Representations and primitive representations as sums of two squares

A representation of a nonnegative integer $n$ as a sum of two squares is an ordered pair $(x,y)\in {\mathbb{Z}}^2$ such that $n=x^2+y^2$.

It is primitive when $\gcd (x,y)=1$ (Common divisor, and the greatest common divisor $\gcd (a,b)$, with the convention $\gcd (0,0):=0$). Two representations are equivalent up to signs and order when one is obtained from the other by independently changing coordinate signs and possibly interchanging the coordinates, and essentially different when they are not so equivalent.

For a positive odd integer, a representation $(x,y)$ is normalized when $x$ and $y$ are positive, $x$ is odd, and $y$ is even.

The ordered-pair convention retains signs and order when a correspondence is being counted. Equivalence up to signs and order is invoked only when those symmetries are deliberately discarded.