{"id":"openai-math/Hilberts-tenth-problem-over-the-rational-numbers","pdf":"https://github.com/openai/math/blob/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/Hilberts-tenth-problem-over-the-rational-numbers-October-6-2026","tags":["mathematical-logic","computability","diophantine-equations","number-theory"],"score":0,"title":"Hilbert’s tenth problem over the rational numbers","domain":"Number theory","family":"004","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":3,"published":"2026-10-06","reactions":{},"discussion":[{"url":"https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/#:~:text=As%20for%20Hilbert%E2%80%99s%20tenth%20problem%20itself%2C%20it%20seemed%20plausible%20to%20me%20that%20it%20was%20actually%20solvable%20over%20the%20rational%20numbers.","kind":"web","site":"galoisrepresentations.org","text":"As for Hilbert’s tenth problem itself, it seemed plausible to me that it was actually solvable over the rational numbers. I don’t think I would have put money on that, however. I still certainly hope that Hilbert’s tenth problem has a positive answer over the rational numbers for algebraic curves,…","added":"2026-10-24","title":"A Brief Digest of Some New OpenAI Results | Persiflage","author":"Persiflage"},{"url":"https://scottaaronson.blog/?p=10169#:~:text=this%20was%20arguably%20the%20biggest%20open%20problem%20in%20computability%20theory","date":"2026-10-07","kind":"web","site":"Shtetl-Optimized","text":"this was arguably the biggest open problem in computability theory (note that uncomputability of solving Diophantine equations, i.e. polynomial equations over the integers , was proved in the 1970s, giving a negative answer to Hilbert’s 10th Problem) Any of the above, alone, could easily have been…","added":"2026-10-24","title":"The Mathocalypse"},{"url":"https://arxiv.org/abs/2610.20691","date":"2026-10-14","kind":"web","site":"arXiv.org","text":"Recently, Hilbert's Tenth Problem over the field $\\mathbb Q$ of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction. In this paper, we prove further that there is no algorithm to decide for any polynomial $P(x_1,\\ldots,x_{86})$…","added":"2026-10-24","title":"86 unknowns for Hilbert's tenth problem over the rationals","author":"Zhi-Wei Sun","record":"arxiv/2610.20691"}],"formalized":false,"family_title":"Hilbert’s tenth problem over ℚ"}