Abstract

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [3], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results such as the following are achieved: Whereas the fundamental constant log 2 = P n2Z + 1=(n2 n ) is not yet known to be 2-normal (i.e. normal to base 2), we are able to establish b-normality (and transcendency) for constants of the form P 1=(nb n ) but with the index n constrained to run over certain subsets of Z + . In this way we demonstrate, for example, that the constant 2;3 = P n=3;3 2 ;3 3 ;::: 1=(n2 n ) is 2-normal. The constants share with ; log 2 and others the property that isolated digits can be directly calculated, but for the new class such computation is extraordinarily rapid. For example, we find that the googol-th (i.e. 10 100 - th) binary bit of 2;3 is 0. We also present a collection of other results -- such as density results and irrationality proofs based on PRNG ideas -- for various special numbers.

Citations