Number Theory Packages

Mathematica not only has broad and deep coverage of state-of-the-art number theory, but also immediately integrates number theory into the complete Mathematica computation and interface environment. Mathematica's unequaled web of symbolic, numeric and other algorithms has made possible a new generation of number theoretic functionality, with many original algorithms developed at Wolfram Research. Mathematica's number theoretic functions are fully scalable, and can immediately be used in high-performance computing, and with gridMathematica.

Number Theory Package Features Built into Mathematica:

Efficient implementation of all standard integer and number theory functions »
Deep handling of algebraic, analytic, multiplicative and additive number theory »
Seamless handling of all types of exact and approximate numbers »
Efficient handling of multimillion-digit numbers »
Immediate scalability to high-performance computing
Optimized on all standard and emerging computer platforms »

Key Advantages of Mathematica for Number Theory:

Full integration of number theory with symbolic algebraic computation »
Convenient symbolic representations of algebraic numbers, etc. »
Flexible built-in graphics, visualization and computational geometry »
Ability to do symbolic operations with number theoretic functions
Broad treatment of Diophantine equations and sets »
Powerful built-in algorithms based on interplay between symbolic computation, numerics and discrete mathematics
Immediate access to a full range of special functions »
Built-in number recognition functions »
Full multiparadigm math-oriented programming language »
Document-centered interface for complete record of investigations »
Instant interface creation for experimentation and compelling demonstrations »
Immediate export of publication-quality graphics, typesetting, etc. »
Highly readable programs suitable for publishing algorithm descriptions
Data extraction from tabular formats, web pages, etc. »
Widespread number theory textbooks based on Mathematica
The Wolfram Demonstrations Project with hundreds of interactive number theory examples »

Interoperability with Number Theory Packages:

MathLink API allowing arbitrary data and function connectivity
Ability to call Mathematica from external programs »

Interesting Tidbits:

Many significant number theory discoveries have been made with Mathematica
Wolfram Research has developed many original number theory algorithms
Mathematica won the "Many Digits" Friendly Competition by a large margin »
Mathematica holds many records for number theoretic computation