List of number theory topics

From Wikipedia, the free encyclopedia

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

• list of recreational number theory topics,
• Topics in cryptography.

Contents 1 Factors 2 Fractions 3 Modular arithmetic 4 Arithmetic functions 5 Analytic number theory: additive problems 6 Algebraic number theory 7 Quadratic forms 8 L-functions 9 Diophantine equations 10 Diophantine approximation 11 Sieve methods 12 Named primes 13 Combinatorial number theory 14 Computational number theory 14.1 Primality tests 14.2 Integer factorization 14.3 Pseudo-random numbers 15 History

[edit] Factors

• Composite number
  • Highly composite number
• Even and odd numbers
  • Parity
• Divisor, aliquot, aliquant
  • 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
• 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

[edit] 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

[edit] 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

[edit] Arithmetic functions

• Multiplicative function
• Additive function
• Dirichlet convolution
• Erdős-Kac theorem
• 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
• Bell series
• Lambert series

[edit] 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
• 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

[edit] Algebraic number theory

See list of algebraic number theory topics

[edit] Quadratic forms

• Unimodular lattice
• Fermat's theorem on sums of two squares
  • Proofs of Fermat's theorem on sums of two squares

[edit] 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
  • 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-Polya 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
  • 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

[edit] 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
• 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

[edit] Diophantine approximation

• 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
  • 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

[edit] Sieve methods

• Large sieve

[edit] 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

[edit] 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

[edit] Computational number theory

• Algorithmic number theory
• Residue number system
• Cunningham project
• Quadratic residuosity problem

[edit] 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
• Miller-Rabin primality test
• Lucas-Lehmer primality test
• Lucas-Lehmer test for Mersenne numbers
• AKS primality test
• NewPGen

[edit] 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
  • FAFNER

[edit] 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

[edit] History

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