Perfect powers generated by the twisted Fermat cubic

TL;DR

This paper investigates the occurrence of perfect powers within elliptic divisibility sequences derived from the twisted Fermat cubic, establishing finiteness results and specific conditions for the presence of perfect powers.

Contribution

It proves finiteness of perfect powers in these sequences and characterizes cases with potential perfect powers based on divisibility and point multiplication conditions.

Findings

Finitely many perfect powers exist in the sequence with first term > 1.

No perfect powers occur if the first term is divisible by 6 and the point is triple another point, except possibly certain powers.

Conditions involving divisibility and point multiplication determine the presence of perfect powers.

Abstract

On the twisted Fermat cubic, an elliptic divisibility sequence arises as the sequence of denominators of the multiples of a single rational point. It is shown that there are finitely many perfect powers in such a sequence whose first term is greater than 1. Moreover, if the first term is divisible by 6 and the generating point is triple another rational point then there are no perfect powers in the sequence except possibly an lth power for some l dividing the order of 2 in the first term.