The abc conjecture and nonWieferich primes

For elMath.org: Miroslav Kures 
Institute of Mathematics, Brno University of Technology 
Contact: 
Creation date: 20071220 

The Wieferich prime is a prime number for which
is satisfied and the nonWieferich prime is a prime number for which
2^{p−1} ≠ 1 ( mod p^{2}). 

The known Wieferich primes are 1093 and 3511. It is not known, if there exist finitely or infinitely many Wieferich primes; Even it is not known, if there exist finitely or infinitely many nonWieferich primes. However, we have a result allied to socalled abcconjecture, which can be formulated for positive integers a,b,c as follows.
For each ε > 0, there is a constant K
_{ε} > 1 such that if a and b are coprime and c=a+b, then
c ≤ K_{ε} rad(a,b,c)^{1+ε}, 

where rad(a,b,c)is the product of the distinct prime numbers dividing a,b and c
J. H. Silverman has proved the following assertion.
If the abc conjecture holds, then there exist infinitely many nonWieferich primes.
Author: Miroslav Kures