The book is available on our E-store, here.
It all started with the number 1. This e-book offers a trip deep into the most elusive and fascinating multi-century old conjecture in number theory: are the binary digits of the fundamental math constants evenly distributed? No one even knows if the proportions of ‘0’ and ‘1’ exist, for any of them: it could oscillate indefinitely between 0% and 100%.
After a quick read, you might be convinced that I solved it. Yet very smooth functions visibly converging to the expected 50% ratio, time after time, are all but smoke and mirrors for the experienced practitioner. They are known to have unexpected, large singularities in extremely rare cases. And if you look at the images in the book with a microscope, there is still a bit of lingering randomness even after using advanced chaos removal techniques. Throughout this book, I guide the reader, warning about these caveats and dead ends while pointing to promising directions to work on the conjecture.
My novel approach sets a clear path towards solving the problem, for the first time ever. Building on solid foundations spanning across multiple disciplines, I share a number of spectacular victories along the way. For instance, this new and unexpected result: all numbers that can be written as
where (an) is an increasing sequence of positive integers with sub-linear growth rate, have a proportion of `1′ equal to zero in their infinite binary digit expansion.
I hope this document becomes a reference on the topic and serves as a launchpad towards a final resolution. It is written in simple English, jargon free, even when covering advanced topics. With enterprise-grade Python code along with efficient algorithms not taught in any classroom or textbook. To benefit from the material, you need beginner experience with Python, and the equivalent of a first-year college course on calculus. The book is targeted to professionals in AI, machine learning, engineering, physics, scientific computing, operations research, computer science, and Fintech. Students with an analytical mindset will discover original, useful material to enhance their creativity, learning experience, while gaining strong exposure to professional code and applications.
This book also opens up new research areas in theoretical and computational number theory, numerical approximation, dynamical systems, quantum dynamics, and the physics of numbers. With a strong emphasis on applications: automated pattern detection and theorem proving with AI, agent-based modeling, building a universal unbiased pattern-rich synthetic dataset, cryptography (fast, strong random number generators based on irrational numbers), dynamical systems with chaos detection and isolation, computer intensive simulations, and high-performance computing to handle numbers such as 2n+1 at power 2n with n=106.
Each chapter is self-contained and can be read separately from the others. Chapters are listed in chronological order. The last one features the most recent research and discoveries. The first one is more technical, focusing on the foundations; it can be skipped initially if you are mainly interested in the applications.
Purchase the book, here. 56 pages. Published by MLTechniques.com in May 2025. See table of contents, 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.