This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)
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
Prime  ▪ NextPrime  ▪ PrimePi  ▪ EulerPhi  ▪ MoebiusMu  ▪ JacobiSymbol  ▪ ...
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
Integers  ▪ Rationals  ▪ Reals  ▪ Algebraics  ▪ Primes
Number Representations
IntegerDigits  ▪ RealDigits  ▪ FromDigits  ▪ DigitCount  ▪ ...
Divisors  ▪ DivisorSigma  ▪ DivisorSum  ▪ MangoldtLambda  ▪ ...
DirichletL Dirichlet L-functions
Zeta  ▪ DirichletCharacter  ▪ LogIntegral  ▪ ZetaZero  ▪ ...
PrimePi  ▪ PrimeOmega  ▪ PrimeNu  ▪ MangoldtLambda  ▪ LiouvilleLambda  ▪ ...
IntegerPartitions restricted and unrestricted partitions of integers
PartitionsP  ▪ PartitionsQ  ▪ FrobeniusNumber  ▪ SquaresR  ▪ ...
PowersRepresentations representations of integers as sums of powers
ToNumberField operate in a given algebraic number field