Wieferich's criterion and the abc-conjecture
The following result is proved. The so called ``abc-conjecture'' of Masser and Oesterlé implies that the number of primes less than \(X\) for which \(\alpha^{p-1}\not\equiv 1 \pmod{p^2}\) where \(\alpha\) is a fixed rational number and \(\alpha \neq \pm 1,0\), is at least \(O(\log X)\). An analogous result is also proved for points of infinite order on elliptic curves having certain \(j\)-invariants. The proofs base on several skillful lemmas.
scientific article; zbMATH DE number 4067006