Random number generators and rare events in the continued fraction of π

Abstract

Failure of pseudo-random number generators in producing reliable random numbers as described by Knuth (Knuth, D.E., 1981, The Art of Computer Programming, Vol. 2, Addison-Wesley) gave birth to a new generation of random number generators such as billions decimals of π. To show that these decimals satisfy all criterion of being random, Bailey and Crandall (Bailey, D.B. and Crandall, R.E., 2003, Random generators and normal numbers, to appear in Experimental Mathematics) provided a proof toward the normality of π. In this article, we try to show similar results by considering the continued fraction of π, which appears random as opposed to other supposed normal numbers whose continued fractions are not random at all. For this purpose, we analyze the continued fraction of π and discuss the randomness of its partial quotients. Some statistical tests are performed to check whether partial quotients follow the Khinchin distribution. Finally, we discuss rare elements in the continued fraction of π.