Pierluigi Vellucci, Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Universita di Roma

Nested square roots of 2 and Gray code
In this talk we describe a "mathematical experiment" that highlights some of the connections between nested square roots of 2 and the Gray code, a binary code employed in Informatics. The nested square roots considered here are obtained from the zeros of a particular class of orthogonal polynomials, which can be related to the Chebyshev polynomials. The expression of the zeros is

+-sqrt(2+-sqrt(2+-sqrt(+-2 ... sqrt(2))))

These polynomials are created by means of the iterative formula, L_n(x) = L^2_{n-1}(x)- 2, employed in Lucas-Lehmer primality test. The experiment is structured in the following steps: 1) we wonder what is the effect of + and - signs in the above expression, giving some preliminary considerations, conjectures and results; 2) we apply a "binary code" that associates bits 0 and 1 to + and - signs in the nested form above, and we numerically evaluate all the zeros of associated polynomial for n=2, n=3 (and so on); 3) an internet search on OEIS database for the first terms of this sequence reveals an ordering of nested square roots previously described, according to the Gray code; the decimal equivalent of Gray code for n on OEIS, is A003188; 4) we give the proof of the iterative formula.