List of number theory topics - Definition

This is a list of number theory topics, by Wikipedia page. See also list of cryptology topics.

Factors

• Composite number
  • Highly composite number
• Even and odd numbers
  • Parity
• Divisor
  • 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 factor
  • Table of prime factors
• Formula for primes
• Factorization
  • RSA number
• Fundamental theorem of arithmetic
• Square-free
• Perfect square
• Power of two
• Integer-valued polynomial

Fractions

• Rational number
• Unit fraction
• Irreducible fraction = in lowest terms
• Dyadic fraction
• Recurring decimal
• Farey sequence
  • Ford circle
  • Stern-Brocot tree
• 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
• 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 lemma
• Congruence of squares
• Luhn formula
• Mod n cryptanalysis

Primality tests and factorization

Main articles:Computational number theory, algorithmic number theory

• Trial division
• Sieve of Eratosthenes
• Primality test
  • Probabilistic algorithm
• Fermat primality test
  • Pseudoprime
  • Carmichael number
  • Euler pseudoprime
  • Euler-Jacobi pseudoprime
  • Fibonacci pseudoprime
  • Probable prime
• Miller-Rabin primality test
• Lucas-Lehmer primality test
• Lucas-Lehmer test for Mersenne primes
• AKS primality test
• NewPGen
• Integer factorization, prime factorization algorithm
  • 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
  • FAFNER

Arithmetic functions

• Multiplicative function
• Additive function
• Dirichlet convolution
• Möbius function
  • Möbius inversion formula
• Sigma function
• Divisor function
• Liouville function
• Partition function (number theory)
  • Integer partition
  • Bell numbers
  • Landau's function
  • Pentagonal number theorem

Analytic number theory: additive problems

• Twin prime
  • Twin prime conjecture
  • Brun's constant
• Cousin prime
• Prime quadruplet
• Sexy prime
• Sophie Germain prime
• Cunningham chain
• Goldbach's conjecture
  • Goldbach's weak conjecture
• Second Hardy-Littlewood conjecture
• Schinzel's hypothesis H
• Waring's problem
  • Brahmagupta's 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

Algebraic number theory

See list of algebraic number theory topics

Quadratic forms

L-functions

• Riemann zeta function
  • Basel problem on ζ(2)
  • Hurwitz zeta function
• Dirichlet series
• Euler product
• Prime number theorem
  • 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
  • Hilbert-Polya conjecture
  • Generalized Riemann hypothesis
  • Mertens function, Mertens conjecture, Meissel-Mertens constant
  • De Bruijn-Newman constant
• Dirichlet character
• Dirichlet L-series
• Dirichlet's theorem
  • Linnik's theorem
• Functional equation (L-function)
• Chebotarev's density theorem
• Local zeta function
  • Weil conjectures
• Modular form
  • modular group
  • Congruence subgroup
  • Hecke operator
  • Cusp form
  • Modular curve
  • Ramanujan-Petersson conjecture
• Birch and Swinnerton-Dyer conjecture
• Automorphic form
• Selberg trace formula
• Artin conjecture
• Sato-Tate conjecture
• Langlands program
• Taniyama-Shimura theorem

Diophantine equations

• Pythagorean triple
• Pell's equation
• Elliptic curve
  • Nagell-Lutz theorem
  • Mordell-Weil theorem
  • Arithmetic of abelian varieties
• Fermat's last theorem
• Mordell conjecture
• Euler's conjecture
• abc Conjecture
• Catalan's conjecture
• Pillai's conjecture
• Hasse principle
• Diophantine set
• Matiyasevich's theorem
• One thousand seven hundred and twenty nine

