Star Fleet Math Solves 19 Erdős Problems, Including $250 Prize Challenge, With Lean-Verified Codex AI Proofs

A research collective named "Star Fleet Math" has published results claiming to have solved 19 open Erdős problems, with 13 of these solutions including formal proofs verified in the Lean theorem prover. This achievement was made possible by running 20 parallel Codex AI accounts to explore proof strategies and generate over 750,000 words of mathematical reasoning.

AI's Role in Tackling Decades-Old Mathematical Challenges

The Erdős problems, named after the prolific Hungarian mathematician Paul Erdős, are a collection of open questions that have challenged mathematicians for decades, some for over 50 years. These problems often come with monetary prizes for their solutions, underscoring their difficulty and importance in number theory and other mathematical fields.

Star Fleet Math's methodology involved a sophisticated interplay between AI and formal verification. Codex AI, OpenAI's coding agent, was instrumental in generating and exploring potential proof strategies. This exploratory phase produced an immense volume of mathematical reasoning, totaling over 750,000 words across the parallel AI sessions. Following this, the Lean theorem prover was used to formally verify the correctness of the proposed solutions, ensuring their mathematical rigor and eliminating potential human error.

Solving Erdős Problem #123: A $250 Prize Challenge

Among the 19 problems reportedly solved, Erdős Problem #123 stands out. This particular problem, a $250 prize challenge in number theory, has long eluded a definitive solution. The team's success in providing a formal proof for this problem, verified in Lean, represents a concrete demonstration of AI's capability in advanced mathematical research. The formal proof for Erdős Problem #123 has been submitted to the official Erdős Problems database, maintained by Thomas F. Bloom, for review and validation.

The Synergy of AI and Formal Verification

This project highlights a powerful synergy between large language models like Codex and formal verification systems such as Lean. While AI can rapidly generate and test hypotheses, formal provers provide the absolute certainty required in mathematics. This combination could accelerate the pace of discovery in fields traditionally reliant on human intuition and painstaking manual verification.

• Exploration: Codex AI efficiently explored vast solution spaces and generated potential proof structures.
• Verification: The Lean theorem prover ensured the logical soundness and correctness of the proposed solutions.
• Scale: Running 20 parallel AI accounts allowed for an unprecedented scale of mathematical reasoning.

Implications for Future Mathematical Research

The successful application of AI to solve long-standing mathematical problems suggests a significant shift in how research might be conducted. This approach could empower mathematicians to tackle even more complex problems by offloading the laborious and repetitive aspects of proof generation and verification to AI systems. It also opens new avenues for exploring mathematical conjectures that were previously too intricate for human teams alone.

This development is a significant milestone in AI news, demonstrating the growing capabilities of AI tools in areas requiring deep logical reasoning and creativity. As these technologies mature, we can expect to see further advancements in autonomous mathematics research.

Key Takeaways

• Star Fleet Math claims to have solved 19 open Erdős problems using AI.
• 13 solutions include formal proofs verified by the Lean theorem prover.
• The team utilized 20 parallel OpenAI Codex AI accounts.
• Erdős Problem #123, a $250 prize problem, was among those solved.
• This project generated over 750,000 words of mathematical reasoning.

Sources

• GitHub - neelsomani/gpt-erdos: A repo to document candidate solutions for Erdős problems produced via LLM-driven proof search · GitHub
• Towards Autonomous Mathematics Research
• AI contributions to Erdős problems · teorth/erdosproblems Wiki · GitHub
• AI contributions to Erdős problems · teorth/erdosproblems Wiki · GitHub
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