In the book by Gelbaum and Olmsted on counterexamples in mathematics, they quote the following as being an open question as of 1990.
The problem comes from a long series of papers by Feit and Thompson on the solvability of groups of odd order (I couldn’t find where exactly, but for example their lemma 34.2 of chapter 5 certainly looks related). Not that I know anything about it all of course. The question as stated by Gelbaum and Olmsted is:
given two different primes and , are the integers and relatively prime ?
and they comment that solving it in the affirmative would make the Feit-Thompson proof “considerably shorter”.