Gaussian integers are complex numbers whose real and imaginary parts are both integers. Give the Gaussian Integer a=8-53i, show that it cannot be expressed in the form $w^2+z^2$ where w and z are GI's. I came up with a rather complicated demonstration, would like to know if there is a simpler one
$a= -(-2+3j)^2(4+j)$. If a can be factored as $(w+iz)(w-iz)$ then the possible values for these two factors are either $(-2+3j)^2$ and $-4-j$ or $-(-2+3j)$ and $(-2+3j)(-4-j)$.
Solving for w gives, for the first factorization, u=-4.5 and for the second, u=4.5
In either case $u+kv$ is not a Gaussian integer
Render $(x+yi)^2=(x^2-y^2)+i(2xy)$. So how can a Gaussian integer squared (or a sum of two of them) give an odd imaginary part?