Primitive Divisors of Certain Elliptic Divisibility Sequences

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.

version accepted for publication. Difference of heights result moved to http://arxiv.org/abs/1104.4645 and improved. Proof simplified to remove need for special cases when n>20

Cited by in corpus (4)

• Algebraic divisibility sequences over function fields
• Lang's Conjecture and Sharp Height Estimates for the elliptic curves $y^2=x^3+ax$
• Some effectivity results for primitive divisors of elliptic divisibility sequences
• The greatest common valuation of ${\varphi }_n$ and ${\psi }_n^2$ at points on elliptic curves