On consecutive perfect powers with elementary methods

Catalan’s conjecture states that the equation xp−yq=1 admits the unique solution 32−23=1 in integers x,y,p,q≥2. The conjecture has been proved by Mihăilescu in 2002 using the theory of cyclotomic fields and Galois modules. Here, relying only on elementary methods, we prove several instances of this result. In particular, we show it in the following cases: p even, q is even, x divides q, y divides x−1, y is a power of a prime, and y≤pp/2.