Question

I am aware that there are a number of primality testing algorithms used in practice (Sieve of Eratosthenes, Fermat’s test, Miller-Rabin, AKS, etc). However, they are either slow (e.g. sieve), probabalistic (Fermat and Miller-Rabin), or relatively difficult to implement (AKS).

What is the best deterministic solution to determine whether or not a number is prime?

Note that I am primarily (pun intended) interested in testing against numbers on the order of 32 (and maybe 64) bits. So a robust solution (applicable to larger numbers) is not necessary.

Answer 1

Up to ~2^30 you could brute force with trial-division.

Up to 3.4*10^14, Rabin-Miller with the first 7 primes has been proven to be deterministic.

Above that, you’re on your own. There’s no known sub-cubic deterministic algorithm.

EDIT : I remembered this, but I didn’t find the reference until now:

http://reference.wolfram.com/legacy/v5_2/book/section-A.9.4

PrimeQ first tests for divisibility using small primes, then uses the
Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses
a Lucas test.

As of 1997, this procedure is known to be correct only for n < 10^16,
and it is conceivable that for larger n it could claim a composite
number to be prime.

So if you implement Rabin-Miller and Lucas, you’re good up to 10^16.