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erdos problems ai mathematics

Of the recent wave of AI-assisted mathematics claims, a disproportionate number involve problems drawn from the enormous catalogue of questions attributed to Paul Erdős, the twentieth-century Hungarian mathematician who posed thousands of them over his career, many with small cash prizes attached and almost all in combinatorics, number theory and graph theory. Several of OpenAI's ten and Anthropic's five reported results this summer trace back to entries on Thomas Bloom's maintained online database of Erdős problems. That is not a coincidence, and understanding why tells you more about the limits of current AI mathematical reasoning than the headline solve counts do.

• Deep theory-building fields such as algebraic geometry, arithmetic geometry or large parts of analysis require constructing new conceptual frameworks, not finding a single object matching a specification.
• Many resolved Erdős problems were already known to be 'probably true' from partial results or computational evidence, narrowing the search considerably before any model was involved.
• Difficulty within the Erdős catalogue varies enormously; some entries are closer to exercises than to genuinely hard open questions, and announcements rarely specify which end of that range was targeted.
• A model finding one clever construction says little about its ability to sustain a multi-year, multi-paper research programme of the kind most genuine mathematical advances require.