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.

ReferenceReference

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, roots and inverses

Mod ▪ PrimitiveRoot ▪ MultiplicativeOrder ▪ ChineseRemainder ▪

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 ▪ 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 ▪ ...

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