Computational number theory

{{short description|Study of algorithms for performing number theoretic computations}}

In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of

computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to diophantine equations, and explicit methods in arithmetic geometry.{{r|pcm}}

Computational number theory has applications to cryptography, including RSA, elliptic curve cryptography and post-quantum cryptography, and is used to investigate conjectures and open problems in number theory, including the Riemann hypothesis, the Birch and Swinnerton-Dyer conjecture, the ABC conjecture, the modularity conjecture, the Sato-Tate conjecture, and explicit aspects of the Langlands program.{{r|pcm}}{{r|bachshallit}}{{r|cohen}}

Software packages

Peter Giblin (1993): Primes and Programming: An Introduction to Number Theory with Computing, Cambridge University Press, ISBN 0-521-40988-8
Nigel P. Smart (1998): The Algorithmic Resolution of Diophantine Equations, Cambridge University Press, ISBN 0-521-64633-2
Ramanujachary Kumanduri and Cristina Romero (1998): Number Theory with Computer Applications, Prentice Hall, ISBN 0-13-801812-X
Fernando Rodriguez Villegas (2007): Experimental Number Theory, Oxford University Press, ISBN 978-0-19-922730-3
Harold M. Edwards (2008): Higher Arithmetic: An Algorithmic Introduction to Number Theory, American Mathematical Society, ISBN 978-1-4704-2153-3
Lasse Rempe-Gillen and Rebecca Waldecker (2014). Primality Testing for Beginners. American Mathematical Society. ISBN 978-0-8218-9883-3

{{reflist|refs=

{{citation

| author1 = Carl Pomerance | author1-link = Carl Pomerance

| editor1 = Timothy Gowers | editor1-link = Timothy Gowers

| title = Computational Number Theory

| work = The Princeton Companion to Mathematics

| year = 2009

| url = https://math.dartmouth.edu/~carlp/PDF/pcm0049.pdf

}}

{{cite book

| author1 = Eric Bach | author1-link = Eric Bach

| author2 = Jeffrey Shallit | author2-link = Jeffrey Shallit

| title = Algorithmic Number Theory, Volume 1: Efficient Algorithms

| publisher = MIT Press

| year = 1996

| isbn = 0-262-02405-5

}}

{{cite book

| author = Henri Cohen | author-link = Henri Cohen (number theorist)

| title = A Course In Computational Algebraic Number Theory

| year = 1993

| isbn = 0-387-55640-0

| volume = 138

| doi = 10.1007/978-3-662-02945-9

}}