Are the digits of e random-like? A theoretical framework to solve this mystery

I invite readers to check my progress on this topic and provide feedback. The deep results are published as paper 51, here.

Any solution to this mythical problem has remained elusive for centuries, yet its formulation can be understood by kids in elementary school. This paper 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 this problem.

The goal is to transition from experimental to theoretical number theory. Yet, I use illustrations with numbers that are so big that no amount of computing power will ever be able to handle them. 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, computer-intensive simulations, operations research, and agent-based modeling.

Read the executive summary and access the free paper, from here. The picture below is central to the proof and explained in the paper. It shows how I get to the first 10,000 digits of e while the vertical axis shows the number of binary digits equal to 1, among 10,000 digits of various numbers, with the one at abscissa 10,000 consisting of the first 10,000 binary digits of e.