Are primes $p$ such that $2^p-1$ is prime for seemingly random?

Are the prime exponents $p$ such that $2^p-1$ is prime for seemingly random?

So suppose you were choosing two 300-digit primes $p, p_2$ for encryption purposes. If these $p, p_2$ were chosen from, say a list of primes $p$ such that $2^p-1$ is prime for, would these be cryptography secure? (I know finding primes these large and also $2^p-1$ is prime would be basically impossible, but assume this is the case that only you know the list.) Thanks for explanations and help.

• At the risk of sounding snarky, I think if there were a pattern to such primes, someone would have surely noticed it by now... – 2012ssohn Aug 4 '17 at 3:08
• I don't understand the question. If your rsa key is $2^{30}$ bits long then the attacker will doubt you used Mersenne primes, as the Miller Rabin pseudo-prime generation needs something like $\mathcal{O}(2^{120})$ operations at that range. – reuns Aug 4 '17 at 3:36
• The list of such $p$ starts 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667. Seems fairly random to me. – Gerry Myerson Aug 4 '17 at 3:44
• Regardless of whether $2^p-1$ is sufficiently random, you don't want your prime numbers for cryptography to come from any, at least to an outsider, discernible family of choices. – Chickenmancer Aug 6 '17 at 2:57
• To date it is not known whether or not there are infinitely many Mersenne primes. – DanielWainfleet Aug 6 '17 at 4:27

The word "seemingly" makes this question very difficult. Or maybe very easy. Those exponents seem random to me, so yeah, okay, they are seemingly random.

Just to be perfectly clear, though, they are not actually random. Last year, two mathematicians from Stanford found that among the first billion primes, a prime with 1 as its least significant digit is much more likely to be followed with a prime ending in 3, 7 or 9 than by another one ending in 1. I read about this in Scientific American, there's also an ArXiV preprint.

This also holds for the known Mersenne prime exponents. 31 is followed by 61 but 61 is followed by 89, 521 is followed by 607, 2281 is followed by 3217, 9941 is followed by 11213, 21701 is followed by 23209, 216091 is followed by 756839, 2976221 is followed by 3021377, 20996011 is followed by 24036583 and 25964951 is followed by 30402457.

Currently, the three highest known exponents end in 7. And even though there is also 107 followed by 127, there is still that general aversion to repeating the final digit.

In short: they are not actually random but they sure seem to be.