List of number theory topics explained

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

quack

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 - 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