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Mathematicians address artificial intelligence

By David H Bailey, on August 9th, 2026<br>Illustration of the planar unit distance problem. Credit: Noga Alon, et al., 2026, OpenAI

Introduction

By several measures, the past year has been a banner year for advances in artificial intelligence in many fields, and perhaps most remarkably in mathematics. While these software systems still have many shortcomings, and some perform significantly better than others, nonetheless the overall picture has been one of rather stunning progress.

A sample of recent results

In May 2026, a model developed by OpenAI found a counter-example to a conjecture first proposed by legendary mathematician Paul Erdős in 1946 (see Here or Here for more details). Colloquially speaking, the problem can be described as follows (see graphic on right): If one draws a pattern of dots on a paper, what is the maximum number of equal-sized lines that can be drawn between them? Erdős conjectured that the optimal arrangement was with points in a regular grid. The OpenAI model found, to the contrary, that less symmetric patterns can yield a much larger number of pairs. The result involved usage of sophisticated principles from advanced number theory, among other things.

Then on 1 August 2026, OpenAI announced results on 10 other problems, including three problems originally proposed by Erdős. Here is a brief summary, as presented in the OpenAI report:

High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold.

Binary and spherical codes. Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes.

Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory.

Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras.

Arithmetic circuit complexity. New lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic-formula lower bound of order n4/log n.

Quantum parallel repetition. An exponential parallel repetition theorem for general two-player quantum games, extending a foundational principle from classical complexity theory.

Closest vector problem. Polynomial-factor hardness of approximation for the closest vector problem, a foundational lattice question related to post-quantum cryptography.

Ehrhart’s volume conjecture. Determining, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.

Multicolor Ramsey numbers. A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.

Extremal number conjectures. Results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.

The Leiden Declaration

In the wake of such developments, a group of mathematicians have called for a new set of principles to govern mathematical research in an era of AI, encapsulated in The Leiden Declaration on Artificial Intelligence and Mathematics. This declaration declares, among other things:

Mathematicians have a choice about whether and how to adopt artificial intelligence in the conduct of their research. They also have a responsibility to ensure the continued flourishing of the discipline. This Declaration calls upon mathematicians to exercise this responsibility, and provides recommendations for individuals, institutions, government, and industry.

Among the concerns raised in the declaration are that current software systems sometimes produce plausible but incorrect "proofs" that are not easily distinguished from correct mathematical proofs. Also, concern was raised that software drawing extensively on published mathematical work may undermine proper attribution (indeed, one common complaint of current systems is that they do not properly document and attribute their work). There is also the risk that private technology companies may come to dominate the mathematical research enterprise, and may give preference to results that are amenable to automated methods, rather than to results with more intrinsic significance.

Here are some recommendations of the Declaration for individual researchers:

Disclose tool use.

Support the need of reviewing.

Adhere to principles of open science.

Retain the responsibility for correctness.

Affirm the humanity of authorship.

Put effort into proper attribution.

Participate in public discourse.

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