Category
page 1Theorems in number theory
Fermat's Last Theorem
theorem in number theory that there are no nontrivial integer solutions of xⁿ+yⁿ=zⁿ for integer n>2
Chinese remainder theorem
theorem for solving simultaneous congruences
Euler's theorem
generalization of Fermat's little theorem, that given coprime positive integers 𝑛 and 𝑎, then the φ(𝑛)-th power of 𝑎 is congruent to 1 modulo 𝑛, where φ is Euler’s totient function
quadratic reciprocity
theorem
Catalan's conjecture
theorem about consecutive powers
Lagrange's four-square theorem
theorem
modularity theorem
theorem in mathematics
Fermat's theorem on sums of two squares
theorem that an odd prime is a sum of two squares iff it is of the form 4𝑛+1
Lindemann–Weierstrass theorem
theorem of number theory
Fermat polygonal number theorem
theorem
Gelfond–Schneider theorem
theorem which establishes the transcendence of a large class of numbers
Zeckendorf's theorem
theorem that every positive integer can be represented uniquely as the sum of one or more distinct Fibonacci numbers in such a way that the sum doesn’t include any two consecutive Fibonacci numbers
Faltings' theorem
theorem
Szemerédi's theorem
theorem that long dense subsets of the integers contain arbitrarily large arithmetic progressions
Euclid–Euler theorem
theorem characterizing the even perfect numbers
Weil conjectures
theorem
Legendre's three-square theorem
theorem that every nonnegative integer is the sum of three perfect squares unless it is a product of a power of four and a number that is 7 modulo 8
Dirichlet's approximation theorem
theorem in number theory
Beatty sequence
a sequence of integers formed by rounding down the integer multiples of a positive irrational number
Minkowski's theorem
symmetric convex set
pentagonal number theorem
theorem
Hurwitz's theorem
theorem in number theory
Zsigmondy's theorem
On primes dividing the difference of nth powers of coprime integers, but not powers < n
Jacobi triple product
mathematical identity found by Jacobi in 1829
Thue–Siegel–Roth theorem
theorem that algebraic numbers do not have unusually accurate rational approximations
Fermat's right triangle theorem
non-existence proof in number theory, the only complete proof left by Pierre de Fermat
Ribet's theorem
Result concerning properties of Galois representations associated with modular forms
Erdős–Graham problem
Theorem on the existence of finite sets of positive integers >1 whose inverses sum to 1
Carmichael's theorem
On prime divisors Fibonacci numbers and Lucas sequences, more generally
Baker's theorem
lower bound for absolute value of linear combinations of logarithms of algebraic numbers
Hasse–Minkowski theorem
two quadratic forms over a number field are equivalent iff they are equivalent locally
Sophie Germain's theorem
On the divisibility of solutions to Fermat's Last Theorem for prime exponent
Von Staudt–Clausen theorem
determines the fractional part of Bernoulli numbers
Midy's theorem
theorem
Equidistribution theorem
Integer multiples of many irrational mod 1 are uniformly distributed on the circle
15 and 290 theorems
theorem
Kummer's theorem
Gives the p-adic valuation of a binomial coefficient
Tijdeman's theorem
theorem
Jacobi's four-square theorem
How many ways that a positive integer can be represented as the sum of 4 squares
Cubic reciprocity
conditions under which the congruence x3 equals p (mod q) is solvable
Apéry's theorem
theorem
Lochs' theorem
theorem
Erdős–Szemerédi theorem
Glaisher's theorem
On the number of partitions of an integer into parts not divisible by another integer
Thue equation
type of equation with integer coefficients
Serre's modularity conjecture
theorem
Birch's theorem
statement about the representability of zero by odd degree forms
Wiener–Ikehara theorem
Tauberian theorem introduced by Shikao Ikehara (1931)
Freiman's theorem
On the approximate structure of sets whose sumset is small
Lamé's theorem
analysis of the complexity of Euclidean algorithm
Moessner's theorem
theorem in number theory
sum of two squares theorem
theorem in number theory