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In a footnote to the list of known Mersenne prime numbers which can be found here, we read that the "ranking" therein is a provisional one since not all possible exponents between $37 \, 156 \, 667$ and $77 \, 232 \, 917$ have been eliminated/tested.

If we may infer from this that the individual(s) who recently found the largest known prime number had not previously discarded (quite understandably!) the possible compositeness of the $\pi(77 \, 232 \, 917)-\pi(74 \, 207 \, 281) - 1 = 166 \, 801$ members of the set $$\{M_{p} \colon p \in (74 \, 207 \, 281, 77 \, 232 \, 917), p \mbox{ is a prime number}\},$$ do we know what it was that prompted them to try their hand at establishing the primality of the very specific Mersenne number $M_{77 \, 232 \, 917}$ ?

Clearly enough, an analogous question can be formulated regarding the discovery of the 46th, 47th, 48th, and 49th known Mersenne prime numbers. If you know the answer to any of those allied questions, do not hesitate to enter it below (it might shed light on the case of the 50th known Mersenne prime).

Thanks in advance for your knowledgeable replies.

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    $\begingroup$ You should have a look at how GIMPS works: mersenne.org/various/works.php In particular, the work (factoring or prime checking) a particular user gets is not entirely within their control, and as in any system of distributed work, it is not always efficient to work through a problem in strict linear order. $\endgroup$ – Jan Grabowski Jan 16 '18 at 8:40
  • $\begingroup$ Note that they have not discovered the Mersenne primes "in order". The huge find $M_{43 \, 112 \, 609}$ was done in August 2008. When $M_{37 \, 156 \, 667}$ was found only fourteen days later (September 2008), it was not a record. Both primes were announced in mid September 2008. Then many months later (either in April 2009 or June 2009, depending on whether you define "discovered" as the time when a machine finished the calculation, or as the time when a person becomes aware of the result), $M_{42 \, 643 \, 801}$ was discovered and was also not a record at the time. $\endgroup$ – Jeppe Stig Nielsen Mar 14 '18 at 23:26
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As Jan Grabowski notes, all the primes that you mention have been discovered by GIMPS. GIMPS draws a distinction between testing and double-checking. When a Lucas–Lehmer test is performed on a Mersenne number and delivers a verdict that the number is prime, the computation is immediately verified. Only after the verification checks out is it announced that a new prime has been discovered. However, if the verdict of the LL test is that the number is composite, the double-checking may not be performed immediately. There is a rather complicated set of assignment rules that determines what computations are to be done next on which computers.

Almost all the Mersenne numbers up to the current largest known Mersenne prime have been tested at least once (the small number of exceptions is due to the nature of distributed computing). However, not all the composites have been double-checked. GIMPS does not declare a range to be prime-free until all the composites in that range have been double-checked.

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    $\begingroup$ Are there any known cases (in the last decade, say) of a double check of a Mersenne number conflicted with the verdict (prime or composite) delivered by the test computation? $\endgroup$ – Aaron Meyerowitz Jan 16 '18 at 17:28
  • $\begingroup$ Do you know if a witness (small factor) of the composite Mersenne numbers (with prime exponent) are computed? That might be as quick or quicker a computation as an LL test (if there is indeed a small factor). Gerhard "Predicts Large Mersenne Prime Gaps" Paseman, 2018.01.16. $\endgroup$ – Gerhard Paseman Jan 16 '18 at 17:31
  • $\begingroup$ @AaronMeyerowitz You could ask your question at mersenneforum, where experts on these matters could quickly answer it. (I believe that at least one of the 17-or-bust primes was found during a double-check.) $\endgroup$ – Pace Nielsen Jan 16 '18 at 17:56
  • $\begingroup$ @Gerhard, any factor of $2^p-1$ must be $1\bmod{2p}$, so can't be really small. $\endgroup$ – Gerry Myerson Jan 16 '18 at 18:05
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    $\begingroup$ @AaronMeyerowitz : Yes, there was a case in 2002 where an alleged Mersenne prime failed verification. More details here: mersenneforum.org/showthread.php?p=6149 $\endgroup$ – Timothy Chow Jan 16 '18 at 18:30
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You can check the current status (updated hourly). Currently among the different stats are:

  • All exponents below 46 251 389 have been tested and verified.
  • All exponents below 82 038 613 have been tested at least once.
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