OpenAI model proves Hilbert's Tenth Problem false over Q, sidestepping 80-year approach
aran_nayebi · x · 2026-10-08
A slept-on result from OpenAI's latest release: a proof that Hilbert's Tenth Problem over Q is false (undecidable).
- For nearly 80 years, human approaches — stemming from Tarski's questions to Raphael Robinson and built on Julia Robinson '49 and Bjorn Poonen '03–09 — first defined the integers inside Q via existential conditions, then transferred undecidability from Z.
- This proof sidesteps that obstacle entirely, achieving undecidability without ever defining integers in Q. As the author understands it, it sets up a sequence of finite rational tests that, if all pass, produce an "integer-like" solution in a nonstandard setting; elliptic-curve/size constraints then force an ordinary integer solution, transferring undecidability from Z to Q.
- The author calls it remarkable out-of-the-box thinking.
Related event: OpenAI Model Proves Hilbert's Tenth Problem Undecidable over Rationals(4 posts)→
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