AI Proves a Mathematical Conjecture: This Time, It’s Real.
OpenAI’s latest model, GPT-5.2 Pro, has independently proved an Erdős conjecture.
The proof was verified by Fields Medalist Terence Tao and described as “the clearest first-of-its-kind result (with primary AI contribution) to date.”
This problem, numbered 281 in the Erdős Problem Database, was jointly proposed by legendary mathematicians Paul Erdős and Ronald Graham in 1980. It concerns the deep relationship between congruence covering systems and natural density.
For 45 years, this problem remained unresolved in the database, waiting for a solution.
That changed on January 17, 2026, when researcher Neel Somani submitted the problem to GPT-5.2 Pro.
The Proof Relies Solely on GPT-5.2 Pro
The Erdős Problem website has now archived the AI-generated proof.
The argument unfolds over the ring of infinite adelic integers, leveraging Haar measure and pointwise ergodic theorems. Combined with compactness arguments, it achieves a transition from pointwise convergence to uniform convergence.
According to Tao, this is a variant of the “Furstenberg correspondence principle,” a standard tool at the intersection of ergodic theory and combinatorics.
However, GPT-5.2 Pro’s approach differs slightly from typical arguments, relying more heavily on Birkhoff’s theorem.
What impressed Tao most was not the method itself, but the fact that the AI made no errors.
What surprised me more is its avoidance of common pitfalls, such as swapping limits or misordering quantifiers—precisely where this problem is most prone to error. Previous generations of large language models would almost certainly have stumbled on these subtleties.
To verify the proof, Tao manually translated the entire ergodic theory argument into combinatorial language, replacing Birkhoff’s theorem with the Hardy-Littlewood maximal inequality and re-running all derivations.
Conclusion: The proof holds.
An Unexpected Discovery
As discussions about GPT-5.2 Pro’s proof unfolded, a user named KoishiChan posted an unexpected finding in the comments section:
The problem actually has a simpler solution, relying on two theorems that had already existed since 1936 and 1966.
The first is the density convergence theorem, proved collaboratively by Harold Davenport and Erdős himself in 1936.
The second is Rogers’ Theorem, first published in Chapter 5 of Halberstam and Roth’s monograph Sequences (1966). Combining these two classical results makes Problem 281 almost a direct corollary.
This raises a puzzling question: Erdős was a co-author of the 1936 paper, yet when he posed this problem in 1980, he did not realize the answer was within reach.
Tao specifically emailed French mathematician Gérald Tenenbaum to seek clarification on this matter.
Tenenbaum confirmed that “as long as the two classical results you mentioned (the Davenport-Erdős theorem and Rogers’ Theorem) are satisfied, the problem is immediately resolved.” However, he also speculated that “the formulation of the problem may have been altered at some point.” No other versions of the statement have been found to date, so it must be treated as originally stated.
More interestingly, in 2007, five leading experts—Filaseta, Ford, Konyagin, Pomerance, and Yu—were solving another Erdős problem without knowing about Rogers’ Theorem. They only added the citation after Tenenbaum alerted them.
Tao remarked: “Rogers’ Theorem has not received the dissemination it deserves. It appears only in Halberstam and Roth’s book, was never published separately, and has very few citations. Perhaps this discussion will bring more attention to this result among researchers working on sieve methods and congruence covers.”
The problem now has two proofs: one via GPT-5.2 Pro’s ergodic theory approach, and another derived by KoishiChan from classical literature in combinatorics.
Tao confirmed that these are “different proofs,” although they share some conceptual overlap.
How to Assess the True Success Rate of AI in Mathematics
After the news spread, various AI models were brought in for cross-validation.
Gemini 3 Pro stated that the proof was correct. Another researcher used GPT-5.2 Pro repeatedly to check the argument’s details; the AI identified only one area needing stricter rigor in step two, suggesting it could bypass ergodic theory using Fatou’s Lemma.
However, Tao pointed out that the direction of Fatou’s Lemma had been applied incorrectly: “I just taught graduate measure theory, and I’ve seen this type of error too many times.”
Subsequent verification confirmed that Fatou’s Lemma was actually being applied to the complement set, which corrected the direction issue, leaving the argument valid.
Nevertheless, Tao issued a sobering reminder. He wrote:
When assessing the true success rate of AI tools, the most significant statistical bias comes from strong reporting bias; negative results are rarely disclosed.
If an individual or an AI company applies a tool to open problems without progress, they have no incentive to report that negative outcome. Even if reported, it is unlikely to spread on social media as widely as positive results.
Although most successes cluster at the easier end of the difficulty spectrum, this does not yet indicate that medium-difficulty Erdős problems are within AI’s reach.
He recommended an open-source project initiated by Paata Ivanisvili and Mehmet Mars Seven, which systematically records both positive and negative results from frontier large language models on Erdős problems.
Data shows that the true success rate of these tools on Erdős problems is only about 1% to 2%.
However, given that there are over 600 unsolved problems in the database, this percentage still represents a substantial and non-trivial number of AI contributions.
References
- 281 Discussion Thread | Erdős Problems — www.erdosproblems.com/forum/thread/281
- 2012695714187325745 — x.com/neelsomani/status/2012695714187325745
- 115911902186528812 — mathstodon.xyz/@tao/115911902186528812