On the greatest prime factor of some divisibility sequences

TL;DR

This paper investigates the behavior of the greatest prime factor in divisibility sequences over function fields and elliptic curves, extending known results from integers under the ABC conjecture.

Contribution

It extends the study of prime factors in divisibility sequences to function fields and elliptic curves, providing new analogues and insights.

Findings

Results for divisibility sequences over function fields

Analogues for sequences related to elliptic curves

Connections to classical integer results under conjectures

Abstract

Let $P(m)$ denote the greatest prime factor of $m$. For integer $a>1$, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, $$P(a^n-1)\gg_{\epsilon, a} n^{2-\epsilon}$$ for any $\epsilon>0$. We study analogues results for the corresponding divisibility sequence over the function field $\mathbb{F}_q(t)$ and for some divisibility sequences associated to elliptic curves over the rational field $\mathbb{Q}$.