User:ArnoldReinhold/Category-Theorems in Number Theory SD test

Category:Theorems in number theory

From Category:Theorems in number theory:

• 0–9
  • 15 and 290 theorems – On when an integer positive definite quadratic form represents all positive integers
• A
  • Alpha-beta conjecture – In additive number theory, a way to measure how dense a sequence of numbers isPages displaying short descriptions of redirect targets
  • Apéry's theorem – Sum of the inverses of the positive integers cubed is irrational
  • Artin–Verdier duality – Theorem on constructible abelian sheaves over the spectrum of a ring of algebraic numbers
  • Ax–Kochen theorem – On the existence of zeros of homogeneous polynomials over the p-adic numbers
• B
  • Baker's theorem – On algebraic independence of logarithms
  • Beatty sequence – Integers formed by rounding down the integer multiples of a positive irrational number
  • Behrend's theorem – On subsets of the integers in which no member of the set is a multiple of any other
  • Birch's theorem – Statement about the representability of zero by odd degree forms
  • Brauer's theorem on forms – On the representability of 0 by forms over certain fields in sufficiently many variables
  • Brumer bound – Bound for the rank of an elliptic curve
• C
  • Carmichael's theorem – On prime divisors in Fibonacci and Lucas sequences
  • Catalan's conjecture – The only nontrivial positive integer solution to x^a-y^b equals 1 is 3^2-2^3
  • Chinese remainder theorem – Theorem for solving simultaneous congruences
  • Chowla–Mordell theorem – When a Gauss sum is the square root of a prime number, multiplied by a root of unity
  • Chowla–Selberg formula – Evaluates a certain product of values of the Gamma function at rational values
  • Congruum – Spacing between equally-spaced square numbers
  • Cubic reciprocity – Conditions under which the congruence x^3 equals p (mod q) is solvable
• D
  • Davenport–Schmidt theorem – How well a certain kind of real number can be approximated by another kind
  • Davenport–Erdős theorem – Equivalence of notions of density for sets of multiples of integers
  • Dedekind discriminant theorem – Finite degree (and hence algebraic) field extension of the field of rational numbersPages displaying short descriptions of redirect targets
  • Dirichlet's approximation theorem – Any real number has a sequence of good rational approximations
  • Division theorem – Division with remainder of integersPages displaying short descriptions of redirect targets
• E
  • Eichler–Shimura congruence relation – Expresses the local L-function of a modular curve at a prime in terms of Hecke operators
  • Eisenstein's theorem – On power series with rational coefficients that are algebraic functions
  • Equidistribution theorem – Integer multiples of any irrational mod 1 are uniformly distributed on the circle
  • Erdős–Fuchs theorem – On the number of ways numbers can be represented as sums of elements of an additive basis
  • Erdős–Graham problem – Theorem on the existence of finite sets of integers >1 whose reciprocals sum to 1
  • Erdős–Szemerédi theorem – For every finite set of real numbers, the pairwise sums or products form a bigger set
  • Erdős–Tetali theorem – Existence theorem for economical additive bases of every order
  • Euclid–Euler theorem – Characterization of even perfect numbers
  • Euler's theorem – Theorem on modular exponentiation
• F
  • Faltings's product theorem – On when a subvariety of a product of projective spaces is a product of varietiesPages displaying short descriptions of redirect targets
  • Faltings's theorem – Curves of genus > 1 over the rationals have only finitely many rational points
  • Fermat polygonal number theorem – Every positive integer is a sum of at most n n-gonal numbers
  • Fermat's Last Theorem – 17th-century conjecture proved by Andrew Wiles in 1994
  • Fermat's right triangle theorem – Rational right triangles cannot have square area
  • Fermat's theorem on sums of two squares – Condition under which an odd prime is a sum of two squares
  • Freiman's theorem – On the approximate structure of sets whose sumset is small
  • Fundamental lemma (Langlands program) – Theorem in abstract algebra
• G
  • Gelfond–Schneider theorem – On the transcendence of a large class of numbers
  • Glaisher's theorem – On the number of partitions of an integer into parts not divisible by another integer
• H
  • Hasse–Minkowski theorem – Two quadratic forms over a number field are equivalent iff they are equivalent locally
  • Heegner's lemma – Criterion for existence of polynomial roots
  • Hilbert's irreducibility theorem – Result in number theory, concerning irreducible polynomials
  • Hurwitz's theorem (number theory) – Theorem in number theory that gives a bound on a Diophantine approximation
• I
  • Ihara's lemma – On when the kernel of the sum of the two p-degeneracy maps is Eisenstein
• J
  • Jacobi triple product – Mathematical identity found by Jacobi in 1829
  • Jacobi's four-square theorem – How many ways a positive integer can be represented as the sum of four squares
• K
  • Kaplansky's theorem on quadratic forms – Result on simultaneous representation of primes by quadratic forms
  • Katz–Lang finiteness theorem – On kernels of maps between abelianized fundamental groups of schemes and fields
  • Krasner's lemma – Relates the topology of a complete non-archimedean field to its algebraic extensions
  • Kronecker's congruence – Theorem on a polynomial involving the elliptic modular function
  • Kummer's congruence – Result in number theory showing congruences involving Bernoulli numbers
  • Kummer's theorem – Describes the highest power of primes dividing a binomial coefficient
