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Conjecture

Feit-Thompson Conjecture

Number Theory

The Feit-Thompson conjecture, suggested by Walter Feit and John Thompson in 1962, states that there are no two distinct primes p and q for which the quotient of p to the power q minus 1 over p minus 1 divides the quotient of q to the power p minus 1 over q minus 1. If true, it would simplify the final step of Feit and Thompson's own proof that every finite group of odd order is solvable. A stronger version, that the two quotients are always coprime, was disproved in 1971.

Facts
Statement
The Feit-Thompson conjecture proposes that for any two distinct prime numbers p and q, the quantity (p to the q minus 1) divided by (p minus 1) never divides the quantity (q to the p minus 1) divided by (q minus 1). 1
Proposed Year
1962 1
Progress Toward Resolution
The conjecture is confirmed for q equals 2 (Stephens 1971) and q equals 3 (Le 2012). A stronger version, that the two numbers are always coprime, was disproved by Stephens in 1971 with the counterexample p equals 17 and q equals 3313. Informal probability arguments suggest the expected number of counterexamples to the original conjecture is close to zero. 1
Chronology
Resolved Year
1971 2
Prize Status
Prize Status (category)
No Prize Offered 1
w-freetextdim2b-0926: category derived from a free-text property; original status/verification detail carried on the source property.
Classification
Resolution Status
Open 1
Open Questions
Prize Status
No prize is recorded.
No source consulted this pass names a monetary prize for this conjecture.
Connections

In Branch

Number Theory, Branches of Mathematics
Source Feit-Thompson Conjecture (Wikipedia)
Sources
1. Feit-Thompson Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    It is known that the conjecture is true for q = 2 (Stephens 1971) and q = 3 (Le 2012).
  • In Branch: Number Theory, Lead sentence
View the Source
2. Feit-Thompson conjecture (Wikipedia)
Wikipedia Feit-Thompson conjecture lead paragraph (w-bbfill-psymath4-0926)
Quote, Wikipedia Feit-Thompson conjecture lead paragraph (w-bbfill-psymath4-0926)
disproved by Stephens (1971
View the Source
Frequently Asked Questions

Has any part of the Feit-Thompson conjecture been settled?

Confirmed for q equals 2 and 3; the stronger coprime version is false.

The conjecture itself is still open, but it is known to be true for q equals 2 (Stephens 1971) and q equals 3 (Le 2012). A stronger version, that the two quotients are always coprime, was disproved in 1971.
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