Claude Mythos Tackles OpenAI's Erdős Problem with a Simple Proof

A New Milestone in AI-Driven Math

Anthropic’s Claude Mythos, an advanced language model (LLM) with agentic capabilities, has reportedly solved the Erdős unit-distance conjecture-a long-standing open problem in combinatorial geometry. This achievement follows OpenAI's recent success in disproving the same conjecture, highlighting a significant milestone in AI-driven mathematical reasoning.

The Erdős unit-distance conjecture, proposed by Paul Erdős in 1946, has been a challenge for mathematicians for over seven decades. OpenAI’s breakthrough earlier this year set a new benchmark in automated reasoning, and now Anthropic's Claude Mythos has matched that feat with a "cute, simple proof," according to Sholto Douglas, an engineer at Anthropic.

## Under the Hood

To achieve this result, Anthropic employed a test system similar to the one used by OpenAI. The process involves isolated instances of Claude Code, each equipped with Mythos access, receiving the problem and developing potential solution paths. These instances then summarize and distribute their findings to other independent instances for further refinement.

• Test System Setup:
  • Isolated Instances: Each instance operates independently to ensure a diverse range of approaches.
  • Mythos Access: Claude Code instances are given access to Mythos, allowing them to leverage its advanced reasoning capabilities.
  • Solution Paths: Multiple paths are developed and summarized before being distributed for further refinement.

Mathematician Daniel Litt noted that while the result from Mythos is "a bit worse" than OpenAI’s, it still demonstrates significant progress. The proof version prepared by Opus 4.7, an earlier model, was also published by Anthropic, providing transparency into their methodology.

Comparisons with Other AI Systems

Google DeepMind recently announced that its AI-assisted system solved nine Erdős problems using the formal proof language Lean. While this approach is less impressive from a pure LLM perspective, it highlights the versatility and effectiveness of combining different AI techniques.

• DeepMind’s Approach:
  • Formal Proof Language: Relies on Lean, which is more structured and rigorous.
  • Cost Efficiency: Solved problems for "a few hundred dollars," showcasing cost-effective solutions.

Claude Code, being an agentic harness rather than a pure LLM, leverages both language modeling and agent-based capabilities to tackle complex problems. This hybrid approach allows it to explore multiple solution paths more efficiently.

## Key Takeaways

• Significant Milestone: Both OpenAI and Anthropic have demonstrated the capability of AI models to solve long-standing mathematical problems.
• Diverse Approaches: Different AI systems are using various methods, from pure LLMs to formal proof languages, each with its own strengths.
• Future Implications: These advancements could lead to more efficient and cost-effective solutions in mathematics and other fields, potentially democratizing access to advanced AI tools.

The rapid progress in AI-driven mathematical reasoning is not only a testament to the capabilities of modern language models but also a promising step towards solving even more complex problems in the future. As researchers continue to refine these systems, we can expect to see more groundbreaking achievements in both mathematics and beyond.