On the greatest prime factor of some divisibility sequences

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 $\mathbb{Q}$.