Some new primality criteria based on Lucas sequences

TL;DR

This paper introduces new primality criteria derived from Lucas sequences, specifically analyzing the behavior of certain sequence ratios modulo sequence terms, offering novel methods for primality testing.

Contribution

The paper presents original primality criteria based on Lucas sequences, expanding the theoretical tools available for primality testing beyond existing literature.

Findings

New primality criteria based on Lucas sequences

Behavior of sequence ratios modulo sequence terms analyzed

Criteria not previously documented in literature

Abstract

In this paper, we provide some novel results concerning the behavior of $\frac{U_{kn}}{U_k}$ modulo ${U_n}$, where $(U_n)_{n\in\mathbb{N}}$ is the Lucas sequence of the first kind. As a consequence, we obtain some primality criteria which do not seem to appear in the literature.