Sums of squares and orthogonal integral vectors
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 twincomplete. 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2008
 arXiv:
 arXiv:0806.3943
 Bibcode:
 2008arXiv0806.3943G
 Keywords:

 Mathematics  Number Theory;
 11R52;
 52C07
 EPrint:
 23 pages, final version to appear in Journal of Number Theory