The Decimal Expansion of π Is Not Statistically Random

Abstract

This paper pursues the question whether the decimal expansion of π behaves as though it were the realization of a sequence of mutually independent, identically distributed random variables with uniform distribution on {0, 1, … , 9}. So far, no deviation from reasonable statistical norms has been found. We introduce an extended runs test of symbolic dynamics that π fails at a high level of statistical significance.

2000 AMS Subject Classification::

• 11Y60
• 11K45
• 60G57
• 65-05

Keywords:

• Pi
• normal numbers
• random number generator
• symbolic dynamics
• runs test

Notes

¹Implementation from A. Fog, http://www.agner.org/random/ (download packages 2008–2010, version 2.01); GNU general public license.