AI Revolutionizes Solutions to Erdos Problems

In recent months, artificial intelligence (AI) has made significant strides in solving longstanding mathematical conjectures known as Erdos problems, named after the 20th-century mathematician Paul Erdos. These problems, totaling 1,179, range from minor curiosities to central issues in number theory and combinatorics. Mehtaab Sawhney, a mathematician at Columbia University, discovered problem #339 while exploring the Erdos problems website. Intrigued by its simplicity, he utilized ChatGPT to assist in his research, leading to the identification of references that had previously eluded him.

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Sawhney collaborated with Mark Sellke, a former academic now at OpenAI, to prompt ChatGPT in uncovering solutions to nine additional Erdos problems and partial solutions to eleven more. This collaboration has contributed to a surge in activity on the Erdos problems website, with AI tools reportedly helping to classify around 100 problems as "solved" since October. The AI's role has primarily involved conducting extensive literature searches and synthesizing existing theorems, sometimes resulting in original proofs with minimal human input.

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Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology, noted that while AI tools have proven useful as research assistants, they are not without limitations. Despite the excitement surrounding AI's capabilities, no major mathematics journal has yet published a peer-reviewed proof that cites the use of large language models (LLMs). However, the landscape may soon change as the integration of AI in mathematical research continues to evolve.

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The First Proof initiative aims to further test AI's mathematical abilities by challenging LLMs to produce proofs for ten selected problems within a week. This project has generated considerable enthusiasm, with numerous claimed solutions flooding social media and email. However, the verification process poses challenges, as many submissions lack accuracy. Daniel Litt, a mathematician at the University of Toronto, expressed concern over the quality of the generated proofs, noting that while some are impressive, a significant portion is erroneous.

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Critics like Litt and Carlo Pagano, who has worked with Google’s DeepMind on Erdos problems, caution against overhyping individual results. They emphasize the need for AI to tackle problems of broader significance beyond the Erdos conjectures. Sawhney, who has taken a leave from Columbia to work with OpenAI, believes that the integration of AI in mathematics will fundamentally alter the discipline. As the field progresses, the potential for AI to reshape mathematical research practices becomes increasingly evident.

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• “There’s a lot of excitement, which is really great to see,” — Lauren Williams, Harvard University Mathematician

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• “It’s absolutely very impressive that the models are sometimes able to generate correct answers to some of the problems,” — Daniel Litt, University of Toronto Mathematician

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• “It’s clear that this will change how we do math,” — Mehtaab Sawhney, Columbia University Mathematician

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While many AI-generated proofs are being claimed, the accuracy of these solutions remains uncertain. Litt has observed that a majority of the submissions are incorrect, highlighting the need for careful verification. The ongoing developments in AI-assisted mathematics warrant close attention as researchers navigate the balance between innovation and accuracy.

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