# Primitive divisors of sequences associated to elliptic curves

## TL;DR

This paper investigates primitive divisors in sequences derived from elliptic curves over number fields, establishing their existence under certain conditions and exploring connections to the Lang-Trotter conjecture.

## Contribution

It proves the existence of primitive divisors for large terms in sequences associated with elliptic curves when Q is a torsion point of prime order and links this to the Lang-Trotter conjecture.

## Key findings

- Primitive divisors exist for large n when Q has prime order torsion.
- Established a connection between primitive divisors and the Lang-Trotter conjecture.
- Provided new insights into the denominators of x-coordinates in elliptic curve sequences.

## Abstract

Let $\{nP+Q\}_{n\geq0}$ be a sequence of points on an elliptic curve defined over a number field $K$. In this paper, we study the denominators of the $x$-coordinates of this sequence. We prove that, if $Q$ is a torsion point of prime order, then for $n$ large enough there always exists a primitive divisor. Later on, we show the link between the study of the primitive divisors and the Lang-Trotter conjecture.