On a criterion for Catalan's Conjecture

TL;DR

This paper presents a new proof of Mihailescu's theorem, which characterizes the solutions to the Catalan's Conjecture equation involving prime exponents, by establishing specific congruence conditions.

Contribution

The paper introduces a novel proof technique for Mihailescu's theorem, providing new insights into the conditions under which the equation has solutions.

Findings

Proves that the equation has no solutions unless certain prime-based congruences hold.

Establishes specific modular conditions necessary for potential solutions.

Offers a new perspective on Catalan's Conjecture through congruence analysis.

Abstract

We give a new proof of a theorem by P. Mihailescu which states that the equation $x^p-y^q=1$ is unsolvable with $x, y$ integral and $p, q$ odd primes, unless the congruences $p^q \equiv p\pmod{q^2}$ and $q^p\equiv q \pmod{p^2}$ hold.