Elliptic Divisibility Sequences, Squares and Cubes

TL;DR

This paper investigates elliptic divisibility sequences, providing explicit formulas for their terms and periods, and characterizes which terms can be perfect squares or cubes using elliptic curve theory and Mazur's theorem.

Contribution

It derives explicit formulas for all EDS terms and periods with zero initial term, and determines conditions under which terms are squares or cubes, using Tate normal form and Mazur's theorem.

Findings

Explicit formulas for EDS terms and periods

Characterization of square and cube terms in EDS

Application of elliptic curve theory to divisibility sequences

Abstract

Elliptic divisibility sequences (EDSs) are generalizations of a class of integer divisibility sequences called Lucas sequences. There has been much interest in cases where the terms of Lucas sequences are squares or cubes. In this work, using the Tate normal form having one parameter of elliptic curves with torsion points, the general terms and periods of all elliptic divisibility sequences with a zero term are given in terms of this parameter by means of Mazur's theorem, and it is shown that which term of h_{n} of an EDS can be a square or a cube by using the general terms of these sequences.