Chebyshev polynomials and higher order Lucas Lehmer algorithm

TL;DR

This paper generalizes the Lucas Lehmer primality test for Mersenne and Wagstaff numbers using Chebyshev polynomials, extending the iteration to all bases and negative bases, and introduces a Chebyshev polynomial primality test.

Contribution

It introduces a new primality test based on Chebyshev polynomials applicable to generalized Mersenne and Wagstaff numbers, extending previous methods to all bases and negative bases.

Findings

Extended Lucas Lehmer iteration to all bases and negative bases.

Developed a Chebyshev polynomial primality test based on Lucas pairs.

Connected Chebyshev polynomial properties with primality testing algorithms.

Abstract

We extend the necessity part of Lucas Lehmer iteration for testing Mersenne prime to all base and uniformly for both generalized Mersenne and Wagstaff numbers(the later correspond to negative base). The role of the quadratic iteration $x \rightarrow x^2-2$ is extended by Chebyshev polynomial $T_n(x)$ with an implied iteration algorithm because of the compositional identity $T_n(T_m(x))=T_{nm}(x)$. This results from a Chebyshev polynomial primality test based essentially on the Lucas pair $(\omega_a,\overline{\omega}_a)$, $\omega_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. It seems interesting that the arithmetic are all coded in the Chebyshev polynomials $T_n(x)$.