OpenAI says internal model has disproved 80-year-old Erdős unit-distance conjecture

TL;DR:

  • OpenAI says an internal general-purpose reasoning model has autonomously disproved Erdős’s 1946 unit-distance conjecture, providing an infinite family of point configurations that yield a polynomial improvement over the long-believed near-linear upper bound.
  • The proof has been verified by a group of external mathematicians and a companion paper has been written by Fields medalist Tim Gowers, who described the result as “a milestone in AI mathematics” he would have accepted for the Annals of Mathematics without hesitation.
  • A refinement by Princeton’s Will Sawin gives an explicit improvement exponent of δ = 0.014, and the construction draws on algebraic number theory — class field towers and Golod-Shafarevich theory — to attack what had been thought a purely geometric problem.

The result, announced on Tuesday, lands in the same week Anthropic co-founder Jack Clark told an Oxford lecture that AI would help make a Nobel prize-winning discovery within 12 months. OpenAI’s claim moves that argument from prediction to evidence — a concrete and externally verified case of an AI model resolving a longstanding open problem in mathematics, not assisting a human who did. The two stories together reframe what UK research-funding bodies and universities will need to plan for in the next research-assessment cycle.

A research-acceleration data point, not just a publicity moment

The substantive claim is that a general-purpose reasoning model — not a system trained specifically for mathematics or scaffolded to search through proof strategies — generated a proof that brought “unexpected, sophisticated ideas from algebraic number theory” to bear on an elementary geometric question. Noga Alon, the Princeton combinatorialist who described the unit-distance problem as one of Erdős’s favourites, called the outcome “an outstanding achievement, settling a long-standing open problem”. Arul Shankar said the model’s chain of thought shows “good intuition, willingness to try approaches considered long-shot by the community, and a predisposition to attempt constructions” — language usually reserved for human mathematicians.

For UK research universities, the result raises immediate questions. Who reviews AI-generated proofs in mathematical journals? How is authorship attributed when no human formulated the construction? What does the EPSRC’s research evaluation framework do with AI-generated contributions? Thomas Bloom, writing in the companion note, was characteristically careful: “this shows that there is a lot more that number theoretic constructions have to say about these sorts of questions than we suspected” — but warned that the more general lesson is that “the coming months and years will see similar successes in many other areas of mathematics”. The same week, the King’s College London study found 60% of UK university students expect AI to make their job market much tougher by the time they graduate; mathematics undergraduates working towards PhDs now also need to factor AI-assisted research into their career planning.

Looking forward

Watch for whether the result triggers a wave of AI-assisted attacks on other Erdős problems and whether the major UK research universities — Oxford, Cambridge, Imperial, Edinburgh — formalise policies on AI-derived publications before the autumn term. For UK SMEs and research-intensive corporates, the practical signal is narrower but real: capability that holds together complex arguments end-to-end is the kind of capability that transfers into engineering, biology, materials science and physics R&D. Jack Clark’s Nobel timeline is no longer a distant frame; OpenAI has provided one data point in its favour, and the EPSRC’s incoming AI-research-and-research-on-AI strategy will need to address what that means for the UK science base.

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