The abc conjecture and non-Wieferich primes in arithmetic progressions
If a prime \(p\) satisfies the congruence \(2^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime and if it satisfies the congruence \(a^{p-1} \equiv \pmod{p^2}\), then it is called a Wieferich prime for base \(a\). As was noted by the authors it can be shown by elementary tools that under the \textbf{abc conjecture} there are infinitely many non-Wieferich primes. In this paper the authors show that under the abc~conjecture the number of non-Wieferich primes is \(\ll \frac{\log x}{\log\log x}\).
scientific article; zbMATH DE number 6154092