Which is the fastest algorithm to find prime numbers?

Question

Which is the fastest algorithm to find out prime numbers using C++? I have used sieve\'s algorithm but I still want it to be faster!

Answer 1

Rabin-Miller is a standard probabilistic primality test. (you run it K times and the input number is either definitely composite, or it is probably prime with probability of error 4^-K. (a few hundred iterations and it's almost certainly telling you the truth)

There is a non-probabilistic (deterministic) variant of Rabin Miller.

The Great Internet Mersenne Prime Search (GIMPS) which has found the world's record for largest proven prime (2^74,207,281 - 1 as of June 2017), uses several algorithms, but these are primes in special forms. However the GIMPS page above does include some general deterministic primality tests. They appear to indicate that which algorithm is "fastest" depends upon the size of the number to be tested. If your number fits in 64 bits then you probably shouldn't use a method intended to work on primes of several million digits.

Answer 2

A very fast implementation of the Sieve of Atkin is Dan Bernstein's primegen. This sieve is more efficient than the Sieve of Eratosthenes. His page has some benchmark information.