Feit–Thompson conjecture

In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by. The conjecture states that there are no distinct prime numbers p and q such that
 * $$\frac{p^{q} - 1}{p - 1}$$ divides $$\frac{q^{p} - 1}{q - 1}$$.

If the conjecture were true, it would greatly simplify the final chapter of the proof of the Feit–Thompson theorem that every finite group of odd order is solvable. A stronger conjecture that the two numbers are always coprime was disproved by  with the counterexample p = 17 and q = 3313 with common factor 2pq +&thinsp;1 = 112643.

It is known that the conjecture is true for q = 2 and q = 3.

Informal probability arguments suggest that the "expected" number of counterexamples to the Feit–Thompson conjecture is very close to 0, suggesting that the Feit–Thompson conjecture is likely to be true.