OpenAI now says its model disproved that assumption by constructing an infinite family of point configurations that achieves a stronger polynomial improvement. According to the company, the proof demonstrates that for infinitely many values of n, the number of unit-distance pairs can grow at least as fast as n^{1+\delta} for some fixed \delta > 0.
The company said the proof was independently checked by external mathematicians, who also produced a companion paper explaining the argument and its broader mathematical significance.
Unlike systems trained specifically for theorem proving or guided through structured proof-search pipelines, OpenAI said the result came from a general-purpose reasoning model tested on a collection of Erdős problems as part of broader research into whether AI systems can contribute to frontier mathematics.
OpenAI described the result as “the first time that a prominent open problem, central to a subfield of mathematics, has been solved autonomously by AI.”
Several prominent mathematicians endorsed the significance of the work. Princeton combinatorialist Noga Alon called it “an outstanding achievement,” adding that “every mathematician working in Combinatorial Geometry thought about this problem.”
Fields Medal winner Tim Gowers described the proof as “a milestone in AI mathematics,” while number theorist Arul Shankar said the result shows current AI systems “are capable of having original ingenious ideas, and then carrying them out to fruition.”
The proof itself draws from algebraic number theory rather than traditional geometric constructions. OpenAI said the argument replaces earlier approaches based on Gaussian integers with more advanced number field constructions involving tools such as infinite class field towers and Golod–Shafarevich theory.
That crossover surprised mathematicians because the unit distance problem had long been treated primarily as a geometric and combinatorial question. The AI-generated proof instead connected it to deep structures in algebraic number theory that researchers had not previously linked to the problem.
The breakthrough also overturns one of the central assumptions surrounding the conjecture. Earlier lower bounds, dating back to Erdős’s original 1946 construction, had only marginally improved over time. The best-known upper bound, established in the 1980s, also remained largely unchanged for decades.
OpenAI said subsequent refinement work by Princeton mathematician Will Sawin established an explicit value of \delta = 0.014, strengthening the original construction.
The company framed the result as part of a broader push toward AI systems capable of sustained reasoning across complex research problems. OpenAI argued that mathematics provides a useful testing ground because proofs can be rigorously verified and long chains of reasoning must remain logically consistent from start to finish.
Beyond pure mathematics, OpenAI said similar reasoning capabilities could eventually support research in fields including biology, engineering, physics, materials science, and medicine.
The company also emphasized that human expertise remains central to the process. While the model generated the proof, mathematicians verified the argument, expanded on its implications, and connected the result to existing mathematical theory.
As mathematician Thomas Bloom wrote in the companion paper, the result suggests “there is a lot more that number theoretic constructions have to say about these sorts of questions than we suspected.”
This analysis is based on reporting from OpenAI.
Image courtesy of OpenAI.
This article was generated with AI assistance and reviewed for accuracy and quality.
