In mathematics, the Feit-Thompson conjecture is a conjecture in number theory, suggested by Walter Feit & John G. Thompson (1962). The conjecture states that there are no distinct prime numbers p and q for which

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 (and rather implausible) 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.

See also

* Cyclotomic polynomials

Links

* Eric W. Weisstein, Feit-Thompson Conjecture at MathWorld.

References

* Feit, Walter & Thompson, John G. (1962), "A solvability criterion for finite groups and some consequences", Proc. Nat. Acad. Sci. U.S.A. 48: 968–970, <http://links.jstor.org/sici?sici=0027-8424%2819620615%2948%3A6%3C968%3AASCFFG%3E2.0.CO%3B2-Q> MR0143802

* Feit, Walter & Thompson, John G. (1963), "Solvability of groups of odd order", Pacific J. Math. 13: 775–1029, ISSN 0030-8730 MR0166261

* Stephens, Nelson M. (1971), "On the Feit-Thompson conjecture", Math. Comp. 25: 625, ISSN 0025-5718, <http://links.jstor.org/sici?sici=0025-5718%28197107%2925%3A115%3C625%3AOTFC%3E2.0.CO%3B2-H> MR0297686

Number Theory