Bounded Representations by $x^2+y^2-z^2$

Przemyslaw Chojecki

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arXiv · 2603.18087

Bounded Representations by $x^2+y^2-z^2$

Przemyslaw Chojecki

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Abstract

We prove that every sufficiently large integer $n$ can be written in the form $n=x^2+y^2-z^2$ with $\textrm{max}(x^2,y^2,z^2)\le n$. The proof converts the problem into finding a primitive binary quadratic form of positive discriminant $4n$ inside a fixed relatively compact open patch of the real hyperboloid $b^2-4ac=4n$. This is then supplied by Duke's theorem in the precise point-counting form deduced from the measure-theoretic duality of Einsiedler-Lindenstrauss-Michel-Venkatesh. A finite parity correction returns to the original ternary variables. This settles Erd\H{o}s Problem 1148.

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BibTeX RIS

Przemyslaw Chojecki. 2026-03-18. Bounded Representations by $x^2+y^2-z^2$. https://arxiv.org/abs/2603.18087

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