Let \( P(n) \) be the largest prime factor of \( n \) and \( s_y(n) \) be the largest divisor \( d \) of \( n \) with \( P(d)\le y \). Thus, \( s_y(n) \) is the \( y \)-smooth part of \( n \). Given a linearly recurrent sequence \( u=(u_n)_{n\ge 0} \) of positive integers, the authors of the paper under review are interested in finding whether there exist \( c>1 \) and \( K \) such that the set\begin{align*}\mathcal{A}_{K,c,u}=\{n: s_{Kn}(u_n)>c^n\},\end{align*}contains many elements. For instance, if \( u_n=\binom{2n}{n} \) is the sequence of middle binomial coefficients, then \( \mathcal{A}_{2,2,u} \) contains all the positive integers. In this paper the authors we address the problem in the simplest case namely \( u_n=a^n-1 \) for some positive integer \( a \). Their main results are the following.
Theorem 1. Assume the \( ABC \)-conjecture. Then for any \( K>0 \), \( c>1 \), the set \( \mathcal{A}_{K,c,u} \) is finite.
This is consistent with:
Theorem 2. We have\begin{align*}\#\left(\mathcal{A}_{K,c,u} \cap [1,N]\right)\ll N\exp\left(-\dfrac{\log N}{156\log \log N}\right).\end{align*}
Their results are easily extendable to all Lucas sequences, in particular, the sequence of Fibonacci numbers. To prove Theorem 1 and Theorem 2, the authors use a clever combination of results in analytic number theory, the \( ABC \)-conjecture, the usual properties of the linear recurrent sequence \( u_n \), as well as bounds from \( p \)-adic valuations.
