Primes for which Powers to Themselves minus 1 have Common Factors

Theorem

Let $p$ and $q$ be prime numbers such that $p^p - 1$ and $q^q - 1$ have a common divisor $d$.

The only known $p$ and $q$ such that $d < 400 \, 000$ are $p = 17, q = 3313$.

Proof

This is of course rubbish, because $p^p - 1$ and $q^q - 1$ have the obvious common factor $2$.

Investigation ongoing, as it appears the author of the work this was plundered from must have been thinking about the Feit-Thompson Conjecture and got it badly wrong.

Sources

• July 1971: N.M. Stephens: On the Feit-Thompson Conjecture (Math. Comp. Vol. 25: p. 625)  www.jstor.org/stable/2005226
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $17$