Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.
Manuscripts (2)
Hilbert’s tenth problem over the rational numbers
October 6, 2026 · source · cite as \cite{OAI:Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026}
We give a negative answer to Hilbert's tenth problem over the rational numbers: no algorithm decides whether a polynomial with integer coefficients has a rational zero. The number of variables is part of the input.
A pointwise 2-converse for elliptic curves with rational two-torsion
October 7, 2026 · source · cite as \cite{OAI:A-pointwise-2-converse-for-elliptic-curves-with-rational-two-torsion-October-7-2026}
We prove a pointwise 2-converse for elliptic curves over \mathbf Q with nonzero rational two-torsion: if the 2^\infty-Selmer corank is zero or one, then the analytic rank and Mordell–Weil rank equal that corank, and the Shafarevich–Tate group is finite. The result allows arbitrary reduction at 2.
Result 004 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md