List of number theory topics

This is a list of number theory topics. See also:


 * List of recreational number theory topics
 * Topics in cryptography

Divisibility

 * 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
 * Method of successive substitution
 * 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
 * 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 &minus; 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