Category: Theorems in number theory

Tijdeman's theorem
In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine
Erdős–Fuchs theorem
In mathematics, in the area of additive number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of elements of a given additive basis, s
Kaplansky's theorem on quadratic forms
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky.
Division theorem
No description available.
Faltings' product theorem
In arithmetic geometry, Faltings' product theorem gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introduce
Mihăilescu's theorem
No description available.
Mestre bound
In mathematics, the Mestre bound is a bound on the analytic rank of an elliptic curve in terms of its conductor, introduced by Mestre.
Romanov's theorem
In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base b, the set of numbers that are t
Six exponentials theorem
In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a
Waldspurger's theorem
In mathematics, Waldspurger's theorem, introduced by Jean-Loup Waldspurger, is a result that identifies Fourier coefficients of modular forms of half-integral weight k+1/2 with the value of an L-serie
Turán–Kubilius inequality
The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a s
Meyer's theorem
In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other word
Siegel–Weil formula
In mathematics, the Siegel–Weil formula, introduced by Weil as an extension of the results of Siegel , expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, wher
Quadratic reciprocity
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
Roth's theorem on arithmetic progressions
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers. It was first proven by Klaus Roth
Erdős–Tetali theorem
In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order. More specifically, it states that for every fix
Quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x4 ≡ p (mod q) is solvable; the word "reciproc
Beatty sequence
In mathematics, a Beatty sequence (or homogeneous Beatty sequence) is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are
Cubic reciprocity
Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from t
Quintuple product identity
In mathematics the Watson quintuple product identity is an infinite product identity introduced by Watson and rediscovered by and . It is analogous to the Jacobi triple product identity, and is the Ma
Freiman's theorem
In additive combinatorics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if is small, then can be contained in a
Mahler's compactness theorem
In mathematics, Mahler's compactness theorem, proved by Kurt Mahler, is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite s
Siegel's theorem on integral points
In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there a
Sum of two squares theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer n > 1 to whether it can be written as a sum of two squares, such that n = a2 + b2 for some integers a, b
Zsigmondy's theorem
In number theory, Zsigmondy's theorem, named after Karl Zsigmondy, states that if are coprime integers, then for any integer , there is a prime number p (called a primitive prime divisor) that divides
Serre's modularity conjecture
In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form.
Wiener–Ikehara theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by Shikao Ikehara. It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969).
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of th
Ax–Kochen theorem
The Ax–Kochen theorem, named for James Ax and Simon B. Kochen, states that for each positive integer d there is a finite set Yd of prime numbers, such that if p is any prime not in Yd then every homog
Matiyasevich's theorem
No description available.
Midy's theorem
In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with
Hasse–Minkowski theorem
The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places,
Apéry's theorem
In mathematics, Apéry's theorem is a result in number theory that states the Apéry's constant ζ(3) is irrational. That is, the number cannot be written as a fraction where p and q are integers. The th
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number of variables and having rational numb
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any i
Skolem–Mahler–Lech theorem
In additive and algebraic number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers satisfies a linear difference equation, then with finitely many exceptions the positions at
Ramanujan's congruences
In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The mathematician Srinivasa Ramanujan discovered the congruences This means that: * If a numbe
Dedekind discriminant theorem
No description available.
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers and , with , there exist integers and such that and
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by André Weil. They led to a successful multi-decade program to prove them, in which many leading researchers developed the frame
Erdős–Graham problem
In combinatorial number theory, the Erdős–Graham problem is the problem of proving that, if the set of integers greater than one is partitioned into finitely many subsets, then one of the subsets can
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis o
Szemerédi's theorem
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with posit
Brauer's theorem on forms
In mathematics, Brauer's theorem, named for Richard Brauer, is a result on the representability of 0 by forms over certain fields in sufficiently many variables.
Tunnell's theorem
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.
In number theory, a congruum (plural congrua) is the difference between successive square numbers in an arithmetic progression of three squares.That is, if , , and (for integers , , and ) are three sq
Thue equation
In mathematics, a Thue equation is a Diophantine equation of the form ƒ(x,y) = r, where ƒ is an irreducible bivariate form of degree at least 3 over the rational numbers, and r is a nonzero rational n
Euclid–Euler theorem
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form 2p−1(2p − 1), where 2p −
Ribet's theorem
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jea
15 and 290 theorems
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix re
Legendre's three-square theorem
In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers if and only if n is not of the form for nonnegative integers a a
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers.
Lochs's theorem
In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by . The theorem states th
Proizvolov's identity
In mathematics, Proizvolov's identity is an identity concerning sums of differences of positive integers. The identity was posed by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Stud
Minkowski's theorem
In mathematics, Minkowski's theorem is the statement that every convex set in which is symmetric with respect to the origin and which has volume greater than contains a non-zero integer point (meaning
Shimura's reciprocity law
In mathematics, Shimura's reciprocity law, introduced by Shimura, describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of
Kronecker's congruence
In mathematics, Kronecker's congruence, introduced by Kronecker, states that where p is a prime and Φp(x,y) is the modular polynomial of order p, given by for j the elliptic modular function and τ run
Marsaglia's theorem
In computational number theory, Marsaglia's theorem connects modular arithmetic and analytic geometry to describe the flaws with the pseudorandom numbers resulting from a linear congruential generator
Kummer's congruence
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by Ernst Eduard Kummer. used Kummer's congruences to define the p-adic zeta function.
