Are Subsequences of Decimal Digits of PI Random?

Abstract

A lot has been done on the randomness of the decimal expansion of Pi with extensive tests of randomness that are used to distinguish good from not-so-good random number generators when applied to the decimal digits of Pi. Pi seems to pass these tests as well as some of the best random number generator (RNG) and could well serve as an RNG provided that the digits of Pi could be easily and quickly produced in the computer [Mar06]. We make an interesting study in the same context in which random substring of arbitrary length are extracted from arbitrary positions a large number of times and each sample is tested for randomness. Our results confirm the randomness of Pi and a recent claim that “Pi is less random than we thought” [TF05] stands refuted. George Marsaglia [Mar06] has also independently refuted the claim but in Marsaglia’s work, the randomness is established on the whole for the first 960 million digits of pi. Our study confirms the randomness for arbitrary subsequences also. Finally, the investigation of some functions of pi-rather than pi itself-is proposed.