Data Science for Number and Coding Theory

Data Science for Number and
Coding Theory
Eldar Sultanow, Capgemini Germany
Divisibility, Periodic Sequences and Discrete Logarithm

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© 2021 Capgemini. All rights reserved.
Data Science for Coding Theory | Eldar Sultanow | 4/12/2021
Agenda
Why Data Science for Mathematics?
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How do you make it?
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Let's demonstrate this with an example!
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Usually it was vice versa: you rather need math for data science.
https://medium.com/s/story/essential-math-for-data-science-why-and-how-e88271367fbd

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Increasingly, Data Science becomes important for pure mathematics.
https://arxiv.org/abs/1911.02008 https://arxiv.org/abs/1912.01412
“This relatively new area of data science is a hot topic not only for its
successes in applied problems, but for the unexpected marriage of
pure mathematics (topology) to the applied world of data.” [p. 10]
https://doi.org/10.1038/s41586-021-03229-4

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How do you make it?
Collect,
generate
Transform
Analyze,
visualize
Prove*
*Proving is the only thing you still have to do yourself to a large extent.
Images: Flaticon.com

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How do you make it?
pandas Collect, import, select, transpose, pivot (…) data
Python Libraries for collecting, generating data
itertools Count, slice, group, iterate, get combinations, permutations
collections Queue, stack, shift, rotate (…) data

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How do you make it?
numpy
Logical (and, or, xor) and binary (bitwise) operations,
mathematical functions
math Math constants, more mathematical functions
SymPy
Symbolic arithmetic, differential/integral calculus, algebra,
discrete mathematics, quantum physics.
Python Libraries for transforming data

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How do you make it?
Python Libraries for analyzing, visualizing data
matplotlib Plot 2D, 3D charts, subplots, export to PDF, SVG, PNG, …
statsmodels Estimations, statistical tests, statistical data exploration
plotly
Interactive, scientific graphs, export to HTML (available for
JavaScript too)

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We illustrate the left-rotation (circular left shift) of a binary number B
(of length l) by r bits using the example B=10110101=181:
Left-rotate B=10110101 by r=2 bits 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 10110101,2 = 11010110 = 214
Left-rotate B=10110101 by r=3 bits 𝜆𝜆𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 10110101,3 = 10101101 = 173
When left-rotating B, we pick a minimum and maximum number from
the output of the rotations:
Bmin=01011011=91 Bmax=11011010=218

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Generally, it applies:
𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚 � 2𝑙𝑙−𝑟𝑟
𝑚𝑚𝑚𝑚𝑚𝑚 2𝑙𝑙
− 1
𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐵𝐵𝑚𝑚𝑚𝑚𝑚𝑚 � 2𝑟𝑟
𝑚𝑚𝑚𝑚𝑚𝑚 2𝑙𝑙
− 1
Darrell Cox et al. found out that the divisibility for special cases holds,
where U is the Hamming weight of Bmax (and Bmin):
91 = 218 � 28−3
𝑚𝑚𝑚𝑚𝑚𝑚 255
218 = 91 � 23
𝑚𝑚𝑚𝑚𝑚𝑚 255
In our example:
Darrell Cox et al.: https://www.isr-publications.com/…
𝑙𝑙|𝑈𝑈 � 𝑟𝑟 + 1

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Let us define the following function (see Darrell Cox et al.):
𝒛𝒛(𝑩𝑩) = �
𝒊𝒊=𝟏𝟏
𝑼𝑼
𝟑𝟑𝑼𝑼−𝒊𝒊
𝟐𝟐𝒙𝒙𝒊𝒊
where again U is the Hamming weight of the binary number B and
0 ≤ 𝑥𝑥1 < 𝑥𝑥2 < ⋯ < 𝑥𝑥𝑈𝑈 ≤ 𝑙𝑙 − 1
are the positions (indexing is zero-based) in B occupied by 1.

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In our case: 𝒛𝒛 𝑩𝑩𝒎𝒎𝒎𝒎𝒎𝒎 = 𝒛𝒛 11011010 = 𝟑𝟑𝟒𝟒
𝟐𝟐𝟎𝟎
+ 𝟑𝟑𝟑𝟑
𝟐𝟐𝟏𝟏
+ 𝟑𝟑𝟐𝟐
𝟐𝟐𝟑𝟑
+ 𝟑𝟑𝟏𝟏
𝟐𝟐𝟒𝟒
+ 𝟑𝟑𝟎𝟎
𝟐𝟐𝟔𝟔
= 𝟑𝟑𝟑𝟑𝟑𝟑
The five positions in Bmax occupied by 1 are (x1,x2,x3,x4,x5)=(0,1,3,4,6).
Similarly, we obtain z(Bmin): 𝒛𝒛 𝑩𝑩𝒎𝒎𝒎𝒎𝒎𝒎 = 𝒛𝒛 01011011 = 𝟖𝟖𝟖𝟖𝟖𝟖
Both numbers, 319 and 842 belong to the 3n+13 cycle given by the function:
𝒇𝒇𝒄𝒄(𝒏𝒏) = �
𝟑𝟑𝟑𝟑 + 𝒄𝒄
𝟐𝟐
𝒊𝒊𝒊𝒊 𝟐𝟐 ∤ 𝒏𝒏
�
𝒏𝒏
𝟐𝟐 𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐𝒐
The parameter c results from
the difference c=2l-3U.
The cycle in our example is: 𝟑𝟑𝟑𝟑𝟑𝟑, 𝟒𝟒𝟒𝟒𝟒𝟒, 𝟕𝟕𝟕𝟕𝟕𝟕, 𝟑𝟑𝟑𝟑𝟑𝟑, 𝟓𝟓𝟓𝟓𝟓𝟓, 𝟖𝟖𝟖𝟖𝟖𝟖, 𝟒𝟒𝟒𝟒𝟒𝟒, 𝟔𝟔𝟔𝟔𝟔𝟔

