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
- divides .
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.
- Feit, Walter; Thompson, John G. (1962), "A solvability criterion for finite groups and some consequences", Proc. Natl. Acad. Sci. U.S.A., 48 (6): 968–970, JSTOR 71265, doi:10.1073/pnas.48.6.968 MR0143802
- Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific J. Math., 13: 775–1029, ISSN 0030-8730, MR 0166261, doi:10.2140/pjm.1963.13.775
- Stephens, Nelson M. (1971), "On the Feit–Thompson conjecture", Math. Comp., 25: 625, JSTOR 2005226, MR 0297686, doi:10.2307/2005226
(This article confuses the Feit–Thompson conjecture with the stronger disproved conjecture mentioned above.)