Feit-Thompson conjecture

Statement

For any two distinct primes $p,q$, the number ${\mathrm{\Phi }}_p(q)=(q^p-1)/(q-1)$ does not divide ${\mathrm{\Phi }}_q(p)=(p^q-1)/(p-1)$ are relatively prime.

A stronger form of the conjecture is that the numbers ${\mathrm{\Phi }}_p(q)$ and ${\mathrm{\Phi }}_q(p)$ are relatively prime, and a counterexample to this stronger form is provided by $p=17,q=3313$. This is the only counterexample for $p,q<400000$.

Related facts and conjectures

• 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.
• Goormaghtigh conjecture: This conjecture states that the equation:

$\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}$

has only two solution pairs.

External links

Other subject wikis

• Groupprops:Feit-Thompson conjecture