How Random are the Digits of π? 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 […]
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Pi Day in the age of AI — The Missing $1m Millenium Prize
Pi Day is celebrated every year on 3/14. Enjoy and share the video I created with our “Pi Day AI agent” at BondingAI, generating hundreds of webpages, turning each one in a screenshot (a frame in the video). All done in Python with source code available here. The video is also on YouTube, here. For […]
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Checking for Randomness: Replacing Test Batteries with a Single Test
In cybersecurity applications where replicability is critical, or when building pseudo-random number generators, it is typical to perform a large number of various tests to check if a sequence of bits is random enough for practical purposes. This is also true in scientific research, to assess whether or not the digits of π or other […]
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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 […]
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New Book: Breakthroughs on the Digit Distribution of Classic Constants
Since the first edition entitled “0 and 1 — From Elemental Math to Quantum AI” and released in early 2025, a lot of progress has been made. Fascinating new results have been uncovered and proved by the author, many still leading to interesting quantum dynamics. In 100 pages, the new material presented here goes far […]
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Short Introduction to Signal Processing and Convolution
In less than 3 pages, this tutorial covers signal processing and convolution quite thoroughly, even advanced concepts. I illustrate the techniques with the Riemann zeta function and the kernel method, along with short, home-made Python code that shows all the detailed steps, rather than based on Blackbox Python libraries. The document looks like a cheat […]
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Simple Normality Test with Application to Random Number Generation
Numbers such as π, e, log 2 or √2 have binary digits (bits) that look randomly distributed. They are very good candidates to generate randomness especially in cryptography. One way to assess their randomness is by proving that they are normal numbers. Such a proof has remained elusive for centuries. Here I focus on a […]
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Cybersecurity Use Case: AI Agent for Anomaly Detection – Part 1
The case discussed here concerns fraudulent paid clicks to defraud a Google advertiser. The sophisticated click fraud scheme involving clicking viruses, data centers and other means, is undetected by Google. I worked with the law firm involved in the litigation, to build an agent able to pinpoint the sources of fraudulent traffic. The agent processes […]
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Visualizing Trading Strategies that Consistently Outperform the Stock Market
In this short paper, I discuss two topics. First, strategies to trade the S&P 500 index with few trades over long time periods, offering best exit, entry and re-entry points during the journey, to beat the baseline return. The baseline consists of staying long the whole time. The dataset has 40 years’ worth of daily […]
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Fast Random Generators with Infinite Period for Large-Scale Reproducible AI and Cryptography
Modern GenAI apps rely on billions if not trillions of pseudo-random numbers. You find them in the construction of latent variables in nearly all deep neural networks and almost all applications: computer vision, synthetization, and LLMs. Yet, few AI systems offer reproducibility, though those described in my recent book, do. When producing so many random […]
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