Catalan's equation with a quadratic exponent (Q2741210)

The author considers the general case of Catalan's equation, namely \(x^M-y^N=\pm 1\), with \(xy\not=0\) and odd exponents \(1<M<N\). It was proved by the reviewer that \(M\) must be prime and that either \(N\) is prime or \(N=q_1 q_2\) where \(q_1\leq q_2\) are prime numbers. Here the author considers the case \(N=q^2\). In this case he proves the congruences NEWLINE\[NEWLINE x\equiv 0, \quad p^{q-1}\equiv 1 \pmod {q^3}. NEWLINE\]NEWLINE His proof is a generalization of the recent proof of \textit{P. Mihǎilescu} [J. Number Theory 99, 225-231 (2003; Zbl 1049.11036)] for the equation \(x^p-y^q=1\) where \(p\) and \(q\) are prime. More recently, Jon Grantham computed lower bounds on the exponents \(M\) and \(N\) and if these results are compared to upper bounds on \(M\) and \(N\) obtained by the reviewer, it follows that both exponents \(M\) and \(N\) must be prime.