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Number Theory

28 built-in function(s) in this category.

  • ContinuedFraction — ContinuedFraction[x, n] (Stable)
  • CoprimeQ — CoprimeQ[n1, n2, ...] (Stable)
  • Divisible — Divisible[n, m] (Stable)
  • DivisorSigma — DivisorSigma[k, n] gives the divisor function sigma_k(n), the sum of the k-th powers of the divisors of n. DivisorSigma[k, n, GaussianIntegers -> True] sums over Gaussian-integer divisors. (Stable)
  • Divisors — Divisors[n] gives a list of the integers that divide n. Divisors[n, GaussianIntegers -> True] includes Gaussian-integer divisors. (Stable)
  • EulerPhi — EulerPhi[n] gives the Euler totient function phi(n). (Stable)
  • ExtendedGCD — ExtendedGCD[n1, n2, ...] (Stable)
  • FactorInteger — FactorInteger[n] gives a list of the prime factors of the integer n, together with their exponents. (Stable)
  • FromContinuedFraction — FromContinuedFraction[{a1, a2, ..., an}] (Stable)
  • GCD — GCD[n1, n2, ...] (Stable)
  • IntegerPartitions — IntegerPartitions[n] (Stable)
  • JacobiSymbol — JacobiSymbol[n, m] (Stable)
  • LCM — LCM[n1, n2, ...] (Stable)
  • LiouvilleLambda — LiouvilleLambda[n] gives the Liouville function lambda(n) = (-1)^Omega(n), where Omega(n) counts the prime factors of n with multiplicity. Completely multiplicative. A non-real Gaussian-integer argument, or GaussianIntegers -> True, is handled over Z[i]. (Stable)
  • MoebiusMu — MoebiusMu[n] gives the Moebius function mu(n): 0 if n has a squared prime factor, otherwise (-1)^k where k is the number of distinct primes. A non-real Gaussian-integer argument is handled over Z[i]. (Stable)
  • MultiplicativeOrder — MultiplicativeOrder[k, n] (Stable)
  • NextPrime — NextPrime[x] gives the next prime after x. (Stable)
  • PartitionsP — PartitionsP[n] (Stable)
  • PartitionsQ — PartitionsQ[n] (Stable)
  • PowerMod — PowerMod[a, b, m] gives a^b mod m. (Stable)
  • Prime — Prime[n] gives the nth prime number. Listable. Small n is read from a sieve table; large n inverts PrimePi via an asymptotic estimate refined against the exact prime counter. Defined for positive integers up to n ~ 1.4*10^12; non-positive-integer arguments give Prime::intpp. (Stable)
  • PrimeNu — PrimeNu[n] gives the number of distinct prime factors of n, nu(n). PrimeNu[n, GaussianIntegers -> True] (or a non-real Gaussian-integer n) counts distinct Gaussian prime factors over Z[i]. PrimeNu[1] is 0; PrimeNu[0] is left unevaluated. (Stable)
  • PrimeOmega — PrimeOmega[n] gives the number of prime factors of n counted with multiplicity, Omega(n). PrimeOmega[n, GaussianIntegers -> True] (or a non-real Gaussian-integer n) counts Gaussian prime factors over Z[i]. PrimeOmega[1] is 0; PrimeOmega[0] is left unevaluated. (Stable)
  • PrimePi — PrimePi[x] gives the number of primes less than or equal to x, exact for x up to 5*10^13 (larger x is left unevaluated). The option Method -> m selects the algorithm: Automatic (default), "Sieve", "Legendre", "Meissel", "Lehmer", "LMO" (Lagarias-Miller-Odlyzko), "DelegliseRivat", or "LucyHedgehog". (Stable)
  • PrimeQ — PrimeQ[n] (Stable)
  • PrimitiveRoot — PrimitiveRoot[n] (Stable)
  • PrimitiveRootList — PrimitiveRootList[n] (Stable)
  • SquareFreeQ — SquareFreeQ[expr] (Stable)