Download Digit Statistics of the First π Trillion Decimal Digits of π

Digit Statistics of the First πe Trillion
Decimal Digits of π
P. Trueb
DECTRIS Ltd., Baden-Dättwil, Switzerland,
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.
0.8
0.6
Number of Sequences
Number of Sequences
Entries
10
Mean
1.000e−01
Variance 4.967e−15
1
0.4
Entries
100
Mean
1.000e−02
Variance 4.943e−16
4
3.5
3
2.5
2
1.5
1
0.2
0.5
0.0999998
0.1
Number of Sequences
Entries
1000
Mean
1.000e−03
Variance 4.184e−17
25
9.99995
10
10.00005
Frequency
×10
Figure 2 Frequencies of all sequences of length
2 in the decimal representation of π.
Figure 1 Frequencies of the digits 0 to 9 in
the decimal representation of π.
30
−3
0
0.1000002
Frequency
20
15
Number of Sequences
0
Entries
16
Mean
6.250e−02
Variance 3.179e−15
2
1.8
1.6
1.4
1.2
1
0.8
10
0.6
0.4
5
0.99998
1
1.00002
Frequency
Figure 3 Frequencies of all sequences of
length 3 in the decimal representation of π.
Number of Sequences
0.2
7
6
5
4
62.4999
62.5
62.5001
62.5002
Frequency
×10
70
60
50
40
30
2
20
1
Entries
4096
Mean
2.441e−04
Variance 1.279e−17
80
3
0
−3
62.4998
Figure 4 Frequencies of the digits 0 to F in the
hexadecimal representation of π.
Entries
256
Mean
3.906e−03
Variance 2.130e−16
8
0
Number of Sequences
0
−3
×10
10
−3
3.9062
3.90625
3.9063
Frequency
×10
Figure 5 Frequencies of all sequences of
length 2 in the hexadecimal representation of π.
0
−3
0.24413
0.24414
0.24415
Frequency
×10
Figure 6 Frequencies of all sequences of length
3 in the hexadecimal representation of π.
Table 1 Expected and observed variances of the frequencies of all sequences up to length 3 in
the decimal and hexadecimal representations of π.
Base
Sequence
Length
Expected Variance of
Frequencies
10
10
10
16
16
16
1
2
3
1
2
3
(4.0±1.9)×10-15
(4.41±0.63) ×10-16
(4.45±0.20) ×10-17
(3.1±1.1) ×10-15
(2.09±0.18) ×10-16
(1.309±0.029) ×10-17
Observed
Variance of
Frequencies
5.0 ×10-15
4.94×10-16
4.18×10-17
3.2×10-15
2.13×10-16
1.279×10-17
Deviation
[σ]
-0.51
-0.85
1.33
-0.03
-0.24
1.02
III. CONCLUSIONS
The frequencies of all sequences up to length 3 in the first 22’459’157’718’361 decimal and
18’651’926’753’033 hexadecimal digits of π are found to be consistent with the hypothesis of π
being a normal number in base 10 and base 16.
REFERENCES
[1] Sevcik, C. (2016), ‘Fractal analysis of Pi normality’, arXiv:1608.00430.
[2] Bailey, D. H. et al (2012), ‘An Empirical Approach to the Normality of π’, Experimental Mathematics 21(4).
[3] Yee, A. J. and Kondo, S. (2013), ‘12.1 trillion digits of Pi. And we’re out of disk space...’,
http://www.numberworld.org/misc_runs/pi-12t.
[4] Trueb, P. (2016) ‘πe trillion digits of π’, http://www.pi2e.ch,
https://www.dectris.com/successstories.html#success_pi.