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A proof that Z\mathbb{Z} is Diophantine in Q\mathbb{Q} through congruent-number twists

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theorymath.LO — Logic

Contributed by Manvir Jaswal ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

We give a proof of Glazer and collaborators' theorem that Z\mathbb{Z} is Diophantine in Q\mathbb{Q}, which builds on OpenAI's proof that Hilbert's tenth problem over Q\mathbb{Q} has a negative answer. Integers are represented by multiples of a point on an elliptic curve of rank one over Q(2)\mathbb{Q}(\sqrt2). An existential condition on two congruent-number twists forbids poles of odd order at odd primes. A height bound at six rational points, from a quaternionic Shimura curve and Serre's conjecture (a theorem of Khare–Wintenberger and Kisin), and a prime pattern from the Green–Tao–Ziegler theorem turn congruences at chosen primes into identities between multiples, and the growth of a Lucas sequence forces integrality. Our contribution is the existence of the rational points that the condition on twists requires: for coprime squarefree integers β>1\beta>1 and θ≥1\theta\ge1 with β≡θ≡1(mod8)\beta\equiv\theta\equiv1\pmod8, each a square modulo the primes of the other, there is an integer ee such that y2=x3−e2xy^2=x^3-e^2x and y2=x3−e2θ2xy^2=x^3-e^2\theta^2x both have rank one and a rational point with abscissa in βQ×2\beta\mathbb{Q}^{\times2}. Where Glazer and collaborators use a restricted 22-converse theorem, this rests on published theorems: the genus criterion of Tian, Yuan and Zhang, with Gross–Zagier.

Provenance statement

This paper and its results were produced by an AI research system that the author designed, built and directs, running on Anthropic's Claude Opus 5.5 in Claude Code. The author chooses the problems and prioritizes the work. Separate AI agents checked each result, and a second program confirmed the computations the proofs use, with any exceptions listed in the public deposit. The system also searched the literature and checked the citations against reliable sources. The author read the paper, reran the computations from the deposit and takes full responsibility for all of the paper's contents.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude Code
Python
PythonVersion 3.12

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Version history

  1. v1 · 30 pagesSubmitted by Manvir JaswalInitial depositCurrentNov 11, 2026