# Is there an anomaly in the distribution of Mersenne primes?

In a recent press release off the Great Internet Mersenne Prime Search distributed computing project page, it is announced that $$2^{82589933} - 1$$ is the largest known (Mersenne) prime, corresponding to the $$51^{\text{st}}$$ known even perfect number $$2^{82589932}(2^{82589933} - 1).$$

According to GIMPS' press release:

This is GIMPS' $$12^{\text{th}}$$ prime discovery between $$2^{20000000}-1$$ and $$2^{85000000}-1$$, triple the expected number of new primes. One reason to search for new primes is to match actual results with expected results. This anomaly is not necessarily evidence that existing theories on the distribution of Mersenne primes are incorrect. However, if the trend continues it may be worth further investigation.

Here are my questions:

(1) How is the expected number of new Mersenne primes computed? Is there some underlying statistical model that is used?

(2) What existing theories would be incorrect if there was indeed an anomaly in the distribution of Mersenne primes?

The following question appears to be related but this is not a duplicate of that one.

• It is related to the theories derived from conjectures related to Mersenne primes, here is a short article. – rtybase Dec 22 '18 at 9:50
• Thanks for the link, @rtybase! Checking it out now... =) – Jose Arnaldo Bebita-Dris Dec 22 '18 at 10:03
• @rtybase, if you could just further expound on your last comment with an actual (not just a link-only) answer, then I will be more than happy to accept it! =) – Jose Arnaldo Bebita-Dris Dec 22 '18 at 10:25
• It's Christmas coming, I have to entertain my family :). But if you found the link useful, I am glad I was helpful! – rtybase Dec 22 '18 at 10:38
• @JoseArnaldoBebitaDris It is extremely hard to estimate the number of Mersenne primes in a range of huge exponents. Moreover, for events that are very unlikely (for example a huge number to be prime), it is not so unusual that we observe much more events than expected. The formula giving the number of Mersenne numbers will anyway only be a rough guess, we cannot expect that it is "accurate" , in particular the formula cannot prove the conjecture that infinite many Mersenne primes exist. – Peter Dec 22 '18 at 13:10