Mathematical conjecture generation using machine intelligence

TL;DR

This paper introduces a geometric and algorithmic framework for generating mathematical conjectures, focusing on inequalities, by modeling conjecture spaces as Banach manifolds and applying machine learning techniques.

Contribution

It develops a novel geometric approach to conjecture generation, modeling conjecture spaces as Banach manifolds and proposing an algorithmic pipeline for discovering new inequalities.

Findings

Generated conjectures about prime counting and Cayley graph diameters

Some conjectures have been proved through private communications

The approach reveals invariances and symmetries in conjecture spaces

Abstract

Conjectures have historically played an important role in the development of pure mathematics. We propose a systematic approach to finding abstract patterns in mathematical data, in order to generate conjectures about mathematical inequalities, using machine intelligence. We focus on strict inequalities of type f < g and associate them with a vector space. By geometerising this space, which we refer to as a conjecture space, we prove that this space is isomorphic to a Banach manifold. We develop a structural understanding of this conjecture space by studying linear automorphisms of this manifold and show that this space admits several free group actions. Based on these insights, we propose an algorithmic pipeline to generate novel conjectures using geometric gradient descent, where the metric is informed by the invariances of the conjecture space. As proof of concept, we give a toy algorithm to generate novel conjectures about the prime counting function and diameters of Cayley graphs of non-abelian simple groups. We also report private communications with colleagues in which some conjectures were proved, and highlight that some conjectures generated using this procedure are still unproven. Finally, we propose a pipeline of mathematical discovery in this space and highlight the importance of domain expertise in this pipeline.