Abstract—The mathematical constant π has recently been computed up to 22’459’157’718’361
decimal and 18’651’926’753’033 hexadecimal digits. As a simple check for the normality of π, the
frequencies of all sequences with length one, two and three in the base 10 and base 16 representations
are extracted. All evaluated frequencies are found to be consistent with the hypothesis of π being a
normal number in these bases.
I. INTRODUCTION
For a number to be normal in base b, every sequence of k consecutive digits has to appear with a
limiting frequency of b-k in the numbers’ base-b representation. It is supposed that π is normal in
any base, but a proof is still lacking [1]. As a consequence, new record computations of π are often
used to perform empirical consistency checks for the normality of π [2]. Recently a new world
record computation has been performed with the y-cruncher code [3]. This computation
encompasses 22’459’157’718’361 decimal and 18’651’926’753’033 hexadecimal digits [4],
adding about 70% more digits to the former record.
II. RESULTS
Figures 1 to 6 show the distributions of the frequencies of all sequences up to length 3 in the
decimal and hexadecimal representations of π. The red and blue regions show the expected one
and two standard deviation bands around the limiting frequency b-k assuming the occurrences of
the sequences to follow a binomial distribution. All distributions show no significant irregularities.
In particular there is no observed frequency deviating more than four standard deviations from the
expected frequency. The expected and observed variances of the frequency distributions are listed
in Table 1. All observed variances match the expected values, the maximum deviation amounting
to 1.33 standard deviations.
P. Trueb
DECTRIS Ltd., Baden-Dättwil, Switzerland,
Digit Statistics of the First πe Trillion
Decimal Digits of π
