Some effectivity results for primitive divisors of elliptic divisibility sequences

Matteo Verzobio

Vol. 325 (2023), No. 2, 331–351

DOI: 10.2140/pjm.2023.325.331

Abstract

Let $P$ be a nontorsion point on an elliptic curve defined over a number field $K$ and consider the sequence ${\{B_n\}}_{n\in \mathbb{N}}$ of the denominators of $x(nP)$. We prove that every term of the sequence of the $B_n$ has a primitive divisor for $n$ greater than an effectively computable constant that we will explicitly compute. This constant will depend only on the model defining the curve.

Keywords

elliptic curves, primitive divisors, elliptic divisibility sequences

Mathematical Subject Classification

Primary: 11B39, 11G05, 11G50

Milestones

Received: 10 March 2022

Revised: 4 September 2023

Accepted: 4 September 2023

Published: 3 November 2023