Primitive prime factors in second order linear recurrence sequences

TL;DR

This paper proves that for a broad class of Lucas sequences, each term generally has a primitive prime factor with an odd exponent, except for a few small indices, which has important implications in number theory.

Contribution

The paper establishes a new result about the existence of primitive prime factors with odd exponents in Lucas sequences, extending previous knowledge.

Findings

Most terms in the sequence have a primitive prime factor with an odd power.

Exceptions occur only at n=1, 2, 3, or 6.

Results have implications for number theory and sequence analysis.

Abstract

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.