Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

That is, given a positive integer m, is the set

$\{n\mid \gcd(m,2^n-1)=1\}$

where n is a positive integer infinite?

share|cite|improve this question

1 Answer 1

We have $n>0$ and $\gcd(2^n-1,m)>1$ only if $2^n\equiv 1\pmod p$ for some $p|m$ (and only if $p$ is odd). If $k_p>0$ is minimal with $2^{k_p}\equiv 1\pmod p$, then $k_p\ge2$. At least the infinitely many $n$ with $n\equiv 1\pmod {k_p}$ for all odd $p|m$ thus have $\gcd(m,2^n-1)=1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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