Goswick, Lee M.; Kiss, Emil W.; Moussong, Gábor; Simányi, Nándor Sums of squares and orthogonal integral vectors. (English) Zbl 1268.11152 J. Number Theory 132, No. 1, 37-53 (2012). 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. Cited in 2 Documents 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) Keywords:cubic lattice; Euler rotation matrix; Hurwitz integral quaternion PDFBibTeX XMLCite \textit{L. M. Goswick} et al., J. Number Theory 132, No. 1, 37--53 (2012; Zbl 1268.11152) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Conjectured complete list of squarefree numbers that can be written as a sum of at most two positive squares, but not as a sum of three positive squares. References: [1] Borwein, J.; Choi, K. K.S., On the representations of \(x y + y z + z x\), Experiment. Math., 9, 153-158 (2000) [2] Carmichael, R. D., Diophantine Analysis (1915), John Wiley & Sons · JFM 45.0283.11 [3] Chowla, S., An extension of Heilbronnʼs class number theorem, Quart. J. Math. Oxford, 5, 304-307 (1934) · JFM 60.0944.04 [4] Conway, J. H.; Smith, D. A., On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry (2003), A.K. Peters · Zbl 1098.17001 [5] Cox, D. A., Primes of the Form \(x^2 + n y^2 (1989)\), John Wiley & Sons: John Wiley & Sons New York [6] Grosswald, E.; Calloway, A.; Calloway, J., The representation of integers by three positive squares, Proc. Amer. Math. Soc., 10, 451-455 (1959) · Zbl 0092.04303 [7] Grosswald, E., Representations of Integers as Sums of Squares (1985), Springer-Verlag: Springer-Verlag New York · Zbl 0574.10045 [8] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1979), Clarendon Press: Clarendon Press Oxford · Zbl 0423.10001 [9] Hurwitz, A., Vorlesungen über die Zahlentheorie der Quaternionen (1919), Berlin · JFM 47.0106.01 [10] E. Kani, Idoneal numbers and some generalizations, preprint, 2009, available at http://www.mast.queensu.ca/ kani/papers/idoneal.pdf; E. Kani, Idoneal numbers and some generalizations, preprint, 2009, available at http://www.mast.queensu.ca/ kani/papers/idoneal.pdf · Zbl 1253.11049 [11] Mordell, L. J., The representation of integers by three positive squares, Michigan Math. J., 7, 289-290 (1960) · Zbl 0097.03102 [12] Pall, G., On the arithmetic of quaternions, Trans. Amer. Math. Soc., 47, 487-500 (1940) · JFM 66.0111.03 [13] Ribenboim, P., My Numbers, My Friends. Popular Lectures on Number Theory (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0947.11001 [14] Sárközy, A., On lattice-cubes in the three-space, Mat. Lapok (1961), (in Hungarian) · Zbl 0106.26105 [15] Weinberger, P. J., Exponents of the class groups of complex quadratic fields, Acta Arith., 22, 117-124 (1973) · Zbl 0217.04202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.