Quote:
Originally Posted by GP2
It's not the exponent that would be the next Wieferich prime, but the nonsquarefree factor itself.

Oops!
Right. So, q
^{2} divides 2
^{q1}  1, and the multiplicative order of 2 (mod q
^{2}) is a
prime divisor p of q  1. This is not true of the two known Wieferich primes to the base 2:
The multiplicative order of 2 (mod 1093
^{2}) is 1092/3 = 346, and the multiplicative order of 2 (mod 3511
^{2}) is 3510/2 = 1755. Neither 346 nor 1755 is prime.
However, to the base 3, we have the small example
3
^{5}  1 = 242 = 2 * 11
^{2}, so that
3
^{5} == 1 (mod 11
^{2}).