Uniform explicit Stewart's theorem on prime factors of linear recurrences

TL;DR

This paper provides a fully explicit and uniform version of Stewart's theorem, demonstrating that the largest prime divisor of the nth term in a Lucas sequence grows faster than n, advancing understanding of prime factors in linear recurrences.

Contribution

The paper introduces a fully explicit and uniform formulation of Stewart's theorem on prime divisors of Lucas sequences, improving previous asymptotic results.

Findings

Largest prime divisor of Lucas sequence terms grows faster than n

Provides explicit bounds for prime factors

Advances understanding of prime distribution in linear recurrences

Abstract

Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of Erd\H{o}s and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewart's result.