An AI-Generated Proof for Erdős Problem 397

SUMMARY

In 30 seconds

Matthew Berman highlights Neel Somani's report that GPT-5.2 Pro generated a proof for Erdős problem 397, which was submitted and accepted by Terence Tao. He extrapolates from this result to a possible intelligence explosion and scientific advances. A full human solution from 2012 was subsequently identified, so the AI proof was not the first solution to the problem; neither the result nor its acceptance establishes that an intelligence explosion is underway.

1An AI-assisted proofBerman recounts Somani's submitted proof and Tao's acceptance. The later discovery of a 2012 human solution changes the interpretation of the result.
2A wave of math resultsBerman describes several recent AI-assisted results, but their difficulty and novelty vary and require individual review.
3A speculative feedback loopHe imagines AI improving its own algorithms and contributing to medicine and materials. This proof does not demonstrate that such a loop has occurred.

Takeaways

Somani reported generating a proof for Erdős problem 397 with GPT-5.2 Pro and formalizing it with Harmonic's tools.
A complete human solution from 2012 was later found, so the AI-generated proof was not the first solution.
The video does not establish that this was among the hardest mathematical problems or that only a few hundred people could solve it.
Recursive AI improvement and applications to disease and materials remain speculative conclusions rather than demonstrated results.

TopicsMathematics AI research AGI

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The Erdős proof

Berman describes Somani's GPT-5.2 Pro proof for problem 397 and its acceptance. A previous complete human solution was later identified.

Neel Somani reported using ChatGPT to produce a proof for an Erdős problem. The post shown in the video identifies it as problem 397 and says GPT-5.2 Pro generated the proof, which was formalized with Harmonic and accepted by Terence Tao. I describe it as one of the hardest problems on Earth, but that ranking and my estimate of how many people could solve it are not established.

SLIDE 1 / 1Somani's report on the proofThe displayed post identifies problem 397, GPT-5.2 Pro and the formalization step; an earlier human solution was later found.

I say this was the sixth such result in two weeks and see it as evidence that AI is advancing quickly in mathematics. Individual problems and contributions differ, however, and a human solution to problem 397 had already appeared in 2012; it was identified later. The generated proof was therefore not the first solution, even though it was a distinct AI-assisted proof.

Possible implications

Berman speculates about recursively improving AI and scientific applications. The mathematical result alone does not establish those outcomes.

I wonder whether AI that can solve difficult mathematics might design better algorithms for itself, apply those improvements and set off an intelligence explosion. In that imagined future, more powerful AI might help tackle diseases or develop new materials. These are possibilities, not outcomes established by this mathematical result.

Somani, a quantitative researcher, says the model produced the proof in about 15 minutes after prompting. I think this could be the beginning of an intelligence explosion. That remains speculation: a proof, even one accepted after review, does not demonstrate recursive self-improvement or a medical breakthrough.