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.