Primes for which Powers to Themselves minus 1 have Common Factors
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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): $17$