A number theory question related to the Feit-Thompson theorem

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 p and q, are the integers N_p(q):=\sum_{k=0}^{q-1}p^k=\frac{p^q-1}{p-1} and N_q(p):=\sum_{i=0}^{p-1}q^i 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 u,v\in\mathbb{Z} satisfying uN_p(q)+vN_q(p)=1 have some recognizable pattern (and without loss of generality assumed p smaller than q).

For p=2 we obtain for q=3,5,7,11,13 respectively the couples (u,v)=(-1,2), (1,-5), (-1,16), (-5,853), (1,-585),

The numbers N_2(q) 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.

: Apparently N_p(q) 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.


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