Feit-Thompson conjecture

Statement

The conjecture states that if $p,q$ are distinct primes, then ${\mathrm{\Phi }}_p(q)=(q^p-1)/(q-1)$ does not divide ${\mathrm{\Phi }}_q(p)=(p^q-1)/(p-1)$.

The stronger conjecture that ${\mathrm{\Phi }}_p(q)$ and ${\mathrm{\Phi }}_q(p)$ are relatively prime is false. The smallest counterexample is $p=17,q=3313$.

Related facts

• Feit-Thompson theorem: The proof of the theorem by Feit and Thompson that every group of odd order is solvable can be simplified considerably if the Feit-Thompson conjecture is true.

References

Journal references

• A solvability criterion for finite groups and some consequences by Walter Feit and John Griggs Thompson, Proceedings of the National Academy of Sciences, Volume 48, Page 968 - 970(Year 1962): ^{More info}
• Solvability of groups of odd order by Walter Feit and John Griggs Thompson, Pacific Journal of Mathematics, Volume 13, Page 775 - 1029(Year 1963): This 255-page long paper gives a proof that odd-order implies solvable: any odd-order group (i.e., any finite group whose order is odd) is a solvable group.^{Project Euclid pageMore info}
• On the Feit-Thompson conjecture by N. M. Stephens, Mathematics of Computation, Volume 25,Number 115, Page 625 - 625(July 1971): ^{JSTOR linkWeblinkMore info}

External links

Subject wiki links

• Number:Feit-Thompson conjecture