> The primality proof took 31 days of non-stop computing on a PC with an Intel I7-4790 CPU.
How does this work? You just try to divide by every possible number and ensure the result isn't an integer?
How does this work? You just try to divide by every possible number and ensure the result isn't an integer?
[0] https://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality...
They don't know all prime numbers up to this number, but they can easily exclude many numbers (for example even).
They use special form primes (the big primes are all n+1 or n-1 forms, with n being easy to factor).
If a Mersenne Number is prime, then the exponent must also be prime (74,207,281 is also prime). That cuts down on a lot of processing time.
Also, all factors of a Mersenne Number, if it's composite, are of the form 2nk + 1, where the Mersenne number is written 2^n - 1. That also helps factor it. I generated a huge list of prime number indices and did some tests on them a couple of years ago. 25% of the Mersenne numbers have 2*n + 1 as a factor. That's even less computation time to find a large prime.
There are simple tests and there are more complex ones, that use probabilities and other workarounds to avoid having to divide by every possible number.