11
$\begingroup$

Let $m,n$ be natural numbers such that $2^n - 1 \mid 3^m - 1$. By results from Bugeaud-Corvaja-Zannier, say Theorem 3 of this paper , we know that for any constant $C > 0$ we must have $m > Cn$ for sufficiently large $n$. Is it known whether we can do better than that? Heuristically I would expect $m$ to be very large compared to $n$.

I would be happy to see any improvement over $Cn$, or even an average statement that $m$ has to be large compared to $n$ most of the time.

$\endgroup$
2
  • 2
    $\begingroup$ No, it is unknown. $\endgroup$
    – markvs
    Jul 29, 2021 at 20:52
  • $\begingroup$ Can you give a (rigorous) example of such an "average statement"? $\endgroup$ Aug 1, 2021 at 3:11

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.