$\mathbb Z$ is Diophantine in $\mathbb Q$

A polynomial $P(q,w_1,\dots,w_k)$ with integer coefficients defines $\mathbb Z$ in $\mathbb Q$ when, for every rational number $q$ , $q$ is an integer exactly when $P(q,w_1,\dots,w_k)=0$ for some rational numbers $w_1,\dots,w_k$ . The polynomial must be explicit: a public list of its terms, or a public program that writes them out. Results rank by degree first, lower is better; then by the number of variables, $k+1$ ; then by the number of nonzero terms; then by the largest absolute coefficient. A stated upper bound, such as degree below $2\uparrow\uparrow 7$ , counts as that value. Elliot Glazer expects that, with no limit on the degree, a definition with under 100 variables is within reach; the open question is then the fewest variables of a quartic. The chart plots the number of variables. Koenigsmann (2016) showed that $\mathbb Q\setminus\mathbb Z$ is Diophantine in $\mathbb Q$ . Whether $\mathbb Z$ itself is Diophantine in $\mathbb Q$ was open, and Mazur's conjectures (1992) predicted that it is not.

These results are Claimed: their authors published them, and we have not checked them.

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Claimed Human Verified Lean Verified

Rank	Bound	Evidence level	Author	Date	Proof scope
1	$12{,}504$ Exact value and conditions $12504$ Version 2 of the manuscript, a collaboration using the collective agentic notes of the six authors. The polynomial has degree 4, 135,721 nonzero terms and largest absolute coefficient 685,783,949,153,203,534,233,601, which is less than $2^{80}$ . Its complete list of terms is public. An earlier manuscript by Sohail Farhangi with ChatGPT 6 (11 October 2026) assumes four arithmetic inputs and gives no number of variables, so it is not listed.	Claimed	Show authors Gabe Conant Sohail Farhangi James Freitag Elliot Glazer Jenn Park Caroline Terry	Nov 5, 6:59 PM PT	Proof scope A written proof in a manuscript, with no human review yet. A certificate check, published with the polynomial, shows that the polynomial is equivalent to the system of equations of the manuscript; it does not check the arithmetic argument. The Lean development is incomplete, so it is not a Lean proof of our statement.
2	$211{,}983{,}774{,}909$ Exact value and conditions $211983774909$ A pull request in qqwref's copy of Elliot Glazer's repository, merged in that copy only. It simplifies the first construction and replaces its catalogue of primes. The polynomial has degree less than $2^{2^{45}}$ , and each coefficient is less than $2^{2^{2^{56}}}$ in absolute value. The pull request does not state the number of nonzero terms.	Claimed	qqwref	Nov 5, 3:50 PM PT	Proof scope Changes to the written proof and the construction, with finite checks. No human review and no Lean proof.
3	$292{,}890{,}231{,}905$ Exact value and conditions $292890231905$ The manuscript says that it was developed primarily by OpenAI Codex agents (Astra 6 and Sol 6.1) and Anthropic Claude agents (Opus 5.5), with auditing from Fable 5.1. The polynomial has degree less than $2\uparrow\uparrow 7$ , and each coefficient is less than $2\uparrow\uparrow 8$ in absolute value, where $2\uparrow\uparrow n$ is a tower of $n$ twos. The manuscript does not state the number of nonzero terms. It builds on OpenAI's preprint of 6 October 2026 on Hilbert's tenth problem over the rational numbers. The time shown is the first commit of the manuscript to its public repository.	Agent breakthrough	Elliot Glazer	Nov 4, 11:50 AM PT	Proof scope A written proof in a manuscript, with no human review yet. The repository's checks rebuild the polynomial and run finite tests; they do not check the arithmetic argument. The Lean development is incomplete, so it is not a Lean proof of our statement.

$\text{number of variables}$ , lower is better $\text{number of variables}$ log scale, lower is better

ClaimedHuman VerifiedLean Verified

\text{number of variables}

(log scale)

Relevant repos

elliotglazer/diophantine-z-q: The manuscripts, both polynomials and their checks. The README states what the checks cover and that the Lean development is incomplete.
qqwref/diophantine-z-q: qqwref's copy of the repository, with the simplified construction.

Z\mathbb ZZ is Diophantine in Q\mathbb QQ

number of variables\text{number of variables}number of variables, lower is betternumber of variables\text{number of variables}number of variableslog scale, lower is better

Relevant repos

$\mathbb Z$ is Diophantine in $\mathbb Q$

$\text{number of variables}$ , lower is better $\text{number of variables}$ log scale, lower is better