Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers.

Question

Let $n$ be a positive integer. Show that $n$ and $2n$ have the same number of representations as a sum of two squares of non-negative integers.

Hint: $(a^2+b^2)(c^2+d^2)=(ac+bd)^2+(ad-bc)^2$

Answer 1

If a pair of numbers $(x_0;y_0)$ it is a solution of $x^2+y^2=n$, then a pair of numbers $(x_0-y_0; x_0+y_0) -$ it is a solution of $x^2+y^2=2n$.

If a pair of numbers $(x_1;y_1)$ it is a solution of $x^2+y^2=2n$, then $x_1, y_1$ of have the same parity and a pair of numbers $\left(\frac{x_1+y_1}{2};\frac{y_1-x_1}{2} \right) -$ it is a solution of $x^2+y^2=n$.