On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

TL;DR

This paper proves that only finitely many perfect powers can appear in certain elliptic divisibility sequences generated by specific elliptic curves, using advanced number theory techniques involving modularity over quadratic fields.

Contribution

It establishes finiteness of perfect powers in elliptic divisibility sequences for a new class of elliptic curves using modularity over real quadratic fields.

Findings

Finiteness of perfect powers in the specified sequences.

Application of modularity over quadratic fields to Diophantine problems.

Extension of known results to non-integral points on certain elliptic curves.

Abstract

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the from $y^2=x(x^2+b)$, where $b$ is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.