Feit–Thompson conjecture

Template:Short description In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by Template:Harvs. The conjecture states that there are no distinct prime numbers p and q such that

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

If the conjecture were true, it would greatly simplify the final chapter of the proof Template:Harv 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 Template:Harvtxt 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 Template:Harv and q = 3 Template:Harv.

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
• Goormaghtigh conjecture

References

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External links

• Template:MathWorld (This article confuses the Feit–Thompson conjecture with the stronger disproved conjecture mentioned above.)