Tuesday, October 13, 2009

A conjecture on primes

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.