On the Randomness of Pi and Other Decimal Expansions

by George Marsaglia.

Abstract: Extensive tests of randomness used to distinguish good from not-so-good random number generators are applied to the digits of pi,e and sqrt(2), as well as to rationals k/p for large primes p. They seem to pass these tests as well as some of the best RNGs, and could well serve in their stead if the digits could be easily and quickly produced in the computer---and they can, at least for rationals k/p. Simple and fast methods are developed to produce, in reverse order, for large primes p and general bases b, the periodic cycles of the base-b expansions of k/p. Specific choices provide high quality, fast and simple RNGs with periods thousands of orders of magnitude greater than what are curently viewed as the longest. Also included are historical references to decimal expansions and their relation to current, often wrong, website discussions on the randomness of pi.

Key Words: Tests of Randomness, Pi, Diehard Tests, Random Number Generators

George Marsaglia, geo@stat.fsu.edu

Editor: Joseph W. McKean, joe@stat.wmich.edu

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