Graves, Hester; Murty, M. Ram The abc conjecture and non-Wieferich primes in arithmetic progressions. (English) Zbl 1272.11014 J. Number Theory 133, No. 6, 1809-1813 (2013). 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 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 \(\ll \frac{\log x}{\log\log x}\). Reviewer: Mohamed El Bachraoui (Al-Ain) Cited in 1 ReviewCited in 14 Documents MSC: 11A41 Primes 11N13 Primes in congruence classes Keywords:Wieferich primes; arithmetic progressions; abc conjecture Citations:Zbl 1269.11007 PDFBibTeX XMLCite \textit{H. Graves} and \textit{M. R. Murty}, J. Number Theory 133, No. 6, 1809--1813 (2013; Zbl 1272.11014) Full Text: DOI Online Encyclopedia of Integer Sequences: Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.