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

Ordering of nested square roots of 2 according to Gray code