Primitive divisors of Lucas and Lehmer sequences, II

TL;DR

This paper introduces an algorithm to identify terms in Lucas and Lehmer sequences lacking primitive divisors and proves that beyond a certain index, all such terms have primitive divisors for sequences with specific algebraic properties.

Contribution

It presents a new algorithm for detecting primitive divisors in Lucas and Lehmer sequences and extends existing results to sequences with bounded height ratios.

Findings

All sequence terms with index > 30 have primitive divisors when height ratio ≤ 4

Improves Stewart's results on more general sequences

Provides an effective computational method for primitive divisor analysis

Abstract

Let $\al$ and $\be$ be conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\al$ and $\be$ with $\hgt(\be/\al) \leq 4$, the $n$-th element of these sequences has a primitive divisor for $n > 30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.