# List of number theory topics

This is a list of topics in number theory. 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
* 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