Primitive divisors of Lucas and Lehmer sequences

TL;DR

This paper investigates primitive divisors in Lucas and Lehmer sequences, employing Thue equations to identify cases up to n=30 and conjecturing their existence for larger n.

Contribution

It applies advanced Thue equation methods to classify Lucas and Lehmer sequences without primitive divisors for n ≤ 30, proposing a conjecture for all larger n.

Findings

Sequences without primitive divisors are fully classified for n ≤ 30.

Computational evidence supports the conjecture that larger n always have primitive divisors.

The approach links primitive divisor problems to solving specific Thue equations.

Abstract

Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $n$-th element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the $n$-th element of such sequences always has a primitive divisor.