Hilbert's Tenth Problem Over the Rationals: OpenAI Says It's Unsolvable

About this video

Hilbert's tenth problem asks for one procedure that decides whether any polynomial equation with integer coefficients has a solution; for whole-number solutions Matiyasevich (building on Davis, Putnam and Robinson) showed in 1970 that no such algorithm exists, but for fraction (rational) solutions the question stayed open. OpenAI claims a negative answer over the rationals: no algorithm decides whether an integer polynomial, with the number of variables part of the input, has a rational zero. This is OpenAI's claim and has not yet been peer reviewed.

In the video, a search machine tests fractions denominator by denominator: x² + y² + z² = 1 gives exact hits immediately, while x² + y² + z² = 7 has none (a classical fact), so a blind search can never say "no". Every hit and count shown is exact.

The animation illustrates the result; it is not a proof.

Papers:
- Hilbert's tenth problem over the rational numbers: github.com/openai/math/blob/main/preprints/Hilberts-tenth-pro…
- A pointwise 2-converse for elliptic curves with rational two-torsion: github.com/openai/math/blob/main/preprints/A-pointwise-2-conv…

Original visuals (three.js) and synthesized sound: every tick is a batch of denominators tested, every bell a fraction landing exactly on the surface.

Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage

Every result explained: gradientdamage.com/p/004