Packing a large number of sophisticated algorithms—many recent and original—into a powerful collection of functions, the Wolfram Language draws on almost every major result in number theory. A key tool for two decades in the advance of the field, the Wolfram Language's symbolic architecture and web of highly efficient algorithms make it a unique platform for number theoretic experiment, discovery, and proof.

Factoring & Primes »

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 and roots

ModularInverse — modular inverse

Mod  ▪  PrimitiveRoot  ▪  MultiplicativeOrder  ▪  ChineseRemainder  ▪  PrimitivePolynomialQ

Diophantine & Other Equations »

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

ContinuedFraction  ▪  FromContinuedFraction  ▪  Rationalize  ▪  ...

IntegerDigits  ▪  RealDigits  ▪  FromDigits  ▪  DigitCount  ▪  ...

Multiplicative Number Theory »

Divisors  ▪  DivisorSigma  ▪  DivisorSum  ▪  PerfectNumber  ▪  MangoldtLambda  ▪  ...

Analytic Number Theory »

DirichletL — Dirichlet L-functions

Zeta  ▪  DirichletCharacter  ▪  LogIntegral  ▪  ZetaZero  ▪  ...

PrimePi  ▪  PrimeOmega  ▪  PrimeNu  ▪  MangoldtLambda  ▪  LiouvilleLambda  ▪  ...

Additive Number Theory »

IntegerPartitions — restricted and unrestricted partitions of integers

PartitionsP  ▪  PartitionsQ  ▪  FrobeniusNumber  ▪  SquaresR  ▪  ...

PowersRepresentations — representations of integers as sums of powers

Algebraic Number Theory »

AlgebraicNumber  ▪  Root  ▪  GaussianIntegers  ▪  MinimalPolynomial  ▪  ...

ToNumberField — operate in a given algebraic number field

NumberFieldDiscriminant  ▪  NumberFieldIntegralBasis  ▪  ...