• L
  • Lagrange's four-square theorem – Every natural number can be represented as the sum of four integer squares
  • Lambek–Moser theorem – On integer partitions from monotonic functions
  • Legendre's three-square theorem – Says when a natural number can be represented as the sum of three squares of integers
  • Lehmer's conjecture – Proposed lower bound on the Mahler measure for polynomials with integer coefficients
  • Lindemann–Weierstrass theorem – On algebraic independence of exponentials of linearly independent algebraic numbers over Q
  • Local trace formula – On the character of the representation of a reductive algebraic group
  • Lochs's theorem – On the rate of convergence of the continued fraction expansion of a typical real number
• M
  • Mahler's compactness theorem – Characterizes sets of lattices that are bounded in a certain sense
  • Manin–Drinfeld theorem – The difference of two cusps of a modular curve has finite order in the Jacobian variety
  • Mann's theorem – In additive number theory, a way to measure how dense a sequence of numbers isPages displaying short descriptions of redirect targets
  • Marsaglia's theorem – Describes flaws with the pseudorandom numbers from a linear congruential generator
  • Matiyasevich's theorem – Solution of some Diophantine equationPages displaying short descriptions of redirect targets
  • Mestre bound – Bound in mathematics
  • Meyer's theorem – Indefinite quadratic form in > 4 variables over the rationals nontrivially represents 0
  • Midy's theorem – On decimal expansions of fractions with prime denominator and even repeat period
  • Mihăilescu's theorem – The only nontrivial positive integer solution to x^a-y^b equals 1 is 3^2-2^3Pages displaying short descriptions of redirect targets
  • Minkowski's theorem – Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
  • Modularity theorem – Relates rational elliptic curves to modular forms
  • Multiplicity-one theorem – Concerns the representation theory of an adelic reductive algebraic group.
• N
  • Nagell–Lutz theorem – Describes rational torsion points on elliptic curves over the integers
• P
  • Pentagonal number theorem – Relates the product and series representations of the Euler function Π(1-x^n)
  • Proizvolov's identity – On sums of differences between 2 equal sets that partition the first 2N positive integers
• Q
  • Quadratic reciprocity – Gives conditions for the solvability of quadratic equations modulo prime numbers
  • Quartic reciprocity – Collection of theorems in number theory on when the congruence x⁴ ≡ p (mod q) is solvable
  • Quintuple product identity – Infinite product identity introduced by Watson
• R
  • Ramanujan's congruences – Some remarkable congruences for the partition function
  • Ribet's theorem – Result concerning properties of Galois representations associated with modular forms
  • Romanov's theorem – Theorem on the set of numbers that are the sum of a prime and a positive integer power of the base
  • Roth's theorem on arithmetic progressions – On the existence of arithmetic progressions in subsets of the natural numbers
• S
  • Serre's modularity conjecture – Conjecture in number theory
  • Shimura's reciprocity law – On the action of ideles of imaginary quadratic fields on the values of modular functions
  • Siegel–Weil formula – Expresses Eisenstein series as a weighted average of theta series of lattices in a genus
  • Siegel's theorem on integral points – Finitely many for a smooth algebraic curve of genus > 0 defined over a number field
  • Six exponentials theorem – Conditions guaranteeing the transcendence of at least one of a set of exponentials
  • Skolem–Mahler–Lech theorem – The zeros of a linear recurrence relation mostly form a regularly repeating pattern
  • Sophie Germain's theorem – On the divisibility of solutions to Fermat's Last Theorem for prime exponent
  • Størmer's theorem – Gives a finite bound on the number of consecutive pairs of smooth numbers that exist
  • Subspace theorem – Points of small height in projective space lie in a finite number of hyperplanes
  • Sum of two squares theorem – Characterization by prime factors of sums of two squares
  • Szemerédi's theorem – Long dense subsets of the integers contain arbitrarily large arithmetic progressions
• T
  • Thomae's formula – Relates theta constants to the branch points of a hyperelliptic curve
  • Three-gap theorem – On distances between points on a circle
  • Thue equation – Diophantine equation involving an irreducible bivariate form of deg > 2 over the rationals
  • Roth's theorem – Algebraic numbers are not near-rational
  • Tijdeman's theorem – There are at most a finite number of consecutive powers
  • Torsion conjecture – Conjecture in number theory
  • Tunnell's theorem – On the congruent number problem: which integers are the area of a rational right triangle
  • Turán–Kubilius inequality – Theorem in probabilistic number theory on additive complex-valued arithmetic functions
• V
  • Von Staudt–Clausen theorem – Determines the fractional part of Bernoulli numbers
• W
  • Waldspurger's theorem – Identifies Fourier coefficients of some modular forms with the value of an L-series
  • Weil conjectures – On generating functions from counting points on algebraic varieties over finite fields
  • Wiener–Ikehara theorem – Tauberian theorem introduced by Shikao Ikehara (1931).
• Z
  • Zeckendorf's theorem – On the unique representation of integers as sums of non-consecutive Fibonacci numbers
  • Zolotarev's lemma – Ties Legendre symbols to permutation signatures
  • Zsigmondy's theorem – On prime divisors of differences two nth powers