In my recent paper 51 on cracking the deepest mathematical mystery, available at https://mltblog.com/3zsnQ2g, I paved the way to solve a famous multi-century old math conjecture. The question is whether or not the digits of numbers such as ฯ are evenly distributed. Currently, no one knows if the proportion of '1' even exists in these binary digit expansions. It could oscillate forever without ever converging. Of course, mathematicians believe that it is 50% in all cases. Trillions of digits have been computed for various constants, and they pass all randomness tests. In this article, I offer a new framework to solve this mystery once for all, for the number e.
Rather than a closure on this topic, it is a starting point opening new research directions in several fields. Applications include cryptography, dynamical systems, quantum dynamics, high performance computing, LLMs to answer difficult math questions, and more. The highly innovative approach involves iterated self-convolutions of strings and working with numbers as large as 2^n + 1 at power 2^n, with n larger than 100,000. No one before has ever analyzed the digits of such titanic numbers!
To read the full article, participate in the AI & LLM challenge, get the very fast Python code, read about ground-breaking research, and see all the applications, visit https://mltblog.com/3DgambA
Tools such as OpenAI can on occasion give the impression that they are able to prove theorems and even generalize them. Whether this is a sign of real (artificial) intelligence or simply combining facts retrieved from technical papers and put things together without advanced logic, is irrelevant. A correct proof is a correct proof, no matter how the author โ a human or a bot โ came to it.
My experience is that tools like OpenAI make subtle mistakes, sometimes hard to detect. Some call it hallucinations. But the real explanation is that usually, it blends different results together and add transition words between sentences in the answer, to make it appear as a continuous text. Sometimes, this creates artificial connections between items that are loosely if at all related, without providing precise reference to the source, and the exact location within each reference. It makes it hard to double check and make the necessary corrections. However, the new generation of LLMs (see https://mltblog.com/4g2sKTv) offers that capability: deep, precise references.
Likewise, mathematicians usually make mistakes in the first proof of a new, challenging problem. Sometimes these are glitches that you can fix, sometimes the proof is fundamentally wrong and not fixable. It usually takes a few iterations to get everything right.
โก๏ธ Read full article and learn how I proved a difficult result with the help of AI, at https://mltblog.com/4jqUiUD
