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
Factoring
FactorInteger — find the factors of an integer
Factoring-Related Functions »
PrimeNu — number of distinct primes
PrimeOmega — number of primes including multiplicities