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Sums of squares and orthogonal integral vectors. (English) Zbl 1268.11152

Summary: Two vectors in \(\mathbb Z^{3}\) are called 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.

MSC:

11R52 Quaternion and other division algebras: arithmetic, zeta functions
11E25 Sums of squares and representations by other particular quadratic forms
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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