In this paper, we discuss some relations between zeros of Lucas–Lehmer polynomials and the Gray code. We study nested square roots of 2 applying a “binary code” that associates bits 0 and 1 to “plus” and “minus” signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas–Lehmer polynomials, which take the form of nested square roots of 2.
In previous papers we introduced a class of polynomials which follow the same recursive
formula as the Lucas–Lehmer numbers, studying the distribution of their zeros
and remarking that this distribution follows a sequence related to the binary Gray code.
It allowed us to give an order for all the zeros of every polynomial Ln. In this paper,
the zeros, expressed in terms of nested radicals, are used to obtain two formulas for
π: the first can be seen as a generalization of a well known formula
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