Catalan's Conjecture over Number Fields

TL;DR

This paper explores the generalization of Catalan's conjecture to number fields, proposing a formulation of Cassles criterion in this context and proving it for specific cases, advancing understanding of solutions over algebraic integers.

Contribution

It introduces a formulation of Cassles criterion over number fields and proves it in particular cases, extending the classical Catalan problem to algebraic number theory.

Findings

Proposed a formulation of Cassles criterion over number fields.

Proved the criterion for specific classes of number fields.

Laid groundwork for characterizing solutions of Catalan's equation in algebraic integers.

Abstract

Catalan conjecture/Mihailescu theorem is a theorem in number theory that was conjectured by Mathematician Eugene Charles Catalan in 1844 and was proved completely by Preda Mihailescu in 2005. Some form of problem dates back atleast to Gersonides who seems to have proved a special case of the conjecture in 1343. The note stating the problem was not given the due imprtance at the begining and appeared among errata to papers which had appeared in the earlier volume of Crelle journal, however the problem got its due considration after work of Cassles and Ko Chao in 1960s. The Catalan problem asks that the equation $x^m-y^n=1$ has no solution for x,y,m,n in +ve integers other than the trivial solution $ 3^2-2^3=1 $. An important and first ingredient for the proof is Cassles criteria which says that whenever we have a solution of $x^p-y^q=1$ with p,q primes then $q|x$ and $p|y$ . Here we look a generalization of the problem, namely we will consider the equation $x^p-y^q=1$ where x,y takes value in ring of integers ${O}_K$ of a number field K and p,q are rational primes. In this article we supply a possible formulation of Cassles criterion and a proof for that in some particular cases of number fields. After this work one can expect to follow Mihailescu and Characterize solutions of Catalan over number fields.