The Feit-Thompson conjecture: there are no distinct prime numbers p and q for which (p^q-1)/(p-1) divides (q^p-1)/(q-1). Note that a stronger statement stating that (p^q-1)/(p-1) and (q^p-1)/(q-1) are relatively prime whenever p and q are distinct primes does not hold.

Indeed, 112643 = GCD((3313^17 -1)/3312, (17^3313-1)/16) - see Stevens (1971).

Note that (3313^17 - 1)/3312 factors as

78115430278873040084455537747447422887 * 23946003637421 * 112643,

while (17^3313-1)/16 modulo (3313^17 -1)/3312 equals...

149073454345008273252753518779212742886488244343395482423

The 1970 Fields medalist and 2008 Abel prize winner John Griggs Thompson was born on Ottawa, KS on October 13.