This is a list of topics in number theory. See also:
List of recreational number theory topics Topics in cryptography
Divisibility rule Composite number Highly composite number Even and odd numbers Parity Divisor, aliquot part Greatest common divisor Least common multiple Euclidean algorithm Coprime Euclid's lemma Bézout's identity, Bézout's lemma Extended Euclidean algorithm Table of divisors Prime number, prime power Bonse's inequality Prime factor Table of prime factors Formula for primes Factorization RSA number Fundamental theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial
Fractions Rational number Unit fraction Irreducible fraction = in lowest terms Dyadic fraction Recurring decimal Cyclic number Farey sequence Ford circle Stern–Brocot tree Dedekind sum Egyptian fraction
Modular arithmetic Montgomery reduction Modular exponentiation Linear congruence theorem Successive over-relaxation Chinese remainder theorem Fermat's little theorem Proofs of Fermat's little theorem Fermat quotient Euler's totient function Noncototient Nontotient Euler's theorem Wilson's theorem Primitive root modulo n Multiplicative order Discrete logarithm Quadratic residue Euler's criterion Legendre symbol Gauss's lemma (number theory) Congruence of squares Luhn formula Mod n cryptanalysis
Arithmetic functions Multiplicative function Additive function Dirichlet convolution Erdős–Kac theorem Möbius function Möbius inversion formula Divisor function Liouville function Partition function (number theory) Integer partition Bell numbers Landau's function Pentagonal number theorem Bell series Lambert series
Analytic number theory: additive problems Twin prime Brun's constant Cousin prime Prime triplet Prime quadruplet Sexy prime Sophie Germain prime Cunningham chain Goldbach's conjecture Goldbach's weak conjecture Second Hardy–Littlewood conjecture Hardy–Littlewood circle method Schinzel's hypothesis H Bateman–Horn conjecture Waring's problem Brahmagupta–Fibonacci identity Euler's four-square identity Lagrange's four-square theorem Taxicab number Generalized taxicab number Cabtaxi number Schnirelmann density Sumset Landau–Ramanujan constant Sierpinski number Seventeen or Bust Niven's constant
Algebraic number theory See list of algebraic number theory topics
Quadratic forms Unimodular lattice Fermat's theorem on sums of two squares Proofs of Fermat's theorem on sums of two squares
L-functions Riemann zeta function Basel problem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture Von Staudt–Clausen theorem Dirichlet series Euler product Prime number theorem Prime-counting function Meissel–Lehmer algorithm Offset logarithmic integral Legendre's constant Skewes' number Bertrand's postulate Proof of Bertrand's postulate Proof that the sum of the reciprocals of the primes diverges Cramér's conjecture Riemann hypothesis Critical line theorem Hilbert–Pólya conjecture Generalized Riemann hypothesis Mertens function, Mertens conjecture, Meissel–Mertens constant De Bruijn–Newman constant Dirichlet character Dirichlet L-series Siegel zero Dirichlet's theorem on arithmetic progressions Linnik's theorem Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem Local zeta function Weil conjectures Modular form modular group Congruence subgroup Hecke operator Cusp form Eisenstein series Modular curve Ramanujan–Petersson conjecture Birch and Swinnerton-Dyer conjecture Automorphic form Selberg trace formula Artin conjecture Sato–Tate conjecture Langlands program modularity theorem
Diophantine equations Pythagorean triple Pell's equation Elliptic curve Nagell–Lutz theorem Mordell–Weil theorem Mazur's torsion theorem Congruent number Arithmetic of abelian varieties Elliptic divisibility sequences Mordell curve Fermat's Last Theorem Mordell conjecture Euler's sum of powers conjecture abc Conjecture Catalan's conjecture Pillai's conjecture Hasse principle Diophantine set Matiyasevich's theorem Hundred Fowls Problem 1729
Diophantine approximation Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number Salem number Transcendental number e (mathematical constant) pi, list of topics related to pi Squaring the circle Proof that e is irrational Lindemann–Weierstrass theorem Hilbert's seventh problem Gelfond–Schneider theorem Erdős–Borwein constant Liouville number Irrationality measure Simple continued fraction Mathematical constant (sorted by continued fraction representation) Khinchin's constant Lévy's constant Lochs' theorem Gauss–Kuzmin–Wirsing operator Minkowski's question mark function Generalized continued fraction Kronecker's theorem Thue–Siegel–Roth theorem Prouhet–Thue–Morse constant Gelfond–Schneider constant Equidistribution mod 1 Beatty's theorem Littlewood conjecture Discrepancy function Low-discrepancy sequence Illustration of a low-discrepancy sequence Constructions of low-discrepancy sequences Halton sequences Geometry of numbers Minkowski's theorem Pick's theorem Mahler's compactness theorem Mahler measure Effective results in number theory Mahler's theorem
Sieve methods Brun sieve Function field sieve General number field sieve Large sieve Larger sieve Quadratic sieve Selberg sieve Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Turán sieve
Named primes Chen prime Cullen prime Fermat prime Sophie Germain prime, safe prime Mersenne prime New Mersenne conjecture Great Internet Mersenne Prime Search Newman–Shanks–Williams prime Primorial prime Wagstaff prime Wall–Sun–Sun prime Wieferich prime Wilson prime Wolstenholme prime Woodall prime Prime pages
Combinatorial number theory Covering system Small set (combinatorics) Erdős–Ginzburg–Ziv theorem Polynomial method Van der Waerden's theorem Szemerédi's theorem Collatz conjecture Gilbreath's conjecture Erdős–Graham conjecture Znám's problem
Computational number theory Note: Computational number theory is also known as algorithmic number theory.
Residue number system Cunningham project Quadratic residuosity problem
Primality tests Prime factorization algorithm Trial division Sieve of Eratosthenes Probabilistic algorithm Fermat primality test Pseudoprime Carmichael number Euler pseudoprime Euler–Jacobi pseudoprime Fibonacci pseudoprime Probable prime Baillie–PSW primality test Miller–Rabin primality test Lucas–Lehmer primality test Lucas–Lehmer test for Mersenne numbers AKS primality test
Integer factorization Pollard's p − 1 algorithm Pollard's rho algorithm Lenstra elliptic curve factorization Quadratic sieve Special number field sieve General number field sieve Shor's algorithm RSA Factoring Challenge
Pseudo-random numbers Pseudorandom number generator Pseudorandomness Cryptographically secure pseudo-random number generator Middle-square method Blum Blum Shub ACORN ISAAC Lagged Fibonacci generator Linear congruential generator Mersenne twister Linear-feedback shift register Shrinking generator Stream cipher see also List of random number generators.
Arithmetic dynamics Aliquot sequence and Aliquot sum dynamics Abundant number Almost perfect number Amicable number Betrothed numbers Deficient number Quasiperfect number Perfect number Sociable number Collatz conjecture Digit sum dynamics Additive persistence Digital root Digit product dynamics Multiplicative digital root Multiplicative persistence Lychrel number Perfect digital invariant Happy number
History Disquisitiones Arithmeticae "On the Number of Primes Less Than a Given Magnitude" Vorlesungen über Zahlentheorie Prime Obsession