On consecutive perfect powers with elementary methods

TL;DR

This paper proves several cases of Catalan's conjecture using elementary methods, including cases with even exponents, divisibility conditions, and bounds on variables, providing alternative proofs to known results.

Contribution

It offers elementary proofs for multiple instances of Catalan's conjecture, expanding understanding without relying on advanced algebraic number theory.

Findings

Proved Catalan's conjecture for even exponents.

Established the conjecture when y divides x-1.

Confirmed the conjecture for y as a prime power and y ≤ p^{p/2}.

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 $y\le p^{p/2}$.