Feit-Thompson conjecture

Statement

For any two distinct primes $p,q$, the numbers $\Phi _{p}(q)=(q^{p}-1)/(q-1)$ and $\Phi _{q}(p)=(p^{q}-1)/(p-1)$ are relatively prime.

This conjecture is false, and a counterexample is provided by $p=17,q=3313$.

Related facts

• Feit-Thompson theorem: The Feit-Thompson theorem states that any groupprops:odd-order group of odd order is groupprops:solvable group. The proof of this theorem would be considerably simplified if the Feit-Thompson conjecture were true.