Computational number theory explained

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.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.

Software packages

• Magma computer algebra system
• SageMath
• Number Theory Library
• PARI/GP
• Fast Library for Number Theory

Further reading

• Book: Eric Bach . Eric Bach. Jeffrey Shallit . Jeffrey Shallit. Algorithmic Number Theory, Volume 1: Efficient Algorithms. MIT Press. 1996. 0-262-02405-5.
• Book: David Bressoud

. David M. Bressoud. David Bressoud. Factorisation and Primality Testing. Springer-Verlag. 1989. 0-387-97040-1. registration.

• Book: Joe P. Buhler . Joe P. Buhler. Peter Stevenhagen. Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications. 44. Cambridge University Press. 2008. 978-0-521-20833-8. 1154.11002.
• Book: Henri Cohen . Henri Cohen (number theorist). A Course In Computational Algebraic Number Theory . 1993. 0-387-55640-0. Springer-Verlag. Graduate Texts in Mathematics. 138. 10.1007/978-3-662-02945-9.
• Book: Henri Cohen . Henri Cohen (number theorist). Advanced Topics in Computational Number Theory. 2000. 0-387-98727-4. Springer-Verlag. Graduate Texts in Mathematics. 193. 10.1007/978-1-4419-8489-0.
• Book: Henri Cohen . Henri Cohen (number theorist). Number Theory – Volume I: Tools and Diophantine Equations. 2007 . 978-0-387-49922-2. Springer-Verlag. Graduate Texts in Mathematics. 239. 10.1007/978-0-387-49923-9.
• Book: Henri Cohen . Henri Cohen (number theorist). Number Theory – Volume II: Analytic and Modern Tools. 2007 . 978-0-387-49893-5. Springer-Verlag. Graduate Texts in Mathematics. 240. 10.1007/978-0-387-49894-2.
• Book: Richard Crandall . Richard Crandall. Carl Pomerance . Carl Pomerance. Prime Numbers: A Computational Perspective. Springer-Verlag. 2001. 0-387-94777-9. 10.1007/978-1-4684-9316-0.
• Book: Hans Riesel

. Hans Riesel. Hans Riesel. Prime Numbers and Computer Methods for Factorization. second. Birkhäuser. 1994. 0-8176-3743-5. 0821.11001. Progress in Mathematics. 126.

• Book: Victor Shoup. Victor Shoup. A Computational Introduction to Number Theory and Algebra. Cambridge University Press. 2012. 9781139165464. 10.1017/CBO9781139165464.
• Book: Samuel S. Wagstaff, Jr.

. Samuel S. Wagstaff, Jr.. Samuel S. Wagstaff, Jr.. The Joy of Factoring . American Mathematical Society . 2013 . 978-1-4704-1048-3 .