Hilbert's Tenth Problem

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). The upstream release attributes its manuscripts to an internal OpenAI model (OpenAI 2026).

The manuscript Is Diophantine in (2026) reports a Diophantine definition of the integers within the rational numbers. In the claimed construction, a fixed polynomial of total degree four, , has integer coefficients. For each , the equation has a solution in rational witness variables iff . Version 14 gives one construction with 299 total variables, whose coefficients are specified by exact symbolic expressions, and a second with 314, whose coefficients are listed in decimal. Both counts include the input . Such a definition would transfer undecidability from the integer problem to the rational problem by imposing the defining equation on each variable of the integer problem. Glazer (2026) credits Astra 6, Sol 6.1, and Claude Opus 5.5 with developing the proof and Fable 5.1 with auditing it. The Lean development supplies auxiliary results for the original construction, whose main formal declaration still contains a proof placeholder. Version 14 provides finite arithmetic checks of the coefficient tables and algebraic reductions, together with internal reviews of the arithmetic argument.