CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Rational isotropy does not solve an integral representation problem

Statement refuted

Rational isotropy of a quadratic form does not by itself solve an integral representation problem for a related binary form.

Facts & Assumptions

Given: The rational Hasse-Minkowski theorem (Hasse-Minkowski theorem over Q).

Counterexample

technique · direct

1.1

The ternary form $X^{2} - Y^{2} - 2 Z^{2}$ is rationally isotropic, for instance at the integer vector $(3, 1, 2)$ . This is consistent with Hasse-Minkowski theorem over Q, which is only a rational statement.

givenalgebra

2.1

The related integral representation problem $x^{2} - y^{2} = 2$ has no integer solution, because $(x - y) (x + y) = 2$ and the two factors must have the same parity, while every factorization of $2$ uses one odd factor and one even factor. So rational isotropy does not automatically produce an integral representation.

step 1.1givenalgebra∎

Depends on

Hasse-Minkowski theorem over Q

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sam Raskin, Introduction to the Arithmetic Theory of Quadratic Forms, section 4.7 (standard reference, not scraped)