Artin–Verdier duality
In mathematics, Artin–Verdier duality is a duality theorem for constructible abelian sheaves over the spectrum of a ring of algebraic numbers, introduced by Michael Artin and Jean-Louis Verdier, that
Nagell–Lutz theorem
In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers.It is named for Trygve
Lindemann–Weierstrass theorem
In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem
Thomae's formula
In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae relating theta constants to the branch points of a hyperelliptic curve .
Mann's theorem
No description available.
Lambek–Moser theorem
In combinatorial number theory, the Lambek–Moser theorem splits the natural numbers into two complementary sets using any non-decreasing function and its inverse. It extends Rayleigh's theorem, which
Sophie Germain's theorem
In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation of Fermat's Last Theorem for odd prime .
Von Staudt–Clausen theorem
In number theory, the von Staudt–Clausen theorem is a result determining the fractional part of Bernoulli numbers, found independently byKarl von Staudt and Thomas Clausen. Specifically, if n is a pos
Hurwitz's theorem (number theory)
In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively
Behrend's theorem
In arithmetic combinatorics, Behrend's theorem states that the subsets of the integers from 1 to in which no member of the set is a multiple of any other must have a logarithmic density that goes to z
Three-gap theorem
In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there w
Birch's theorem
In mathematics, Birch's theorem, named for Bryan John Birch, is a statement about the representability of zero by odd degree forms.
Fermat's right triangle theorem
Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. I
Alpha-beta conjecture
No description available.
Catalan's conjecture
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderbor
Davenport–Schmidt theorem
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it t
Manin–Drinfeld theorem
In mathematics, the Manin–Drinfeld theorem, proved by Manin and Drinfeld, states that the difference of two cusps of a modular curve has finite order in the Jacobian variety.
Brumer bound
In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by Brumer.
Størmer's theorem
In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a me
Chowla–Mordell theorem
In mathematics, the Chowla–Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity. It was proved and publi
Equidistribution theorem
In mathematics, the equidistribution theorem is the statement that the sequence a, 2a, 3a, ... mod 1 is uniformly distributed on the circle , when a is an irrational number. It is a special case of th
Davenport–Erdős theorem
In number theory, the Davenport–Erdős theorem states that, for sets of multiples of integers, several different notions of density are equivalent. Let be a sequence of positive integers. Then the mult
Kummer's theorem
In mathematics, Kummer's theorem is a formula for the exponent of the highest power of a prime number p that divides a given binomial coefficient. In other words, it gives the p-adic valuation of a bi
Subspace theorem
In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt.
Erdős–Szemerédi theorem
The Erdős–Szemerédi theorem in arithmetic combinatorics states that for every finite set of integers, at least one of , the set of pairwise sums or , the set of pairwise products form a significantly
Roth's theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number a
Torsion conjecture
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian vari
Glaisher's theorem
In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher.
Fermat polygonal number theorem
In additive number theory, the Fermat polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. That is, every positive integer can be written as the sum of th
Chowla–Selberg formula
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadra
Eichler–Shimura congruence relation
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 an
Carmichael's theorem
In number theory, Carmichael's theorem, named after the American mathematician R.D. Carmichael,states that, for any nondegenerate Lucas sequence of the first kind Un(P,Q) with relatively prime paramet
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity: for complex numbers x and y, with |x| < 1 and y ≠ 0. It was introduced by Jacobi in his work Fundamenta Nova Theoriae Functionum
Pentagonal number theorem
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that In other words, The exponents 1, 2, 5, 7, 1
Multiplicity-one theorem
In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question i
Faltings's theorem
In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points.
Local trace formula
In mathematics, the local trace formula is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation of G(F) on the discrete part of L2(G(F)), for G a red
Baker's theorem
In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by
Jacobi's four-square theorem
Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer n can be represented as the sum of four squares.
Modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational n
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if n and a are coprime positive integers, and is Euler's totient function, then a rai
Katz–Lang finiteness theorem
In number theory, the Katz–Lang finiteness theorem, proved by Nick Katz and Serge Lang, states that if X is a smooth geometrically connected scheme of finite type over a field K that is finitely gener
Lehmer's conjecture
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such tha
Zeckendorf's theorem
In mathematics, Zeckendorf's theorem, named after Belgian amateur mathematician Edouard Zeckendorf, is a theorem about the representation of integers as sums of Fibonacci numbers. Zeckendorf's theorem
Eisenstein's theorem
In mathematics, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function with rational number coef
Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as: with x and y integers, if and only if The prime numbers for which this is true are ca