Primitive Divisors of Certain Elliptic Divisibility Sequences

TL;DR

This paper proves that under certain conditions, the $n$-th element of elliptic divisibility sequences generated by a non-torsion point on specific elliptic curves always has a primitive divisor, extending understanding of divisibility properties.

Contribution

It establishes new conditions ensuring the existence of primitive divisors in elliptic divisibility sequences for a broad class of points and curves.

Findings

Primitive divisors exist for even $n>2$ under given conditions.

Primitive divisors exist for odd $n>1$ when $x(P)<0$ or $x(P)$ is a perfect square.

Results extend previous knowledge on divisibility sequences of elliptic curves.

Abstract

Let $P$ be a non-torsion point on the elliptic curve $E_{a}: y^{2}=x^{3}+ax$. We show that if $a$ is fourth-power-free and either $n>2$ is even or $n>1$ is odd with $x(P)<0$ or $x(P)$ a perfect square, then the $n$-th element of the elliptic divisibility sequence generated by $P$ always has a primitive divisor.