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. 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

• Algorithmic Number Theory, Volume 1: Efficient Algorithms. MIT Press. 1996. ISBN 0-262-02405-5. https://cs.uwaterloo.ca/~shallit/ant.html. 

• David M. Bressoud (1989). Factorisation and Primality Testing. Springer-Verlag. ISBN 0-387-97040-1. https://archive.org/details/factorizationpri0000bres. 

• Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography. MSRI Publications. 44. Cambridge University Press. 2008. ISBN 978-0-521-20833-8. https://www.cambridge.org/us/academic/subjects/mathematics/number-theory/algorithmic-number-theory-lattices-number-fields-curves-and-cryptography?format=HB&isbn=9780521808545. 

• A Course In Computational Algebraic Number Theory. Graduate Texts in Mathematics. 138. Springer-Verlag. 1993. doi:10.1007/978-3-662-02945-9. ISBN 0-387-55640-0. 

• Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics. 193. Springer-Verlag. 2000. doi:10.1007/978-1-4419-8489-0. ISBN 0-387-98727-4. 

• Number Theory – Volume I: Tools and Diophantine Equations. Graduate Texts in Mathematics. 239. Springer-Verlag. 2007. doi:10.1007/978-0-387-49923-9. ISBN 978-0-387-49922-2. 

• Number Theory – Volume II: Analytic and Modern Tools. Graduate Texts in Mathematics. 240. Springer-Verlag. 2007. doi:10.1007/978-0-387-49894-2. ISBN 978-0-387-49893-5. 

• Prime Numbers: A Computational Perspective. Springer-Verlag. 2001. doi:10.1007/978-1-4684-9316-0. ISBN 0-387-94777-9. 

• Hans Riesel (1994). Prime Numbers and Computer Methods for Factorization. Progress in Mathematics. 126 (second ed.). Birkhäuser. ISBN 0-8176-3743-5. 

• Victor Shoup (2012). A Computational Introduction to Number Theory and Algebra. Cambridge University Press. doi:10.1017/CBO9781139165464. ISBN 9781139165464. 

• Samuel S. Wagstaff, Jr. (2013). The Joy of Factoring. American Mathematical Society. ISBN 978-1-4704-1048-3. https://www.ams.org/bookpages/stml-68.