In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Walter Feit and John G. Thompson (1962). The conjecture states that there are no distinct prime numbers p and q such that
If the conjecture were true, it would greatly simplify the final chapter of the proof (Feit & Thompson 1963) 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 Stephens (1971) with the counterexample p = 17 and q = 3313 with common factor 2pq + 1 = 112643.
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.