Diophantine approximation

• Irrational number
  • Square root of two
  • Quadratic irrational
  • Integer square root
  • Algebraic number
    • Pisot-Vijayaraghavan 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
  • Erdös-Borwein constant
• Liouville number
• Continued fraction
  • Mathematical constant (sorted by continued fraction representation)
  • Khinchin's constant
  • Generalized continued fraction
• Kronecker's theorem
• Thue-Siegel-Roth theorem
• Prouhet-Thue-Morse constant
• Gelfond-Schneider constant
• 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
• Effective results in number theory
• Mahler's theorem

Sieve methods

Named primes

• Cullen prime
• Euler prime
• Fermat prime
• Mersenne prime
  • New Mersenne conjecture
  • Great Internet Mersenne Prime Search
• Newman-Shanks-Williams prime
• Wagstaff prime
• Wall-Sun-Sun prime
• Wieferich prime
• Wilson prime
• Wolstenholme prime
• Woodall prime
• Prime pages

Combinatorial number theory

• Covering system
• Erdös-Ginzburg-Ziv theorem
• Polynomial method

Pseudo-random numbers

• Pseudorandom number generator
  • Pseudorandomness
  • Cryptographically secure pseudo-random number generator
• Middle-square method
• Blum Blum Shub
• ISAAC
• Lagged Fibonacci generator
• Linear congruential generator
  • Mersenne twister
• Linear feedback shift register
• Shrinking generator
• Stream cipher

History

• Disquisitiones Arithmeticae
• On the Number of Primes Less Than a Given Magnitude
• Vorlesungen über Zahlentheorie

Recreational topics

NB Listing here is not pejorative, since many famous topics in number theory have origins in challenging problems posed purely for their own sake. See recreational mathematics.

• Integer sequence
• Fibonacci sequence
  • Golden mean base
  • Fibonacci coding
• Lucas sequence
• Digital sum
• Persistence of a number
• Normal number
  • Stoneham number
  • Champernowne constant
  • Absolutely normal number

Number sequences

• Figurate numbers
• Polygonal number
  • Triangular number
  • Square number
• Pentagonal number
• Hexagonal number
• Heptagonal number
• Octagonal number
• Nonagonal number
• Decagonal number
• Centered square number
• Centred pentagonal number
• Centered hexagonal number
• Tetrahedral number
• Pyramidal number
  • Triangular pyramidal number
  • Square pyramidal number
  • Pentagonal pyramidal number
  • Heptagonal pyramidal number
• Octahedral number
• Star number
• Perfect number
  • Quasiperfect number
  • Almost perfect number
  • Multiply perfect number
  • Hyperperfect number
  • Semiperfect number
  • Primitive semiperfect number
  • Unitary perfect number
  • Weird number
• Amicable number
• Sociable number
• Abundant number
• Deficient number
• Aliquot sequence
• Super-Poulet number
• Lucky number
• Happy number
• Powerful number
• Primeval number
• Repunit
  • Repunit (factors)
• Repdigit
• Palindromic number
• Automorphic number
• Triangular square number
• Smith numbers
• Polydivisible number
• Harmonic divisor number
• Sphenic number
• Keith number
• Kaprekar number
• Smith number
• Harshad number
• Double Mersenne number
• Zeisel number
• Heteromecic number
• Niven numbers
• Superparticular number
• Untouchable number
• Self number
• Highly totient number
• Practical number

Prime and related sequences

• Semiprime
• Almost prime
• Unique prime
• Factorial prime
• Permutable prime
• Palindromic prime
• Cuban prime
• Lucky prime

Magic squares, etc.

• Ulam spiral
• Magic star
• Magic square
  • Frenicle's standard form
  • Prime reciprocal magic square
  • Trimagic square
  • Multimagic square
  • Panmagic square
  • Diabolic square
  • Satanic square
  • Most-perfect magic square
  • Conway's Lux method for magic squares
• Magic cube
  • Perfect magic cube
  • Semiperfect magic cube
  • Bimagic cube
  • Trimagic cube
  • Multimagic cube
• Magic tesseract
• Perfect magic tesseract
• Semiperfect magic tesseract
• Magic hypercube
• Magic constant
• Squaring the square