Fastest algorithm for primality test [closed]

Answer 1

The best algorithm best of my view is "ALI primality test".

Answer 2

One good method is the Miller-Rabin test. It should be noted however, that this is only a probabilistic test.

Answer 3

If you want to test a long long for primality then the Baillie PSW primality test is a good choice. This test does one strong pseudoprime test and one Lucas test and hence is very fast. It is expected that there exist some composites that pass this test, but so far none are known, and there certainly is no exception below 10¹⁵. A variant of this test is for example used in Mathematica.

Answer 4

I believe that the asymptotically fastest current (non-probabilistic) primality test is the "Lenstra/Pomerance improved AKS", which has complexity that is essentially O(n^6).

However, the range of long long (assuming a typical system where that is a 64 bit integer) is not really that big. In particular, there are only ~200 million primes less than 2^32, so using a fast probabilistic test, followed by trial division with a precomputed list of primes (or just looking the number up in a list of primes, if you have one) would be pretty darn fast in that range, and is probably the right way to go about it.