Elliptic divisibility sequences, squares and cubes

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. It is shown that which term h(n) of an EDS with zero terms can be a square or a cube by using the general terms of these sequences.