big primes
This site uses the Baillie-PSW primality test. It has been proven to always work on numbers with up to 64 bits (18 digits), and no one has ever found a composite number that it thinks is prime. Therefore, if this website ever gives you a non-prime number, let everyone know! It would be a mathematical discovery.

That said, because we want you to feel truly safe and not rely on conjecture, we also apply the Miller-Rabin primality test with 7 rounds for numbers with more than 64 bits (mostly because we like the number 7). The default setting of 50 digits (166 bits) with 7 rounds provably has a chance of failure of one in 5 quadrillion (see this paper by Damgard, Landrock, and Pomerance for how we arrived at that estimate). Numbers with more digits have an even smaller chance of failure.