AI and Erdős Problem 397: A Verified Proof, Prior Work and the Prospect of Self-Improvement

Neel Somani reports that he prompted GPT-5.2 Pro to work on Erdős problem 397 and submitted a proof accepted by Terence Tao for the problem database. The proof was formalized using Harmonic’s Aristotle and checked in Lean. Berman calls it an extremely hard open problem and says the model took about 15 minutes; that timing is a report, not the complete time required for review and formalization.

An important later correction is that Tao’s [AI contributions tracker](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems) lists the January 10, 2026 solution as complete and Lean-checked, but records a similar full human solution from a 2012 Chinese competition found on January 14. Thus it is an AI-assisted valid solution, not a demonstrated first solution to a previously unsolved problem. Erdős problems cover a wide range of difficulty.

Berman points to several recent AI-assisted attempts and argues that models are beginning to contribute to frontier mathematics. Some entries in Tao’s list are full solutions, while others are partial, incorrect or later found to have prior solutions. These outcomes are not interchangeable, and a formal proof verifies the encoded statement rather than proving that no earlier result existed.

He uses an illustrative chart to describe an intelligence explosion: AI that helps discover better methods could improve future AI, creating a feedback loop. That is a possible scenario, not a measured trajectory or proof of unbounded intelligence. The speed and extent of such a loop depend on research, implementation, hardware and other constraints.

Berman points to a model reaching gold-medal-level performance on International Mathematical Olympiad problems as evidence of progress. The IMO is a mathematics competition, not a coding competition. Competition performance and research on open problems are different tests, and a medal-level score does not itself establish broad autonomous mathematical discovery.

He describes an AI-discovered matrix multiplication improvement but calls the system AlphaFold. The result shown is from Google DeepMind’s [AlphaEvolve](https://deepmind.google/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/): its procedure found an algorithm using 48 scalar multiplications for 4×4 complex-valued matrices, compared with 49 in the cited prior method. AlphaFold concerns protein structure; AlphaTensor is another distinct matrix-algorithm system.

Berman argues that even small gains could matter at scale because matrix multiplication is central to AI. But the specific 4×4 complex result does not mean every matrix multiplication has become faster: practical performance depends on hardware, matrix sizes, numerical properties and deployment. He also discusses DeepMind’s reports that AlphaEvolve helped optimize selected data-center scheduling, TPU design and model-training operations, not the entirety of Google’s infrastructure.

Berman mentions Sakana AI’s AI Scientist as another effort to automate parts of research, and suggests that discoveries could eventually feed back into model development. The video includes a paid sponsorship for HPC-AI’s fine-tuning infrastructure and a promotional credit offer; these commercial claims reflect the time of the video.

Berman shows Tao’s public tracker of AI contributions to Erdős problems. It distinguishes full solutions, partial results, failed attempts and instances where literature review later identified prior work. Some solutions combine GPT-5.2 with Aristotle and Lean, while other entries involve different systems or human contributions. A green formal-verification marker has a narrower meaning than novelty or proof that all related questions are settled.

Berman predicts that stronger mathematical reasoning and open-ended scientific discovery could eventually help solve disease and improve AI itself. These are hopeful projections, not evidence that cancer or all diseases will be solved. He argues that AI work can be parallelized, while real systems remain constrained by compute, energy, data, experimental validation and human oversight.