based on an earlier version of the Wolfram Language.
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
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
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