Number Theory
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