How to prove there are an infinite number of squarefree numbers of the form $2^p-1$, where $p$ is prime?

It is conjectured that all numbers of the form $2^p-1$ are squarefree. I've been having trouble proving that there are an infinite number of squarefree numbers of the form $2^p-1$; also I am unable to prove it for numbers of the form $2^n-1$ when the restriction on prime exponents is dropped. I can see that there are an infinite number of primes which divide some number of the form $2^p-1$ (all primes dividing $2^p-1$ are larger than than $p$, so if $p$ is the largest known prime dividing any number of this form, there is an even larger prime dividing $2^p-1$). Also I can prove the related statement that there are an infinite number of squarefree numbers of the form $n^2+1$ by overcounting the squareful values according to squares of primes of the form $4k+1$, and after some fiddling, bounding them below a constant fraction, but I can't figure out how to adapt this idea to the $2^p-1$ case.

Hints as well as full solutions are appreciated.

  • 7
    $\begingroup$ This might be very non-trivial. $\endgroup$ – Alex Becker Jun 2 '12 at 21:47
  • 1
    $\begingroup$ If you start by letting $p$ be a prime, then there is exactly one number of the form $2^p-1$, and this number is either square-free or it isn't, $\endgroup$ – hmakholm left over Monica Jun 2 '12 at 22:32
  • 11
    $\begingroup$ Did you know that it hasn't even been proved that there are an infinite number of composites of the form $2^p-1$? $\endgroup$ – Gerry Myerson Jun 2 '12 at 23:35
  • 2
    $\begingroup$ Well, if you believe that there are infinitely many Mersenne primes, you automatically believe this fact. My guess would be that this is a hard fact. $\endgroup$ – rotskoff Jun 3 '12 at 5:00
  • 3
    $\begingroup$ Unfortunately the problem seems to be open. Incredibly, it is also unknown if there are an infinite number of non-Wieferich primes. $\endgroup$ – Dan Brumleve Jun 4 '12 at 5:09

This is an open problem (mentioned also in a comment) as conjectured by Schinzel.

An interesting consequence, due to Rotkiewicz, is that your open question - if true - would imply there are infinitely many primes $p$ for which $2^{p-1} \not\equiv 1$ (mod $p^2$).

This latter statement was shown by Silverman to be a consequence of the $abc$-conjecture, so it's "probably" true (perhaps someday Mochizuki's work will be verified or refuted...).

Thus, there is little hope of using Rotkiewicz's work to contradict the infinitude conjectured here.

I am re-tagging this as open, but you might find the citation below of interest.

Book: Ribenboim, P., Numbers, M., & Friends, M. (2000). Popular Lectures on Number Theory.

Edit: See also the MO post here.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.