π-Formulas and Gray code
01 Pubblicazione su rivista
Vellucci Pierluigi, Bersani Alberto Maria
ISSN: 1827-3491
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
related to the smallest positive zero of Ln; the second is an exact formula for π achieved
thanks to some identities valid for Ln.