The abc conjecture and non-Wieferich primes
|For elMath.org: Miroslav Kures
|Institute of Mathematics, Brno University of Technology
|Creation date: 2007-12-20
The Wieferich prime is a prime number for which
is satisfied and the non-Wieferich prime is a prime number for which
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
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