Feit-Thompson conjecture

Conjecture (Walter Feit & John Thompson). There are no prime numbers $p$ and $q$ (with $p\ne q$) such that ${\displaystyle \frac{p^q-1}{p-1}}$ is divisible by ${\displaystyle \frac{q^p-1}{q-1}}$.

Feit and Thompson, in regards to the Feit-Thompson theorem, have said that proving this conjecture would simplify their proof of their theorem, “rendering unnecessary the detailed use of generators and relations.” In 1971, Stephens strengthened the conjecture to state that

$$\gcd (\frac{p^q-1}{p-1},\frac{q^p-1}{q-1})=1$$

always, and then found the counterexample $p=17$, $q=3313$. The numbers ${\displaystyle \frac{{17}^{3313}-1}{17-1}}$ and ${\displaystyle \frac{{3313}^{17}-1}{3313-1}}$ do have 112643 as their greatest common divisor, but dividing the former by the latter leaves a remainder of 149073454345008273252753518779212742886488244343395482423. No other counterexamples have been found to Stephen’s stronger version of the conjecture.

Title	Feit-Thompson conjecture
Canonical name	FeitThompsonConjecture
Date of creation	2013-03-22 17:55:38
Last modified on	2013-03-22 17:55:38
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	8
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 20A05
Classification	msc 20E32