The Catalan Equation in Finitely Generated Domains

TL;DR

This paper establishes explicit upper bounds for solutions to the Catalan equation within finitely generated integral domains of characteristic zero, extending previous results and providing clearer bounds in special cases.

Contribution

It provides explicit bounds for exponents in the Catalan equation over finitely generated domains, improving upon and refining earlier results by Brindza.

Findings

Explicit bounds for p and q in the Catalan equation are derived.

The main theorem generalizes previous results to broader algebraic domains.

A simplified proof with explicit bounds is given for the case of S-integers in number fields.

Abstract

We consider the Catalan equation $x^p - y^q = 1$ in unknowns $x, y, p, q$, where $x, y$ are taken from an integral domain $A$ of characteristic $0$ that is finitely generated as a $\mathbb{Z}$-algebra and $p, q > 1$ are integers. We give explicit upper bounds for $p$ and $q$ in terms of the defining parameters of $A$. Our main theorem is a more precise version of a result of Brindza. Brindza also gave inexplicit bounds for $p$ and $q$ in the special case that $A$ is the ring of $S$-integers for some number field $K$. As part of the proof of our main theorem, we will give a less technical proof for this special case with explicit upper bounds for $p$ and $q$.