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For years, there was a simple reason why the largest known prime is of the form $2^{p}-1$: We had the Lucas-Lehmer test which was specific to Mersenne numbers, and faster than all other known methods.

However, for several years now the GIMPS project have ditched the Lucas-Lehmer approach in favor of the Fermat primality test, joined with a verifiable proof method described here.

I have not gone into the details yet, but I'm already wondering - if GIMPS stopped using the test superficially tailored for Mersenne primes, does that mean their approach is applicable to all numbers of the same magnitude? Can we find a new "largest prime" which is not Mersenne? Or are there additional benefits of a number being Mersenne which still makes Mersenne numbers of this magnitude the only viable candidates for primality testing?

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    $\begingroup$ Any documentation on GIMPS abandoning Lucas-Lehmer in favor of Fermat? The link makes no mention of GIMPS nor Lucas-Lehmer nor Fermat nor Mersenne. $\endgroup$
    – Gerry Myerson
    Commented Dec 7 at 8:21
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    $\begingroup$ It seems I was mistaken and didn't read their site carefully enough. It seems they still use LL as the final test, and simply switched to Fermat-with-proofs for the initial screening mersenne.org/various/math.php#prp $\endgroup$
    – Gadi A
    Commented Dec 7 at 12:43

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