Technical background

The unit distance problem posed by Paul Erdős in 1946 is one of the most famous problems in combinatorial geometry: for n points on a plane, what's the maximum number of pairs of points that are exactly distance 1 apart? For decades, the academic consensus held that the square-grid construction was essentially at the theoretical upper limit.

The core breakthrough

In October 2025, OpenAI had an embarrassing moment — the then-GPT-5 claimed to have solved 10 unsolved Erdős problems, only to be found to have merely located existing solutions in the literature, and was publicly mocked by LeCun and others. This time, history did not repeat.

OpenAI's new reasoning model, not specifically trained for math problems, autonomously discovered a completely new proof path — creatively applying deep tools from algebraic number theory (infinite class field tower theory, Golod-Shafarevich theory) to an elementary geometry problem, delivering a proof that mathematicians have verified as correct and milestone-worthy. Princeton's Will Sawin subsequently provided a more precise result: the new construction is asymptotically superior to the square grid — the first such construction in 80 years.

Why it matters

Mathematics is the best litmus test for AI reasoning capability: problems are precisely stated, proofs can be rigorously verified, and long chains of reasoning tolerate no gaps. This proof doesn't just solve a specific problem — it proves that general reasoning models can already produce original breakthroughs in real mathematical research, not merely be literature-review machines.

Gowers calls it a milestone for AI mathematics. Arul Shankar is more direct: AI models are no longer just human mathematicians' assistants — they can generate original, clever ideas and bring them to fruition. The paradigm of mathematical research is quietly changing.

From an industry perspective, this signals that deep collaboration between AI and basic-science research is becoming reality: AI can not only accelerate computation, but also bring fresh perspectives at the conceptual level. This may be one of the most symbolically significant AI advances of 2026.