Spectacular New Discovery about the Digits of π

Everyone believes that the digits of constants such as π or √2 cannot be distinguished from a sequence of random bits. The first few trillion successfully pass all tests of randomness. However, proving that they indeed behave perfectly randomly is arguably one of the oldest and most difficult unsolved math conjectures. So far, nobody succeeded in proving even the most basic facts for any of these constants, for instance:

• In the binary digits of π, is the proportion of 1 above 0%, or do we have fewer and fewer 1 when looking at increasingly long sequences, to the point that the proportion of 0 eventually dwarfs that of 1?
• Does the proportion of 0 or 1 actually exist? Or is it infinitely oscillating between 0% and 100% without ever converging?

In this article, I prove that if the proportion of 1 exists, it must be between 1/4 and 3/4 either for  π or π + 2/3, or both. And if it does not exist, it must hit a ratio between 1/4 and 3/4, infinitely frequently. The same is true for the proportion of 0, and for all other math constants. I then generalize this result to obtain tighter bounds, namely 5/16 and 11/16.

This new theorem with a computer-assisted proof is generic, applicable to almost all constants. I then do a deep dive on specific constants called algebraic numbers (roots of a polynomial with integer coefficients, such as √2). I obtain some fascinating results and then find very strong patterns in the way the digits are generated when using dynamical systems to produce them, in particular for 7 – 4√3. I have yet to confirm the patterns with a mathematical proof. If proved, it would lead to the first established deep result about the digit distribution of these numbers. So far it is based on empirical observations involving insanely large numbers and smart high-performance computing.

Historical context

It is known that almost all numbers have random-like digit sequences and hence called normal numbers. Exceptions are very rare, though there are infinitely many including all rational numbers and many others. Yet, despite this fact, no one has succeeded in proving or disproving that any of the top math constants is a normal number. It may as well be an unprovable statement. My paper features the best proved so far. By contrast, in a recent number theory PhD thesis with results published here, it is shown that the first n binary digits of √2 contain at least √n ones, regardless of n. This was the strongest result proved until now, and far weaker than what I prove in my paper.

Download your copy of the paper

Available as technical paper #61, here, with Python code using the Gmpy2 library and illustrations. It is included in my new eBook about the digit distribution of math constants, available here. Links are clickable in the eBook (PDF). 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.