An elliptic sequence is not a sampled linear recurrence sequence

TL;DR

The paper proves that elliptic divisibility sequences derived from rational points on elliptic curves cannot be represented as sampled linear recurrence sequences, highlighting a fundamental difference in their structure.

Contribution

It establishes that elliptic divisibility sequences are not equivalent to sampled linear recurrence sequences, revealing a key distinction in their algebraic properties.

Findings

Elliptic divisibility sequences cannot be expressed as sampled linear recurrence sequences.

The result clarifies the structural differences between elliptic sequences and linear recurrences.

This contributes to understanding the complexity of sequences generated by elliptic curves.

Abstract

Let $E$ be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let $P=(x_1/z_1^2,y_1/z_1^3)$ be a rational point of infinite order on $E$, where $x_1,y_1,z_1$ are coprime integers. We show that the integer sequence $(z_n)$ defined by $nP=(x_n/z_n^2,y_n/z_n^3)$ for all $n\ge 1$ does not eventually coincide with $(u_{n^2})$ for any choice of linear recurrence sequence $(u_n)$ with integer values.