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

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.

• This might be very non-trivial. – Alex Becker Jun 2 '12 at 21:47
• 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, – hmakholm left over Monica Jun 2 '12 at 22:32
• Did you know that it hasn't even been proved that there are an infinite number of composites of the form $2^p-1$? – Gerry Myerson Jun 2 '12 at 23:35
• 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. – rotskoff Jun 3 '12 at 5:00
• Unfortunately the problem seems to be open. Incredibly, it is also unknown if there are an infinite number of non-Wieferich primes. – 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.