On a criterion for Catalan's conjecture (Q1597195)

This paper gives a new proof of a criterion of Pedra Mihăilescu (not yet published): if \(p\) and \(q\) are odd prime numbers such that the Diophantine equation \(x^p-y^q=1\) has a non-trivial solution then \(p^{q-1}\equiv 1 \pmod {q^2}\) and \(q^{p-1}\equiv 1 \pmod {p^2}\). Mihăilescu's proof uses Stickelberger's theorem and is purely algebraic. The present proof uses a result of Eichler (1965) and needs some computer verification, but it is also quite short.