During the past week, several leading AI labs reported independent breakthroughs on the Erdős unit‑distance conjecture and related problems. According to the accounts, Anthropic, OpenAI and DeepMind each found ways to overcome an obstacle that had stood for roughly eighty years, using different models and approaches.
What the labs did
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Anthropic researcher Levent Alpoge spent a weekend running Claude Mythos air‑gapped and in multiple instances; the model had no access to OpenAI’s published proof. The model produced an independent disproof that, the reports say, was shorter and cleaner than OpenAI’s 125‑page chain‑of‑thought document.
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In the same week OpenAI also reported a breakthrough, using its own approach to clear the roughly 80‑year barrier.
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DeepMind’s AlphaProof Nexus produced further progress: the reports state it resolved nine additional Erdős problems, each at a cost of a few hundred dollars.
The reports emphasize that the three labs did not coordinate with one another; the results emerged independently.
Why this matters
Paul Erdős left behind more than a thousand open problems, many of which served as long‑standing challenges for human mathematicians. That multiple large language/model‑based systems cleared an 80‑year‑old obstacle within a week — and that one system reached for a tool from algebraic number theory (a class field tower) outside the traditional human approaches — suggests the bottleneck in problem solving is shifting. The constraint may no longer be the mathematical problem itself but rather how researchers interpret, validate and integrate machine‑produced proofs.
Implications for research
Independent convergence across systems signals several possible consequences: rapid advancement of model capabilities, increasing automation of mathematical methods, and a reshaping of traditional research roles. Concrete implications include:
- rethinking the role of human research mathematicians when systems can produce full proofs or disproofs quickly;
- heightened need for verification and audit processes, including independent human checks, formal verification methods, or cross‑validation by other models;
- changing cost structures if certain solutions can be reproduced repeatedly for a few hundred dollars.
Summary
The week’s developments — Anthropic’s Levent Alpoge running air‑gapped Claude Mythos, OpenAI’s own solution, and DeepMind’s AlphaProof Nexus resolving multiple problems — show that large language and proof systems can independently tackle long‑standing Erdős problems. The labs say they did not coordinate, and that independent convergence is the key signal: difficult mathematical problems are rapidly becoming solvable by machine systems, prompting a reassessment of human roles and verification practices in mathematical research.
