Terms of Lucas sequences having a large smooth divisor

TL;DR

This paper investigates the smooth divisors of terms in Lucas sequences, demonstrating that for most positive integers n, the smooth part of a^n-1 grows slower than a^{o(n)}.

Contribution

It establishes a new asymptotic bound on the smooth parts of Lucas sequence terms, advancing understanding of their factorization properties.

Findings

The smooth part of a^n-1 is typically small relative to a^{o(n)}.

Most positive integers n yield a small smooth divisor in Lucas sequence terms.

Provides asymptotic bounds for the smooth divisors of sequence terms.

Abstract

We show that the $Kn$--smooth part of $a^n-1$ for an integer $a>1$ is $a^{o(n)}$ for most positive integers $n$.