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2018-03-01, 14:37   #12
Dr Sardonicus

Feb 2017
Nowhere

105338 Posts

Quote:
 Originally Posted by GP2 It's not the exponent that would be the next Wieferich prime, but the non-square-free factor itself.
Oops!

Right. So, q2 divides 2q-1 - 1, and the multiplicative order of 2 (mod q2) 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 10932) is 1092/3 = 346, and the multiplicative order of 2 (mod 35112) is 3510/2 = 1755. Neither 346 nor 1755 is prime.

However, to the base 3, we have the small example

35 - 1 = 242 = 2 * 112, so that

35 == 1 (mod 112).