How do I find $a,b\in\mathbb{Z}$ s.t. $\{ac-bd+i(ad+bc)\mid c, d\in\mathbb{Z}\}$ have real and imaginary parts both even or both odd?

Solution 1:

Conceptually it boils down to: $\,\ \alpha\beta\,$ is even $\iff \alpha\,$ or $\,\beta\,$ is even, $ $ for $\,\alpha,\beta\in\Bbb Z[i],\,$ once we generalize the notion of "even" appropriately for Gaussian integers.

Hint for $\,p := (2,i\!-\!1)$ we have $\Bbb Z[i]/p \cong \Bbb Z/2\,$ so $\,p\,$ is prime & induces a parity structure on $\,\Bbb Z[i]\,$ via $\,\alpha := a+bi\,$ is $\rm\color{#c00}{even}$ $\iff p\mid\alpha \iff 2\mid a+b\iff a,b\,$ are $\rm\color{#c00}{equal\ parity}$. Hence

$$\begin{align} &\alpha\beta\,\ \text{is even},\ {\rm for}\,\ \beta = c+di\\[.2em] \iff\ &p\mid \alpha\beta\\[.2em] \iff\ & p\mid \alpha\,\ {\rm or}\ \,p\mid \beta,\,\ \text{by $\,p\,$ prime}\\[.2em] \iff\ &\alpha\,\ \text{is even or}\,\ \beta\,\ \text{is even}\\[.2em] \iff \ & a\equiv b\,\ {\rm or}\,\ c\equiv d\!\!\pmod{\!2}\end{align}\qquad\qquad$$