The abc conjecture and non-Wieferich primes

For elMath.org: Miroslav Kures
Institute of Mathematics, Brno University of Technology
Contact: kures@fme.vutbr.cz
Creation date: 2007-12-20

The Wieferich prime is a prime number for which

2p−1 ≡ 1    ( mod p2)

is satisfied and the non-Wieferich prime is a prime number for which

2p−1 ≠ 1    ( mod p2).

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 non-Wieferich primes. However, we have a result allied to so-called abc-conjecture, which can be formulated for positive integers a,b,c as follows.

For each ε > 0, there is a constant Kε > 1 such that if a and b are coprime and c=a+b, then

c ≤ Kε 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 non-Wieferich primes.

Author: Miroslav Kures