Prime Numbers

The primes have been a focal point for investigations of numbers for more than two millennia. The Wolfram Language implements state-of-the-art algorithms for handling both primes and the advanced mathematics that has grown up around their study. Use Prime to quickly find the billionth prime, or Zeta to get empirical evidence related to the Riemann hypothesis.

Generating Primes

Prime the n th prime number

NextPrime next, previous, etc. prime

RandomPrime pick a random prime

MersennePrimeExponent exponents for which is prime

Sequence of Primes »

PrimePi the number of primes up to n

Zeta Riemann zeta function

ZetaZero zeros of the zeta function

LogIntegral  ▪  RiemannR  ▪  RiemannSiegelZ  ▪  PrimeZetaP  ▪  ...

Primality Testing

PrimeQ test if a number is prime

PrimePowerQ test if a number is a prime power

CoprimeQ test if numbers are coprime

CompositeQ test if a number is composite

MersennePrimeExponentQ test if a number is an exponent in a Mersenne prime

Theorems & Equations

Primes the domain of primes

Reduce reduce equations over the primes

FindInstance find Diophantine solutions over the primes

FullSimplify simplify assuming numbers are prime


FactorInteger find the factors of an integer

IntegerExponent  ▪  SquareFreeQ

Factoring-Related Functions »

PrimeNu number of distinct primes

PrimeOmega number of primes including multiplicities

LiouvilleLambda  ▪  MangoldtLambda  ▪  ...