How can I check if a large number is prime?

Question

I was looking at a list of prime numbers in binary format and I noticed a pattern, where if 2^n - 1, a Mersenne Prime, is in the list, then (2^(n+1) - 2^(n) - 1) is also in the list.I looked at the list of Mersenne Primes on Wikipedia and checked the primality of (2^43112610 - 2^43112609 - 1) on Wolfram Alpha, but to test for any results larger than that, the results are inconclusive.The largest discovered prime number is a Mersenne Prime (2^82589933 &minus; 1), so I'm wondering how I can go about testing (2^82589934 - 2^82589933 &minus; 1).

Someone · Accepted Answer

> I was looking at a list of prime numbers in binary format and I noticed a pattern, where if 2^n - 1, a Mersenne Prime, is in the list, then (2^(n+1) - 2^(n) - 1) is also in the list.

That’s because both are equal. We have

  2^(n+1) = 2 × 2^(n) = 2^(n) + 2^(n)

so

  2^(n+1) - 2^(n) - 1
  = 2^(n) + 2^(n) - 2^(n) - 1
  = 2^(n) - 1

InterNautic · Answer

Ummm, did you look into https://en.wikipedia.org/wiki/Primality_test ?

trinovantes · Answer

Haven't kept up with cryptography research but I think msieve is still the current state of the art integer factorization software that's open source