HILBERT'S TENTH PROBLEM ON FRACTIONS: 86 UNKNOWNS ARE ENOUGH

Hilbert’s tenth problem, posed in 1900, asks for an algorithm that takes any polynomial equation with integer coefficients, P(x₁, …, xₙ) = 0, and decides whether it has a solution in whole numbers. In 1970, Matiyasevich, building on work by Davis, Putnam and Robinson from 1961, proved that no such algorithm exists. The problem is undecidable.

Mathematicians then started counting. How many unknowns does an equation need before the question becomes undecidable? Zhi-Wei Sun, at Nanjing University, had shown that 11 suffice for whole numbers, and recently brought it down to 10.

The same question, for fractions

Ask the question for rational numbers — fractions — instead. The classic route would go through whole numbers: if you could single out the integers among the rationals using a polynomial equation, Matiyasevich’s theorem would carry over. The paper lists the milestones. Robinson gave a logical definition of the integers inside the rationals in 1949, simplified by Poonen in 2009. Koenigsmann showed in 2016 that the non-integers can be described by such an equation, and later work cut the number of unknowns needed for that to 10 (Daans, 2024), then 7 (Sun).

Then, the paper says, “quite recently, OpenAI released a preprint” proving that Hilbert’s tenth problem over the rationals is undecidable. That preprint, dated 6 October 2026 and made available on a GitHub repository of OpenAI, rests on what Sun calls a “five-contact height estimate” and a construction that excludes “odd poles”. How it was produced is not described in Sun’s paper.

Theorem 1.1

Sun’s contribution is a number. Theorem 1.1: HTP₈₆(ℚ) is undecidable. In words: there is no algorithm that decides, for every polynomial f(x₁, …, x₈₆) with integer coefficients, whether f = 0 has a solution in fractions.

The proof feeds his own 10-unknown result for whole numbers into the machinery of the OpenAI preprint. Ten points on a fixed elliptic curve — here y² = x³ − d₀²x over the numbers a + b√2, with d₀ = 75 − 53√2 — encode ten whole numbers. Special primes, detected by the height estimate, let the equation check whether the input polynomial vanishes at those ten numbers. A finite list of “guards” rules out the false solutions that remain. Run alongside a plain search for whole-number solutions, this turns any oracle for fractions into a decision procedure for whole numbers — which cannot exist.

Saving unknowns

Most of the 42 pages go into economy. Sun lists five strategies. A change of coordinates moves the five “contacts” of the height estimate to finite points. A single “slope” variable serves twenty-one checks at once. One shared group of eleven points on the curve replaces many separate ones — which requires building whole families of primes that fit together, using a generalised Chinese remainder theorem and results by Tao and Ziegler on polynomial patterns in the primes, among others. The checks are reorganised so that 33 conditions replace 42. And at the end, every leftover algebraic choice and every “this must not be zero” condition is folded into one final variable.

The count, written out in the paper:

2(10 + 11) + 1 + 2 · 21 + 1 = 86

— two coordinates for each of the 21 points on the curve, one shared slope, two helper unknowns for each of 21 checks, and one last variable that absorbs the rest.

Not the last word

Sun is explicit that 86 is “an upper bound supplied by this construction”, not a claim that fewer unknowns would fail. On whole numbers, the figure went from 11 to 10; on fractions, the race has just started.

The result also stacks on recent work that has not been through peer review, according to the references: the OpenAI preprint from early October and Sun’s own 10-unknown paper, posted on arXiv this year. Sun’s paper, too, is a preprint. Its 42 pages of number theory — sieves, prime patterns and elliptic curves — will now be checked by other mathematicians.