Ordering of nested square roots of 2 according to Gray code

TL;DR

This paper explores the relationship between zeros of Lucas-Lehmer polynomials, nested square roots of 2, and Gray code, establishing an ordering based on binary coding of signs in the nested structure.

Contribution

It introduces a novel method linking Gray code to the ordering of zeros of Lucas-Lehmer polynomials expressed as nested square roots of 2.

Findings

Established a correspondence between Gray code and zeros of Lucas-Lehmer polynomials.

Provided an ordering scheme for the zeros based on binary coding.

Enhanced understanding of the structure of nested square roots of 2.

Abstract

In this paper we discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits $0$ and $1$ to $\oplus$ and $\ominus$ signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which assume the form of nested square roots of 2.