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The divisibility seems to be true for all these
In our case the bit-length 8 divides 5⋅3+1, where the Hamming
weight is U=5 and the left-rotational distance is r=3, since we
obtain Bmax=11011010 by three left rotates of Bmin=01011011.
𝑙𝑙|𝑈𝑈 � 𝑟𝑟 + 1
cycle cases (Darrell Cox et al.).

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conversions permutations, rotations
We prepare with some basic functions.

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Bmin, Bmax left-rotational distance r

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c, z(B), z-1(B) do the plotting

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Now we generate cycles by rotating binary numbers.
Our introductory example is highlighted.

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dataframe filtered by r∈{3,7} and
sorted by r and (Bmax-Bmin)
dataframe filtered by r∈{9,11} and
sorted by r and (Bmax-Bmin)
Each bar goes from Bmin to Bmax:

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We calculate the difference quotient using Dataframe.diff().

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dataframe sorted by r dataframe sorted by (Bmax- Bmin)
The difference quotient is almost 1: ∆Bmax=1⋅∆Bmin.
We map ∆Bmin to ∆Bmax:

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Discrete mathematics and graph theory provide more examples (see
Kleinnijenhuis et al. and repository of Christian Koch):
Christian Koch: https://github.com/c4ristian/collatz

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Tasks and visualizations in the field of combinatorics can be performed
using data science too.
Alexander Rahn et al.: https://arxiv.org/abs/2101.09719

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References
Binary Rotation
Darrell Cox, Sourangshu Ghosh, and Eldar Sultanow (2021). Generalizing Halbeisen's and
Hungerbühler's optimal bounds for the length of Collatz cycles to 3n+c cycles. Journal of
Mathematics and Computer Science, 24(4), 330–337, http://dx.doi.org/10.22436/jmcs.024.04.05
Cycles
Anant Gupta (2020). On cycles of generalized collatz sequences. arXiv:2008.11103 [math.NT]
Discrete Mathematics and Graph Theory
Jan Kleinnijenhuis, Alissa M. Kleinnijenhuis, Mustafa G. Aydogan (2021). The Collatz tree as a
Hilbert hotel. arXiv:2008.13643v3 [math.GM]
Christian Koch (2020). Collatz Python Library. https://github.com/c4ristian/collatz
Alexander Rahn, Eldar Sultanow, and Idriss J. Aberkane (2021). Collatz convergence is a Hydra
game. arXiv:2101.09719v1 [math.GM]
Binary Sequences and Functions
Jean-Paul Allouche & Jeffrey Shallit, Automatic Sequences (2003). Theory, Applications,
Generalizations. Cambridge University Press
Image: https://pixabay.com/de/vectors/buch-lernen-bildung-805405/

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Abendveranstaltung mit Capgemini
Save the Date und sei bei unserem virtuellen
Abendprogramm am 11.02.2021 von 18:00 Uhr bis 20:00
Uhr dabei.
Es erwartet dich unter anderem eine Keynote „Coding at
Capgemini – Revolution or Evolution for Clients” von unserem
Head of Innovation Thilo Hermann.
Erhalte spannende Einblicke und Beispiele, wie Capgemini
seine Kunden revolutionär aber auch evolutionär mit Software
unterstützt.
Schnapp dir außerdem einen Drink für ein ausgelassenes
Get-Together: lerne dabei unsere Bereiche kennen und
vernetze dich mit Capgemini-Mitarbeitern.
Du hast Interesse?
Dann melde dich bis zum 11.02.2021 unter https://capgemini-
events.de/abendveranstaltung_mit_capgemini/ an.
Wir freuen uns auf dich!

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Let‘s get in contact and visit us now at
our virtual booth/258
here on the OOP!
Architecting Our Future

Contact
Eldar Sultanow
Capgemini
Bahnhofstraße 30
90402 Nürnberg
eldar.sultanow@capgemini.com
Tel.: +49 1514 0251786

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