This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
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Number Theory
Packing a large number of sophisticated algorithms—many recent and original—into a powerful collection of functions, Mathematica draws on almost every major result in number theory. A key tool for two decades in the advance of the field, Mathematica's symbolic architecture and web of highly efficient algorithms make it a unique platform for number theoretic experiment, discovery, and proof.
FactorInteger find the factors of an integer
PrimeQ test whether an integer is prime

Congruences & Modular Arithmetic
PowerMod modular powers, roots and inverses

Reduce find general solutions to Diophantine equations
FindInstance search for particular solutions to Diophantine equations
Element test field, ring, etc. memberships

Number Representations

DirichletL Dirichlet L-functions
IntegerPartitions restricted and unrestricted partitions of integers
PowersRepresentations representations of integers as sums of powers
ToNumberField operate in a given algebraic number field
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