Generalization of Lucas' theorem for Fermat's quotient. II (Q804616)

The article continues a study begun by \textit{H. Osada} and the author [C. R. Math. Acad. Sci., Soc. R. Can. 11, No.4, 115-120 (1989; Zbl 0687.10009)]. The problem is to find all pairs (p,m) with p an odd prime and \(m>1\) such hat the Fermat quotient \((m^{p-1}-1)/p\) is an \(\ell th\) power of an integer, where \(\ell\) is any prime. The main results in the present paper: (1) If \(2| m\) and \(p>3\), the only solution is (7,2), provided that Catalan's conjecture holds (in fact, a weaker assumption about Catalan's equation would suffice); (2) if \(m\equiv 3\) or 5 (mod 8) and \(\ell >2\), the only solution up to \(m<50\) is (3,5). The reviewer would like to point out that \textit{K. Inkeri}'s article [Acta Arith. 21, 299-311 (1972; Zbl 0228.10017)] contains results related to (2).