Academician János Pach, Research Professor at Rényi Institute, on the impact of the refutation of an 80-year-old conjecture by Paul Erdős and on applying the method to solve other problems
"It was clear that this would happen sooner or later, but few thought it would happen so soon. On 20 May 2026, the breakthrough occurred. OpenAI announced that one of its models – one not even specifically developed for scientific research – had refuted a famous conjecture of Paul Erdős, namely that among n points in the plane, the same distance can occur at most approximately as many times as it does in a √n × √n square lattice. The proof, which has been verified by distinguished mathematicians, is publicly available on the internet. The paper is entirely the creation of artificial intelligence and, in its present form, could be published unchanged in any of the world's leading professional journals.
We mathematicians could still imagine ourselves, only a few weeks ago, as Christopher Columbus, who, sailing beyond the Pillars of Hercules, 'first dared to embark upon the unknown path' in order to discover new lands and prove new theorems (Torquato Tasso). If we were fortunate, we might occasionally experience the uplifting feeling of seeing a strip of land appear on the horizon."
These reflections were written by János Pach, Research Professor in the Department of Geometry at Rényi Institute, on the occasion of artificial intelligence refuting one of Paul Erdős's conjectures, made public in 1946, concerning the distribution of distances among so-called lattice points. The solution attracted considerable attention and has been the subject of continuous discussion among mathematicians because the system that achieved the breakthrough – one of OpenAI's models – was not a supermodel specifically designed for mathematical proofs. The result was subsequently verified by a panel of nine mathematicians.
The proof of the conjecture marks an important milestone for both mathematics as well as professional and wider AI communities: it is the first time that artificial intelligence has independently solved a significant open problem that occupies a central place in a branch of mathematics. János Pach's personal essay, entitled A Spectre Is Haunting Mathematics – the Spectre of Artificial Intelligence, can be read online (originally in Hungarian) in the electronic mathematics journal Érintő HERE.
"Artificial intelligence will most likely become much better than humans at solving specific problems, but for quite some time it will not be able to take over the role of mathematicians asking good questions," predicts Research Professor János Pach, who believes that these AI systems do not yet possess mathematical intuition. "The task is not to despair, but to tame artificial intelligence," he adds.
János Pach's interview, broadcast in the science programme Felfedező (Discoverer) on Kossuth Radio (NPR Hungary), can be listened to in Hungarian HERE.
János Pach is a Research Professor at Alfréd Rényi Institute of Mathematics. His research focuses on combinatorial and algorithmic geometry, as well as geometric graph theory. He is one of the world's leading researchers in combinatorial and algorithmic geometry. He applies graph-theoretic and hypergraph-theoretic methods in algorithmic geometry and robotics. Geoscape (From Geometry to Combinatorics and Back: Escaping the Curse of Dimensionality) research group, which he leads, is funded by the European Union through an ERC Advanced Grant. The principal objective of the Geoscape project is to investigate several famous open problems in combinatorics for classes of graphs and hypergraphs that play an important role in geometry, algebra, and practical applications. According to the ERC-funded project proposal, discoveries related to the questions addressed by the project are expected to "bring us closer to resolving several classical problems, such as the Erdős – Hajnal Conjecture, the Danzer – Rogers Conjecture, and the Schur – Erdős Problem, while also contributing to the development of efficient algorithms applicable to clustering and property-testing tasks on large networks." More information about the research group is available on its website HERE. |
Many members of the mathematical community believe that it is only a matter of time before artificial intelligence produces breakthroughs in other scientific disciplines that are unimaginable today. The growing presence and use of AI in mathematical research also inspired the declaration published in June 2026 by a broader international group of scholars – not only mathematicians – working on the subject. The document is entitled the Leiden Declaration on AI and Mathematics. The article on Renyi Institute's own website renyi.hu about the declaration can be read HERE.