Well, I need someone here with programming skills (because I have none of it) to check if this problem that I am proposing is at least true for the known Mersenne primes, and here is the list of the exponents of the known Mersenne primes :
And the problem is:
If the number $M_p=2^p-1$ is prime then it can be written in one of the two following forms:
$M_p=18k+1$ or $M_p=18k+13$, for $p\geq5$, that is, Mersenne primes, when divided with $18$ leave a remainder that is equal to $1$ or equal to $13$.
I feel that this is quite easy to program but since my skills in programming are practically non-existent it would be so nice if someone here would do that job for me (and for himself if he is interested in this kind of problems).
If this is a proven fact about Mersenne primes then please tell me where I can find the proof because I did not find fact of this kind when I was reading about known facts about Mersenne primes, and I am sorry if this is something quite elementary beacuse I already posted some questions which turned out to be homework-type problems and I did not see it in the moment of posting. Thank you.