On the impossibility of discovering a formula for primes using AI

TL;DR

This paper investigates the theoretical limits of machine learning in discovering formulas for prime numbers, demonstrating fundamental impossibilities through information theory and probabilistic number theory.

Contribution

It introduces a formal framework connecting Kolmogorov complexity with ML limits and proves the impossibility of formula discovery for primes using ML techniques.

Findings

Derives the Erdős-Kac Law and Hardy-Ramanujan theorem via Maximum Entropy methods.

Establishes the Prime Coding Theorem showing the limits of ML in prime formula discovery.

Highlights fundamental theoretical barriers rooted in algorithmic information theory.

Abstract

The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erd\H{o}s-Kac Law and Hardy-Ramanujan theorem in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.