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I have been exploring the fascinating world of prime numbers, particularly Mersenne Primes, and have noticed an interesting pattern. It seems to me that $2^n - 1$ is prime as long as $n$ itself is a Mersenne prime.

I already know that $2^n - 1$ is not necessarily prime just because $n$ is a normal prime number, but I have not found a counterexample that to the claim that $2^n - 1$ is a prime whenever $n$ is a Mersenne prime.

Is there a proof that this claim is false?

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  • $\begingroup$ True for the first four Mersenne primes, and false for the fifth one already - see here. $\endgroup$ Commented Nov 6, 2017 at 16:23
  • $\begingroup$ I know some facts don't related directly with your question. Maybe you know those, I cited these because I think that are important. You can see from this The Prime Page, by Professor Chris Caldwell the graph for Mersenne primes in section 6 Where is the next larger Mersenne prime?. On the other hand I like very much also several conjectures due to Professor Farideh Firoozbakht, you can read these in The Prime Puzzles & Problems Connection, by Carlos Rivera. I say for example the Puzzle 434 and Puzzle 517. $\endgroup$ Commented Nov 6, 2017 at 21:33

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$2^{13}-1 = 8191$ is a Mersenne prime, but $2^{8191}-1$ is composite.

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  • $\begingroup$ Excellent, I knew that logically there had to be a counterexample, I just couldn't find one on the internet. If there is an article that goes into depth on this, or where you got this example, please link to it! $\endgroup$ Commented Nov 6, 2017 at 16:23
  • $\begingroup$ @KernelStearns But this is on the internet; the above list says that $2^{8191}-1$ is not a (Mersenne) prime. $\endgroup$ Commented Nov 6, 2017 at 16:25
  • $\begingroup$ @DietrichBurde I see now the small type at the bottom that implies every Mersenne prime between 0 and M37,156,667 is on the list. However, this post was meant to provide a quick answer to other people who are new to primes and have the same question. $\endgroup$ Commented Nov 6, 2017 at 16:28
  • $\begingroup$ That's fine. I only wanted to argue about your statement "I just couldn't find one on the internet." Here the homepage on Mersenne "www.mersenne.org" immediately suggest itself. $\endgroup$ Commented Nov 6, 2017 at 16:33
  • $\begingroup$ In the past I tried to combine some problems as the mentioned in the top, Puzzle 434 and Puzzle 517 by Firoozbakht from The Prime Puzzles & Problems Connection, or Puzzle 774 by Noahiro Nomoto with the equation for integers $n>1$ involving the sum of remainders function (see the article [1] or [2]), but I didn't any remarkable deduction. Maybe you or @DietrichBurde can do experiments in your home about it. [1] T. Cross, A note on almost perfect numbers, Mathematics Magazine, 47 (1974), 230–231. [2] Z. Spivey, The Humble Sum of Remainders Function, Mathematics Magazine, 78, No. 4 (2005). $\endgroup$ Commented Nov 6, 2017 at 22:02

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