya never actually conjectured that the statement was true; rather, he showed that the truth of the statement would imply the Riemann hypothesis. For this reason, it is more accurately called "Pólya's problem". The size of the smallest counterexample is often used to demonstrate the fact that a conjecture can be true for many cases and still fail to hold in general, providing an illustration of the strong law of small numbers. | https://en.wikipedia.org/wiki/Pólya_conjecture |
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. | https://en.wikipedia.org/wiki/Selberg_sieve |
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 conjectures generalize the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function of a number field as the product of a regulator related to S-units of the field and a rational number. When K/k is an abelian extension and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. Rubin (1996) and Cristian Dumitru Popescu gave extensions of this refined conjecture to higher orders of vanishing. | https://en.wikipedia.org/wiki/Stark_conjectures |
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 (1861). Stern was a German number theorist; Brocot was a French clockmaker who used the Stern–Brocot tree to design systems of gears with a gear ratio close to some desired value by finding a ratio of smooth numbers near that value. The root of the Stern–Brocot tree corresponds to the number 1. The parent-child relation between numbers in the Stern–Brocot tree may be defined in terms of continued fractions or mediants, and a path in the tree from the root to any other number q provides a sequence of approximations to q with smaller denominators than q. Because the tree contains each positive rational number exactly once, a breadth first search of the tree provides a method of listing all positive rationals that is closely related to Farey sequences. The left subtree of the Stern–Brocot tree, containing the rational numbers in the range (0,1), is called the Farey tree. | https://en.wikipedia.org/wiki/Stern–Brocot_tree |
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. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section ω: k → O of the natural surjection O → k. The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character. | https://en.wikipedia.org/wiki/Teichmüller_character |
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. | https://en.wikipedia.org/wiki/Turán_sieve |
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 numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number. | https://en.wikipedia.org/wiki/Restricted_divisor_function |
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. | https://en.wikipedia.org/wiki/Class_number_formula |
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 for which there exist various models. | https://en.wikipedia.org/wiki/Classical_modular_curve |
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. | https://en.wikipedia.org/wiki/Classical_modular_curve |
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 as a computer algorithm by Michael A. Morrison and John Brillhart in 1975.The continued fraction method is based on Dixon's factorization method. It uses convergents in the regular continued fraction expansion of k n , k ∈ Z + {\displaystyle {\sqrt {kn}},\qquad k\in \mathbb {Z^{+}} } .Since this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious). It has a time complexity of O ( e 2 log n log log n ) = L n {\displaystyle O\left(e^{\sqrt {2\log n\log \log n}}\right)=L_{n}\left} , in the O and L notations. | https://en.wikipedia.org/wiki/Continued_fraction_factorization |
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 quantity should have. In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties Dyson had hypothesized. | https://en.wikipedia.org/wiki/Freeman_Dyson |
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 properties this yet-to-be-defined quantity should have. In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties hypothesized for it by Dyson. | https://en.wikipedia.org/wiki/Crank_of_a_partition |
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. | https://en.wikipedia.org/wiki/Diamond_operator |
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. | https://en.wikipedia.org/wiki/Gaussian_unitary_ensemble |
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. | https://en.wikipedia.org/wiki/Dirichlet's_divisor_problem |
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 formulate a general Brun theorem (such as Theorem 2.1); as a result there are surprisingly many papers which repeat in considerable detail the steps of Brun's argument. Diamond & Halberstam: 42 attribute the terminology Fundamental Lemma to Jonas Kubilius. | https://en.wikipedia.org/wiki/Fundamental_lemma_of_sieve_theory |
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. | https://en.wikipedia.org/wiki/Fundamental_theorem_of_ideal_theory_in_number_fields |
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 {\displaystyle d} is a divisor of n {\displaystyle n} and φ {\displaystyle \varphi } is Euler's totient function. it also can be written as P ( n ) = ∑ d ∣ n d τ ( d ) μ ( n / d ) {\displaystyle P(n)=\sum _{d\mid n}d\tau (d)\mu (n/d)} where, τ {\displaystyle \tau } is the divisor function, and μ {\displaystyle \mu } is the Möbius function. This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933. | https://en.wikipedia.org/wiki/Pillai's_arithmetical_function |
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 n ) 2 / 3 ) = L n {\displaystyle \exp \left(\left((64/9)^{1/3}+o(1)\right)\left(\log n\right)^{1/3}\left(\log \log n\right)^{2/3}\right)=L_{n}\left} in O and L-notations. It is a generalization of the special number field sieve: while the latter can only factor numbers of a certain special form, the general number field sieve can factor any number apart from prime powers (which are trivial to factor by taking roots). | https://en.wikipedia.org/wiki/Number_Field_Sieve |
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 these values is exponential in the size of n (see below). | https://en.wikipedia.org/wiki/Number_Field_Sieve |
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 achieve this speed-up, the number field sieve has to perform computations and factorizations in number fields. This results in many rather complicated aspects of the algorithm, as compared to the simpler rational sieve. The size of the input to the algorithm is log2 n or the number of bits in the binary representation of n. Any element of the order nc for a constant c is exponential in log n. The running time of the number field sieve is super-polynomial but sub-exponential in the size of the input. | https://en.wikipedia.org/wiki/Number_Field_Sieve |
