Researchers at Google DeepMind have developed an artificial intelligence system called AlphaProof Nexus, which found and formally proved solutions to nine open mathematical problems posed by the Hungarian mathematician Paul Erdős. In total, the system checked 353 problems from his catalog, including two problems that had remained unsolved for more than 50 (!) years.
How AlphaProof Nexus Works
The system combines large language models (LLMs) with formal mathematical proof tools. AlphaProof Nexus does not simply generate a solution, but seeks to present it in a form that allows automatic verification. It has Lean — a special language that makes it possible to formalize mathematical statements and proofs. The verification system checks every logical step and identifies errors or missing justifications, so the correctness of the proof can be confirmed.
AlphaProof Nexus is built on earlier mathematical tools from Google DeepMind, including AlphaProof. In 2024, AlphaProof, together with another of the company's systems — AlphaGeometry 2 — solved four of the six problems at the International Mathematical Olympiad, and researchers have now adapted this approach to work on open research-level problems.
How Such Complex Problems Were Solved
- During the experiment, AlphaProof Nexus examined 353 unsolved problems from the Erdős catalog — and found proofs for nine of them. Two of the problems had remained open for more than half a century!
- Another area of research was conjectures from the Online Encyclopedia of Integer Sequences (OEIS) — a database containing numerical sequences and information about their properties. Of 492 open conjectures, the system was able to prove 44.
- The researchers tested the technology on other complex scientific problems: the system worked in algebraic geometry, optimization, quantum optics, and graph theory.
Why a Human Is Still Needed
However, these results do not mean that AI has solved all of Erdős's unsolved problems, or that a neural network can handle any mathematical problem on its own! Only a select number of solutions within the tests conducted were successful. Solving such problems requires not merely performing calculations, but finding a new, rigorous proof.
One of the main challenges in applying language models to mathematics is the possibility of subtle logical errors: a text may look convincing even if the reasoning contains an unfortunate gap. Formal verification using Lean helps reduce this risk (the system accepts a proof only if its logical structure meets the specified requirements), but not entirely — a qualified mathematician still has the final say.
The authors of the work believe that formal proof search could prove useful not only in pure mathematics: such tools could help researchers verify reasoning and solve problems in related disciplines. But even a breakthrough like this does not eliminate the need for professional mathematicians: specialists are still needed to formulate problems, evaluate results, and advance scientific ideas.
