Mathematician digests OpenAI's number theory results; Hodge papers pulled over sign error
lpachter · x · 2026-10-11
Math blogger Persiflage (UCLA professor lpachter) reviews OpenAI's recent algebraic number theory results:
- Modularity of elliptic curves over imaginary quadratic fields: Caraiani–Newton had largely solved the X₀(15) rank-zero case; OpenAI overcame the remaining case of reducible mod 3 and mod 5 representations. The author sees the key move as "a really nice trick" a human could plausibly find, not a fundamental breakthrough. Bao Le Hung has already posted a paper on Hexagon Math extracting the trick, sufficing for all totally real fields and CM fields not containing ζ₅.
- Retraction: OpenAI has withdrawn some partial results towards the Hodge Conjecture after a sign error was found—"may be the best evidence of sentience yet."
- Footnote: the author dictated the whole post to ChatGPT due to RSI in his hands.
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