## 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Also included are historical references to decimal expansions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

**Author:**

George Marsaglia, geo@stat.fsu.edu

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

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