Perfect powers in elliptic divisibility sequences

TL;DR

This paper proves that elliptic divisibility sequences associated with certain elliptic curves contain only finitely many perfect powers, using advanced number theory techniques.

Contribution

It establishes finiteness results for perfect powers in elliptic divisibility sequences under specific initial conditions, extending previous work.

Findings

Finitely many perfect powers in sequences with initial term divisible by 2 or 3

Finiteness also holds for Mordell curves with initial term greater than 1

Examples show sequences with no perfect power terms

Abstract

It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with corresponding elliptic divisibility sequences having no perfect power terms. The proofs combine primitive divisor results with modular methods for Diophantine equations.