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arXiv · 0806.3943

Sums of squares and orthogonal integral vectors

Lee M. Goswick·Emil W. Kiss·Gabor Moussong·Nandor Simanyi
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Abstract

Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.

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Lee M. Goswick, Emil W. Kiss, Gabor Moussong, Nandor Simanyi. 2011-08-10. Sums of squares and orthogonal integral vectors. https://arxiv.org/abs/0806.3943

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