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arXiv · 2608.24854

Metaconjugation and Quadratic Forms

Adriana Cardoso·António Leite·António Machiavelo·Rafael Moreira·Luís Roçadas
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Abstract

We present a quaternionic proof that the quadratic form $t^2+2x^2+5y^2+10z^2$ represents all positive integers, and that the form $t^2+x^2+7y^2+7z^2$ represents all natural numbers which are neither equal to $3 \cdot 7^\ell$ nor equal to $6 \cdot 7^\ell$, for any $\ell \in \mathbb{N}_0$. To do this we introduce a technique that may be useful for other purposes, and that we call "metaconjugation".

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Adriana Cardoso, António Leite, António Machiavelo, Rafael Moreira, Luís Roçadas. 2026-08-25. Metaconjugation and Quadratic Forms. https://arxiv.org/abs/2608.24854

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