Feit–Thompson conjecture

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

${\displaystyle {\frac {p^{q}-1}{p-1}}}$ divides ${\displaystyle {\frac {q^{p}-1}{q-1}}}$.

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.

It is known that the conjecture is true for q = 2 (Stephens 1971) and q = 3 (Le 2012).

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.

• Cyclotomic polynomials
• Goormaghtigh conjecture

• 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, Bibcode:1962PNAS...48..968F, doi:10.1073/pnas.48.6.968, JSTOR 71265, PMC 220889, PMID 16590960 MR0143802
• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order" (PDF), Pacific J. Math., 13: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, MR 0166261
• Le, Mao Hua (2012), "A divisibility problem concerning group theory", Pure Appl. Math. Q., 8 (3): 689–691, doi:10.4310/PAMQ.2012.v8.n3.a5, ISSN 1558-8599, MR 2900154
• Stephens, Nelson M. (1971), "On the Feit–Thompson conjecture", Math. Comp., 25 (115): 625, doi:10.2307/2005226, JSTOR 2005226, MR 0297686

• Weisstein, Eric W. "Feit–Thompson Conjecture". MathWorld. (This article confuses the Feit–Thompson conjecture with the stronger disproved conjecture mentioned above.)