Feit-Thompson conjecture

Statement

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

A stronger form of the conjecture is that the numbers $\Phi _{p}(q)$ and $\Phi _{q}(p)$ 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