Hilbert's tenth problem asks for an algorithm deciding whether an arbitrary polynomial with integer coefficients has a zero with all coordinates in the integers. It is problem 10 of Hilbert's problems. No such algorithm exists for integer solutions (Sun 2026).
The rational analog asks for a zero with all coordinates in the rational numbers. Sun (2026) reports that this problem is recursively undecidable even for polynomials in 86 variables. This is an upper bound on a sufficient number of variables, not a claim that 86 is minimal. The proof uses arithmetic inputs from OpenAI (2026). Those inputs and the resulting rational undecidability claim have not been independently verified here. The upstream release attributes its manuscripts to an internal OpenAI model (OpenAI 2026).
