Hilbert’s tenth problem over ℚ

What OpenAI claims

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.

Papers (2)

1. Hilbert’s tenth problem over the rational numbers

   6 October 2026 · Preprint, not peer reviewed · PDF

   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.

2. A pointwise 2-converse for elliptic curves with rational two-torsion

   7 October 2026 · Preprint, not peer reviewed · PDF

   We prove a pointwise 2-converse for elliptic curves over Q with nonzero rational two-torsion: if the 2^∞-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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.