In
mathematics and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, computational number theory, also known as algorithmic number theory, is the study of
computational methods for investigating and solving problems in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
and
arithmetic geometry, including algorithms for
primality testing
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating wheth ...
and
integer factorization, finding solutions to
diophantine equations
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates t ...
, and explicit methods in
arithmetic geometry.
Computational number theory has applications to
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
, including
RSA
RSA may refer to:
Organizations Academia and education
* Rabbinical Seminary of America, a yeshiva in New York City
*Regional Science Association International (formerly the Regional Science Association), a US-based learned society
*Renaissance S ...
,
elliptic curve cryptography and
post-quantum cryptography, and is used to investigate
conjectures and
open problems in number theory, including the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pu ...
, the
Birch and Swinnerton-Dyer conjecture, the
ABC conjecture, the
modularity conjecture
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, the
Sato-Tate conjecture, and explicit aspects of the
Langlands program.
Software packages
*
Magma computer algebra system
*
SageMath
SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, nu ...
*
Number Theory Library
*
PARI/GP
*
Fast Library for Number Theory
Further reading
*
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*
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References
External links
*
{{Number theory-footer
Number theory
Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...