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   ▪ ...