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 as 2×5 yielding HP10(1) = 25, 25 factors as 5×5 yielding HP10(2) = HP25(1) = 55, 55 = 5×11 implies HP10(3) = HP25(2) = HP55(1) = 511, and 511 = 7×73 gives HP10(4) = HP25(3) = HP55(2) = HP511(1) = 773, a prime number. Some sources use the alternative notation HPn for the homeprime, leaving out parentheses. | https://en.wikipedia.org/wiki/Home_prime |
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(49) = HP(77) can be calculated in practice. | https://en.wikipedia.org/wiki/Home_prime |
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 the calculation of HP49(117). This followed the factorization of HP49(110) on 8 September 2012 and of HP49(104) on 11 January 2011, and prior calculations extending for the larger part of a decade that made extensive use of computational resources. | https://en.wikipedia.org/wiki/Home_prime |
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 also has lists for the bases 2 through 9. The primes in HP(n) are 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, ... (sequence A037274 in the OEIS)Aside from the computational problems that have had so much time devoted to them, it appears absolute proof of existence of a home prime for any specific number might entail its effective computation. In purely heuristic terms, the existence has probability 1 for all numbers, but such heuristics make assumptions about numbers drawn from a wide variety of processes that, though they are likely correct, fall short of the standard of proof usually required of mathematical claims. | https://en.wikipedia.org/wiki/Home_prime |
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 ring of integers of K. The order of the group, which is finite, is called the class number of K. The theory extends to Dedekind domains and their fields of fractions, for which the multiplicative properties are intimately tied to the structure of the class group. For example, the class group of a Dedekind domain is trivial if and only if the ring is a unique factorization domain. | https://en.wikipedia.org/wiki/Ideal_class_group |
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. | https://en.wikipedia.org/wiki/Integer_complexity |
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 =\lfloor 5.19615242270663...\rfloor =5.} | https://en.wikipedia.org/wiki/Integer_square_root |
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 a large number of up to half of all residue classes. The larger sieve can exploit the deletion of an arbitrary number of classes. | https://en.wikipedia.org/wiki/Larger_sieve |
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 calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x 2 ≡ a mod p {\displaystyle x^{2}\equiv a{\bmod {p}}} for an odd prime p {\displaystyle p} ; that is, to determine the "perfect squares" modulo p {\displaystyle p} . However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required. For example, in the case p ≡ 3 mod 4 {\displaystyle p\equiv 3{\bmod {4}}} using Euler's criterion one can give an explicit formula for the "square roots" modulo p {\displaystyle p} of a quadratic residue a {\displaystyle a} , namely, ± a p + 1 4 {\displaystyle \pm a^{\frac {p+1}{4}}} indeed, ( ± a p + 1 4 ) 2 = a p + 1 2 = a ⋅ a p − 1 2 ≡ a ( a p ) = a mod p . | https://en.wikipedia.org/wiki/Quadratic_Reciprocity |
{\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 quadratic reciprocity theorem was conjectured by Euler and Legendre and first proved by Gauss, who referred to it as the "fundamental theorem" in his Disquisitiones Arithmeticae and his papers, writing The fundamental theorem must certainly be regarded as one of the most elegant of its type. | https://en.wikipedia.org/wiki/Quadratic_Reciprocity |
(Art. 151)Privately, Gauss referred to it as the "golden theorem". | https://en.wikipedia.org/wiki/Quadratic_Reciprocity |
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 leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program. | https://en.wikipedia.org/wiki/Quadratic_Reciprocity |
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. | https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity |
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-dimensional projective algebraic variety over the field Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq.Making the variable transformation u = q−s, gives Z ( V , u ) = exp ( ∑ m = 1 ∞ N m u m m ) {\displaystyle {\mathit {Z}}(V,u)=\exp \left(\sum _{m=1}^{\infty }N_{m}{\frac {u^{m}}{m}}\right)} as the formal power series in the variable u {\displaystyle u} . Equivalently, the local zeta function is sometimes defined as follows: ( 1 ) Z ( V , 0 ) = 1 {\displaystyle (1)\ \ {\mathit {Z}}(V,0)=1\,} ( 2 ) d d u log Z ( V , u ) = ∑ m = 1 ∞ N m u m − 1 . {\displaystyle (2)\ \ {\frac {d}{du}}\log {\mathit {Z}}(V,u)=\sum _{m=1}^{\infty }N_{m}u^{m-1}\ .} In other words, the local zeta function Z(V, u) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nm of solutions of the equation defining V in the degree m extension Fqm. | https://en.wikipedia.org/wiki/Local_zeta-function |
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 {\displaystyle n} . The multiplicative digital root for the first few positive integers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. (sequence A031347 in the OEIS)Multiplicative digital roots are the multiplicative equivalent of digital roots. | https://en.wikipedia.org/wiki/Multiplicative_persistence |
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. | https://en.wikipedia.org/wiki/Pisano_period |
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 congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3). | https://en.wikipedia.org/wiki/Löschian_number |
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. | https://en.wikipedia.org/wiki/Odd_greedy_expansion |
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 is equivalent to a Diophantine equation (a polynomial equation in multiple integer variables). | https://en.wikipedia.org/wiki/Optic_equation |
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 factorization of n {\displaystyle n} . The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers R {\displaystyle \mathbb {R} } , the completion of the rational numbers with respect to the p {\displaystyle p} -adic absolute value results in the p-adic numbers Q p {\displaystyle \mathbb {Q} _{p}} . | https://en.wikipedia.org/wiki/P-adic_order |
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 sieves that make the parity problem less of an obstacle. | https://en.wikipedia.org/wiki/Parity_problem_(sieve_theory) |
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