Erdos problems now seem to be falling with regularity at the hands of AI.
AI has notched another milestone in mathematical research, with GPT 5.2 Pro helping solve Erdős Problem #281—a number theory puzzle that has remained open since it was first posed in 1980. This marks the third Erdős problem solved with AI assistance in recent months, following a pattern of increasingly sophisticated AI contributions to pure mathematics.
The solution was generated by Neel Somani, founder of Eclipse and former quantitative researcher at Citadel, who used GPT 5.2 Pro to tackle the decades-old problem about covering systems and congruence classes. The proof was subsequently verified by Fields Medalist Terence Tao, who called it “perhaps the most unambiguous instance” of AI solving an open mathematical problem to date.
The Problem and Its Solution
Erdős Problem #281 deals with infinite sequences of integers and asks whether certain density conditions can be guaranteed with finitely many congruence classes. The problem sat unsolved for over 44 years, originally appearing in a 1980 work by legendary mathematician Paul Erdős and Ronald Graham.
Somani’s approach, facilitated by GPT 5.2 Pro, employed ergodic theory—specifically working within the profinite integers with Haar measure. The proof leverages the Birkhoff ergodic theorem and Dini’s theorem to show that if avoiding all congruences leads to density zero, then avoiding just the first k congruences must uniformly approach zero as k increases.
Expert Validation and Commentary
The solution underwent rigorous scrutiny from multiple mathematicians on erdosproblems.com. Tao not only verified the proof but also provided an alternative formulation using combinatorial methods and the Hardy-Littlewood maximal inequality, further cementing the result’s validity.
“Previous generations of LLMs would almost certainly have fumbled these delicate issues,” Tao noted, highlighting the AI’s ability to avoid common pitfalls around limit interchanges and quantifier ordering—errors that frequently trip up even experienced mathematicians.
Thomas Bloom, who curates the Erdős Problems website, observed that the solution demonstrates why “passing to the profinite completion and using the Haar measure on that is a useful thing to do,” suggesting that ergodic theory tools may not have been in Erdős and Graham’s usual mathematical toolkit—potentially explaining why the problem remained open for so long.
A Twist: Classical Methods Existed Too
In an intriguing development, user KoishiChan pointed to classical results from a 1966 work by Halberstam and Roth, combined with Rogers’ theorem and a 1936 paper by Davenport and Erdős himself, that could have solved the problem decades ago. This revelation puzzled experts, as Erdős would certainly have known these tools.
“Now I am really puzzled, because Erdős would certainly have known both of these facts in 1980,” Tao commented, speculating that the problem may have been stated without extensive solution attempts, or that relevant connections were simply overlooked.
Part of a Broader Trend
This solution follows two other recent AI-assisted breakthroughs on Erdős problems. In December 2024, Robinhood CEO Vlad Tenev’s math AI startup claimed to have solved Problem #728, which had been open for 30 years. Shortly after, GPT-5.2 and Harmonic appeared to autonomously solve Erdős Problem #124.
The rapid succession of these solutions suggests AI systems are reaching a threshold where they can meaningfully contribute to open problems in pure mathematics—particularly when the solution requires connecting disparate mathematical frameworks or applying tools from fields outside the problem’s traditional domain.
What Makes This “Most Unambiguous”?
Tao’s assessment that this represents “perhaps the most unambiguous instance” of AI solving an open problem reflects several factors: the problem’s clear open status since 1980, the novelty of the ergodic theory approach, independent verification by multiple experts, and the AI’s avoidance of typical mathematical errors.
Multiple AI systems—including ChatGPT DeepResearch, Gemini DeepResearch, and Claude—conducted literature searches and found no prior solutions to this specific infinitary problem, though they did locate work on related finitary problems.
The solution has now been added to the Erdős Problems wiki as a “Section 1” result—a designation for problems solved with novel methods not found in existing literature.
Implications for Mathematical Research
While AI-assisted mathematical discovery is still in its early stages, these developments suggest a future where AI systems serve as powerful collaborators in mathematical research—not by replacing human mathematicians, but by suggesting connections between mathematical frameworks that might not be obvious to researchers working within specific traditions.
The fact that GPT 5.2 Pro avoided limit interchange errors and correctly navigated the subtleties of ergodic theory represents a significant advance over previous AI systems. However, as Tao’s discovery of an error in a proposed Fatou’s lemma argument demonstrates, human verification remains essential.
For now, the mathematical community is adjusting to a new reality where decades-old problems can fall to AI-human collaboration, raising questions about which other long-standing puzzles might be next on the list.