I’ve written lately about two general ways to prove that a number is prime: Pratt certificates for moderately-large primes and elliptic curve certificates for very large primes.
If you can say more about the prime you wish to certify, there may be special forms of certificates that are more efficient. In particular, there are efficient tests to determine whether Fermat number or a Mersenne number is prime.
Pepin’s test, implemented in four lines of Python here, determines whether or not a Fermat number is prime. It can instantly prove that the known Fermat primes are indeed prime. It can quickly show that the next several Fermat numbers are composite, but the time required to run Pepin’s test increases rapidly for larger numbers.
The Lucas-Lehmer test, implemented in six lines of Python here, tests whether a Mersenne number is prime. The largest known primes are Mersenne primes because the Lucas-Lehmer test is relatively efficient, though it still takes a lot of effort to search for new Mersenne primes.
Certifying that a number is composite is potentially much easier than certifying that a number is prime. A famous example of a proof that a number is composite was Euler’s announcement in 1732 that
232 + 1 = 641 × 6700417,
thereby proving that the fifth Fermat number is not prime, disproving Fermat’s conjecture.
Several Fermat numbers have been fully factored, concretely proving that they are composite, but some Fermat numbers have been proven composite via Pepin’s test that have not been factored. The analogous statement is true for Mersenne primes and the Lucas-Lehmer test.