RESEARCHOpenAI
OpenAI's internal reasoning model disproved an open math problem first posed by Paul Erdős in 1946. Cambridge mathematician Tim Gowers (a Fields medalist, the math equivalent of a Nobel) validated the 125-page proof.
First time a general-purpose AI solved a frontier-research problem with no human guidance. Tim Gowers (Fields medalist) validated the proof. That's the credibility marker that matters.
Same capability transfers to any problem with a clear answer and a long verification chain. Drug discovery, materials science, security exploits, algorithm design, compiler optimization. Anywhere humans previously spent months on a single problem, the next 18 months look different.
OpenAI's internal frontier just visibly diverged from its public products. The gap widened. Karpathy joined Anthropic three days before this announcement. Not a coincidence.
⚡ Why this matters
- A general-purpose AI just solved an 80-year-old open math problem with no human guidance.
- First frontier-research-grade reasoning by a general model.
- Anywhere you have a hard problem with a clear answer, the next 18 months look different.
🔍 What happened
- May 20, 2026. OpenAI announces an internal general-purpose reasoning model autonomously solved Erdős's planar unit distance problem.
- Problem: given n points in a plane, what's the maximum number of pairs exactly distance 1 apart?
- For 80 years, the belief was square grids were near-optimal at n^(1+o(1)).
- Model disproved that. Constructions with n^(1+δ) (Will Sawin refined to δ=0.014). Polynomial improvement.
- Proof uses infinite class field towers (Golod-Shafarevich theory) and algebraic number fields embedded in the plane.
- 125-page chain-of-thought proof.
- Model only got the problem statement. No hints. No partial proof.
- Validated by Tim Gowers (Cambridge, Fields medalist), Noga Alon (Princeton), Arul Shankar, Jacob Tsimerman.
💬 Smart takes
- Tim Gowers: "This is a milestone in AI mathematics."
- Noga Alon: "Applies fairly sophisticated tools from algebraic number theory in an elegant and clever way."
- Arul Shankar: "Current AI models go beyond just helpers. They are capable of having original ingenious ideas."
- Skeptic: OpenAI didn't disclose model name, success rate on other Erdős problems, or inference cost. Humans still picked the problem and wrote the companion paper. We don't know if this is one-off or systematic.
🧭 Where this goes
- Anthropic and DeepMind ship competing "open problem solved" announcements within 90 days.
- "Number of named open problems solved" becomes the new headline benchmark by Q4.
- The recursive AI-improves-AI loop accelerates. Karpathy's Anthropic role (May 19, three days earlier) is not a coincidence.
- First AI-discovered drug candidate from Isomorphic Labs enters clinical pipeline within 18 months.
🎯 Implication
- For PMs whose products depend on long verification chains (engineering, science, security, design): scope a Q3 pilot where the model gets an unstructured problem. Measure original-output quality, not benchmark scores.
- For execs: any roadmap assuming "AI is good for first drafts but humans do the hard reasoning" needs revision.