Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sums of squares and orthogonal integral vectors   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.

29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
27
The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin