On the generalized Wieferich criteria
Using similar methods the author extends the result of \textit{A. Granville} and \textit{M. B. Monagan} [Trans. Am. Math. Soc. 306, 329-359 (1988; Zbl 0645.10018)]\ as follows: If \(p\) is an odd prime and \(x\), \(y\), \(z\) are integers with \(p\nmid xyz\) and \(x^p+ y^p+ z^p =0\), then \(a^{p- 1} \equiv 1\pmod {p^2}\) for all positive integers \(a\leq 113\). This implies that \(p> 8.858 \times 10^{20}\). In the cited article it was proved that \(a^{p-1} \equiv 1\pmod {p^2}\) for all integers \(2\leq a\leq 89\) and then \(p> 714, 591, 416, 091, 389\). Note that the investigation of the results of this kind was started by \textit{A. Wieferich} [J. Reine Angew. Math. 136, 293-302 (1909; JFM 40.0256.03)]\ who proved in this case \(2^{p-1} \equiv 1\pmod {p^2}\).
scientific article; zbMATH DE number 723976