Primitive divisors of Lucas and Lehmer sequences, III

TL;DR

This paper proves that for all Lucas or Lehmer sequence elements beyond the index 30,030, each has a primitive divisor, extending the understanding of divisibility properties in these sequences.

Contribution

It establishes a new explicit bound (30,030) beyond which all Lucas or Lehmer sequence elements have primitive divisors, improving previous results.

Findings

All Lucas or Lehmer sequence elements with index > 30,030 have primitive divisors.

The result confirms a long-standing conjecture for large indices.

Provides a concrete bound for primitive divisor existence in these sequences.

Abstract

In this paper we prove that if $n > 30,030$ then the $n$-th element of any Lucas or Lehmer sequence has a primitive divisor.