Editing List of number theory topics

{{Short description|none}}
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 (mathematics)|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]]

==[[Fraction (mathematics)|Fraction]]s==

*[[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 function]]s==

*[[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 form]]s==

*[[Unimodular lattice]]
*[[Fermat's theorem on sums of two squares]]
**[[Proofs of Fermat's theorem on sums of two squares]]

==[[L-function]]s==

*[[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 Gamma|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 (L-functions)|Artin conjecture]]
*[[Sato&ndash;Tate conjecture]]
*[[Langlands program]]
*[[modularity theorem]]

==[[Diophantine equation]]s==

*[[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 sequence]]s
**[[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 (number)|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]]
*[[Equidistributed sequence|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 theory|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]]
*[[Restricted sumset|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 test]]s===

*[[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 test|Lucas–Lehmer primality test]]
*[[Lucas–Lehmer test for Mersenne numbers]]
*[[AKS primality test]]

===[[Integer factorization]]===

*[[Pollard's p &minus; 1 algorithm|Pollard's ''p''&nbsp;&minus;&nbsp;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 number]]s===

*[[Pseudorandom number generator]]
**[[Pseudorandomness]]
**[[Cryptographically secure pseudo-random number generator]]
*[[Middle-square method]]
*[[Blum Blum Shub]]
*[[ACORN_(PRNG)|ACORN]]
*[[ISAAC (cipher)|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]]''

[[Category:Mathematics-related lists|Number theory]]
[[Category:Number theory| ]]
[[Category:Lists of topics|Number theory]]