Any solution to the mythical problem in question has remained elusive for centuries. It is deemed more difficult than proving the Riemann Hypothesis, yet its formulation can be understood by kids in elementary school. The question is whether or not the digits of mathematical constants such as π, behave like a random sequence.
This article sets a significant milestone towards a full resolution. It serves as a blueprint, featuring the architecture of a highly constructive yet difficult proof, based on new theoretical developments and concepts designed specifically to tackle the problem, with success.
The goal is to transition from experimental to theoretical number theory. Yet, I use illustrations with numbers that have more than 210000 digits, that is, larger than 2m where m =210000. No amount of computing power will ever be able to handle such numbers. Obviously, I use some tricks here, and such numbers make the patterns and their complexity easier to detect with the naked eye. Most importantly, these patterns can be explained with theoretical arguments based on the new theory: the iterated self-convolutions of infinite strings of characters and their congruence classes, akin to p-adic numbers, in a special topological space.
Finally, there is a strong connection to the deepest aspects of discrete dynamical systems approaching their chaotic regime. Also, there are practical applications to cryptography, and all computations are doubled-checked using external libraries.
Read the full paper shared below to discover the mystery and how it is demystified. It is about the randomness of the binary digits of one of the most celebrated math constants. With a deep technical dive, I explain why the proportion of ones is 50% (illustrated in the picture below, featuring the first 10000 digits arising when hitting the rightmost vertical dashed line just before chaos starts).
Target audience
The material shared here is accessible to engineers, computer scientists, and AI practitioners, written in a style appealing to them rather than for pure mathematicians. The goal is to encourage these practitioners to use AI to push the limits further and get the strongest possible version of this seminal result. In the end, the problem is as much about number theory as it is about computer science.
For the time being, my $1 million award to solve a related problem (see here) is temporarily withdrew until I fully assess whether or not the new theory leads to an easy solution.
The result, and how access to the paper
The paper establishes, beyond any doubt, the very first deep result related to the digit distribution of any well-known math constant such as 𝜋, e, log 2, or the square root of 2. The theory applies to just one of them. Until today, the only established facts were trivialities, such as the absence of periodicity in the digit sequence, or the fact that the first 20 trillion digits of 𝜋 pass all the randomness tests.
I added the paper as a new chapter to my book on chaotic dynamical systems, available here. The theory is a continuation and culmination of the material presented in the previous chapters. It is also available (for free) as paper 51, here. The figure below (explained in the paper) was the starting point to initiate the theoretical investigations.
It is my hope that the seminal material presented in this document starts a renewed interest on this topic with the use of AI to prove even deeper and more diversified results. Currently, the mathematical community has lost the will to ever solve this problem, considering it as unsolvable, and bringing it to a standstill since 2012: Bailey & Borwein were the last to make theoretical contributions – however modest – related to specific math constants. My paper proves otherwise, setting the path for more deep research to come, not just by mathematicians but also by operations research, AI and computer scientists.
About the Author
Vincent Granville is a pioneering GenAI scientist and machine learning expert, co-founder of Data Science Central (acquired by a publicly traded company in 2020), Chief AI Scientist at MLTechniques.com and GenAItechLab.com, former VC-funded executive, author (Elsevier) and patent owner — one related to LLM. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Follow Vincent on LinkedIn.