On consecutive perfect powers with elementary methods

Abstract

Catalan's conjecture claims that the Diophantine equation $x^p-y^q=1$ admits the unique solution $3^2-2^3=1$ in integers $x,y,p,q \ge 2$. The conjecture has been finally proved by P. Mih\u{a}ilescu (2002) using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary techniques, we prove several instances of this classical result. In particular, we prove the conjecture in the following cases: $p$ even (due to V.A. Lebesgue), $q$ is even (due to L. Euler and Chao Ko), $x$ divides $q$, $y$ divides $x-1$, $y$ is a power of a prime, and $y\le p^{p/2}$.