Discussion:
Are the Digits of Pi Truly Random? - Must Read for Math and Data Geeks
Vincent Granville
2018-02-06 04:57:14 UTC
Raw Message
This article covers far more than the title suggests. It is written in simple English and accessible to quantitative professionals from a variety of backgrounds. Deep mathematical and data science research (including a result about the randomness of Pi, which is just a particular case) are presented here, without using arcane terminology or complicated equations.

The topic discussed here, under a unified framework, is at the intersection of mathematics, probability theory, chaotic systems, stochastic processes, data and computer science. Many exotic objects are investigated, such as an unusual version of the logistic map, nested square roots, and representation of a number in a fractional or irrational base system.

The article is also useful to anyone interested in learning these topics, whether they have any interest in the randomness or Pi or not, because of the numerous potential applications. I hope the style is refreshing, and I believe that you will find plenty of material rarely if ever discussed in textbooks or in the classroom. The requirements to understand this material are minimal, as I went to great lengths (over a period of years) to make it accessible to a large audience. The content (including truly original exercises and applications) is also valuable to college professors looking for off-the-beaten-path material to teach, as well as to students wishing to get exposure and interest to advanced topics usually reserved for 'elite' data scientists, without being discouraged due to the simplicity of the presentation.

The randomness of the digits of Pi is one of the most fascinating, unsolved mathematical problems of all times, having been investigated by many million of people over several hundred years. The scope of this article encompasses this particular problem as part of a far more general framework. More questions are asked than answered, making this document a stepping stone for future research.

1. General Framework

Questions, Properties and Notations about Chaotic Sequences Investigated Here
Potential Applications, Including Random Number Generation

2. Examples of Chaotic Sequences Representing Numbers

Data Science Step
Mathematical Step
Numbers in Base 2, 10, 3/2 or Pi
Nested Square Roots
Logistic Map

3. About the Randomness of the Digits of Pi

The Digits of Pi are Random in the Logistic Map System
Paths to Proving Randomness in the Decimal System
Connection with Brownian Motions

4. Curious Facts

Application to Cryptography, Financial Markets, and HPC
Exercises
Digits of Pi in Base Pi

https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random
Helmut Wabnig
2018-02-06 09:07:54 UTC
Raw Message
On Mon, 5 Feb 2018 20:57:14 -0800 (PST), Vincent Granville
Post by Vincent Granville
This article covers far more than the title suggests. It is written in simple English and accessible to quantitative professionals from a variety of backgrounds. Deep mathematical and data science research (including a result about the randomness of Pi, which is just a particular case) are presented here, without using arcane terminology or complicated equations.
The topic discussed here, under a unified framework, is at the intersection of mathematics, probability theory, chaotic systems, stochastic processes, data and computer science. Many exotic objects are investigated, such as an unusual version of the logistic map, nested square roots, and representation of a number in a fractional or irrational base system.
The article is also useful to anyone interested in learning these topics, whether they have any interest in the randomness or Pi or not, because of the numerous potential applications. I hope the style is refreshing, and I believe that you will find plenty of material rarely if ever discussed in textbooks or in the classroom. The requirements to understand this material are minimal, as I went to great lengths (over a period of years) to make it accessible to a large audience. The content (including truly original exercises and applications) is also valuable to college professors looking for off-the-beaten-path material to teach, as well as to students wishing to get exposure and interest to advanced topics usually reserved for 'elite' data scientists, without being discouraged due to the simplicity of the presentation.
The randomness of the digits of Pi is one of the most fascinating, unsolved mathematical problems of all times, having been investigated by many million of people over several hundred years. The scope of this article encompasses this particular problem as part of a far more general framework. More questions are asked than answered, making this document a stepping stone for future research.
1. General Framework
Questions, Properties and Notations about Chaotic Sequences Investigated Here
Potential Applications, Including Random Number Generation
2. Examples of Chaotic Sequences Representing Numbers
Data Science Step
Mathematical Step
Numbers in Base 2, 10, 3/2 or Pi
Nested Square Roots
Logistic Map
3. About the Randomness of the Digits of Pi
The Digits of Pi are Random in the Logistic Map System
Paths to Proving Randomness in the Decimal System
Connection with Brownian Motions
4. Curious Facts
Application to Cryptography, Financial Markets, and HPC
Exercises
Digits of Pi in Base Pi
https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random
Fine as long as you pronounce it correctly.

Pi is not pie, and if you cannot hold your pee you are incontinent.

w.