Computational number theory

In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of algorithms for performing number theoretic computations. The best known problem in the field is integer factorization.

Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, volume 1: Efficient Algorithms. MIT Press, 1996, ISBN 0-262-02405-5
Buhler, J.P.; P., Stevenhagen, eds. (2008). Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications 44. Cambridge University Press. ISBN 978-0-521-20833-8. Zbl 1154.11002.
Henri Cohen, A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics 138, Springer-Verlag, 1993.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, 2001, ISBN 0-387-94777-9
Riesel, Hans (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics 126 (second ed.). Boston, MA: Birkhäuser. ISBN 0-8176-3743-5. Zbl 0821.11001.
Victor Shoup, A Computational Introduction to Number Theory and Algebra. Cambridge, 2005, ISBN 0-521-85154-8

Number-theoretic algorithms

Primality tests	AKS test APR test Baillie–PSW ECPP test Elliptic curve Pocklington Fermat Lucas Lucas–Lehmer Lucas–Lehmer–Riesel Proth's theorem Pépin's Quadratic Frobenius test Solovay–Strassen Miller–Rabin

Prime-generating	Sieve of Atkin Sieve of Eratosthenes Sieve of Sundaram Wheel factorization

Integer factorization	Continued fraction (CFRAC) Dixon's Lenstra elliptic curve (ECM) Euler's Pollard's rho p − 1 p + 1 Quadratic sieve (QS) General number field sieve (GNFS) Special number field sieve (SNFS) Rational sieve Fermat's Shanks' square forms Trial division Shor's

Multiplication	Ancient Egyptian Long Karatsuba Toom–Cook Schönhage–Strassen Fürer's

Discrete logarithm	Baby-step giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve

Greatest common divisor	Binary Euclidean Extended Euclidean Lehmer's

Modular square root	Cipolla Pocklington's Tonelli–Shanks

Other algorithms	Chakravala Cornacchia Integer relation Integer square root Modular exponentiation Schoof's

Italics indicate that algorithm is for numbers of special forms Smallcaps indicate a deterministic algorithm

Major topics in number theory

Algebraic number theory Analytic number theory Geometric number theory Computational number theory Transcendental number theory Combinatorial number theory Arithmetic geometry Arithmetic topology Arithmetic dynamics

Numbers Natural numbers Prime numbers Rational numbers Irrational numbers Algebraic numbers Transcendental numbers p-adic numbers Arithmetic Modular arithmetic Arithmetic functions

Quadratic forms Modular forms L-functions Diophantine equations Diophantine approximation Continued fractions

List of number theory topics (recreational)

See also