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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?

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

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

Jose Arnaldo Bebita Dris
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    It is related to the theories derived from conjectures related to Mersenne primes, here is a short article. – rtybase Dec 22 '18 at 09:50
  • Thanks for the link, @rtybase! Checking it out now... =) – Jose Arnaldo Bebita Dris Dec 22 '18 at 10:03
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    @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
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    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
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    @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
  • off-topic comment : A bit depressing how fast the records for the largest prime are broken. – Peter Dec 22 '18 at 13:12
  • @Peter I certainly agree with your first comment. Perhaps I was trying to find an answer to the question of or to articulate the anomaly alluded to in the press release. That being said, although I feel a bit sad that I do not get to discover the next Mersenne prime(s), I also feel encouraged to continue pursuing this research area, as this is closely related to my own works on odd perfect numbers. (I invite you to check out this arXiv preprint as well as this MO question, in case you have not yet done so already.) – Jose Arnaldo Bebita Dris Dec 22 '18 at 14:21
  • @JoseArnaldoBebitaDris For the sake of curiousity : How are ODD perfect numers related to Mersenne primes ? – Peter Dec 22 '18 at 14:25
  • @Peter: One of my (intractable) conjectures is that the Euler/special primes are in one-to-one correspondence with the odd perfect numbers in the same way that the Mersenne primes are in one-to-one correspondence with the even perfect numbers. Then another one of my conjectures is that the Descartes-Frenicle-Sorli conjecture is in fact false, so that the Euler/special prime $q$ of an odd perfect number has exponent $k > 1$, while Mersenne primes $2^p - 1$ trivially have exponent one. Then the abundancy indices of $q^k$ and $2^p - 1$ satisfy certain inequalities. – Jose Arnaldo Bebita Dris Dec 23 '18 at 02:19
  • @Peter: I meant to say that "the abundancy indices of (the Euler factor) $q^k$ and (of the Mersenne prime) $2^p - 1$ satisfy certain numerical inequalities" in my last comment. – Jose Arnaldo Bebita Dris Dec 23 '18 at 03:22

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