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
Root — represent an algebraic number
RootReduce — reduce algebraic numbers to canonical form
GaussianIntegers — allow factorization over Gaussian integers
Rationalize — find rational approximations
LatticeReduce — find short bases in integer lattices
IntegerPartitions — find restricted and unrestricted partitions of integers
PowersRepresentations — find representations of integers as sums of powers
ToNumberField — operate in a given algebraic number field
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