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”.
This looks a very natural question to answer even without the group theory background. Since nothing so obvious shows up here I tried to do some reverse engineering, namely to find if the integers satisfying have some recognizable pattern (and without loss of generality assumed p smaller than q).
For we obtain for respectively the couples , , , , ,
The numbers form the sequence 3, 7, 31, 127, 2047, 8191,… of Mersenne numbers, but it’s just a name, and the OEIS unfortunately didn’t help saying anything about the sequence 2, -5, 16, 853, -585…
So, no progress at all on this Feit-Thompson question.
Update: Apparently is called a quantum integer (or q-bracket) and pops up in many areas, see this paper by Nathanson and this book by Kac and Cheung.