Vincent Granville

2018-02-06 04:57:14 UTC

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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.Raw Message

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.

This article is structured as follows:

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

Randomness and The Bad Seeds Paradox

Application to Cryptography, Financial Markets, and HPC

Exercises

Digits of Pi in Base Pi

The full article can be found here.

Read the full article at

https://www.datasciencecentral.com/profiles/blogs/are-the-digits-of-pi-truly-random