Way too late to the party, but hope this helps. This is relevant if you are looking for big primes:
To test large odd numbers you need to use the Fermat-test and/or Miller-Rabin test.
These tests use modular exponentiation which is quite expensive, for n
bits exponentiation you need at least n
big int multiplication and n
big int divison. Which means the complexity of modular exponentiation is O(n³).
So before using the big guns, you need to do quite a few trial divisions. But don't do it naively, there is a way to do them fast. First multiply as many primes together as many fits into a the words you use for the big integers. If you use 32 bit words, multiply 3*5*7*11*13*17*19*23*29=3234846615 and compute the greatest common divisor with the number you test using the Euclidean algorithm. After the first step the number is reduced below the word size and continue the algorithm without performing complete big integer divisions. If the GCD != 1, that means one of the primes you multiplied together divides the number, so you have a proof it's not prime. Then continue with 31*37*41*43*47 = 95041567, and so on.
Once you tested several hundred (or thousand) primes this way, you can do 40 rounds of Miller-Rabin test to confirm the number is prime, after 40 rounds you can be certain the number is prime there is only 2^-80 chance it's not (it's more likely your hardware malfunctions...).