Are the Digits of Pi Random? State of the Art & Free Book on the Topic

Over the last 10 years, I spent a lot of time analyzing the digits of the classic math constants such as π, e, log 2, √2 and so on. Not testing them for randomness but trying to formally prove that they are undistinguishable from random bit streams. And trying to identify which constants are the easiest targets to obtain seminal results.

Trillions of digits (of π in particular) have been computed. They pass all known tests of randomness. Borwein and Pouffe came up with spectacular BBP formula to compute these digits around 2012. More recently, it was proved that among the first n binary digits of √2 and similar numbers, at least √n must be 1. This is the best known to date, in terms of theoretical results. No one even knows if the proportion of 1 in the binary digits of √2 actually exists, or if it oscillates indefinitely between 0% and 100%.

However, I have been secretly working on this topic for years, without publishing in scientific journals. I am still far from solving the famous conjecture about the randomness of the digits in question. Actually, I still offer a $1 million prize for a proof or disproof, similar to the $1 million offered by the Clay Institute to solve the Riemann conjecture. But I made significant progress and recently obtained deep results that dwarf everything published to this date. In particular:

• If the digits of both √2 or √3 are “random”, then they are uncorrelated.
• If the proportion of 1 in the binary digits of √2 actually exists, then for at least one of the numbers √2, √2 + 1/3, √2 +2/3, that proportion must be between 5/16 and 11/16.

These are just two among many results I proved, some involving dynamical systems with applications in cryptography. I am currently the only one actively researching this topic at a theoretical level with focus on specific constants, besides cranks and amateurs who are nowhere close and wrong most of the time. But I would love to see others involved. I don’t care who make the most spectacular advances in the long run, whether it is me or not. So, I decided to share all my work for free, with everyone. It is the starting point, and a must-read to not repeat what I already did, but instead go beyond, possibly far beyond. I set the path to a potential resolution. I invite you to follow it and continue it from where it stops now.

The result about zero correlation between the digits of √2 and √3 can be found here. The full book (106 pages) is available here for free, no sign-up required. It also features material related to AI, quantum dynamics, computer-assisted proofs, universal synthetic data and high-performance computing (HPC) with the Gmpy2 library. HPC to validate my findings deals with insanely large numbers. I did my best to make it accessible to the largest possible audience. Quants, developers & engineers with solid analytic background, physicists, mathematicians and related professionals will enjoy the read.

To no miss future articles, subscribe to my AI newsletter, here.

About the Author

Vincent Granville is a pioneering GenAI scientist, co-founder at BondingAI.io, the LLM 2.0 platform for hallucination-free, secure, in-house, lightning-fast Enterprise AI at scale with zero weight and no GPU. He is also author (Elsevier, Wiley), publisher, and successful entrepreneur with multi-million-dollar exit. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. He completed a post-doc in computational statistics at University of Cambridge.