Factorising $(1+3i)$ into the product of two Gaussian integers. So first I apply the G.I norm on this and obtain $\|1+3i\|=10=2\times 5$, so I expect the first Gaussian integer to have norm $2$ and the second to have norm $5$.
Do I now have to guess? I.e. With norm $2$ there are only $\pm 1 \pm i$ and with norm $5$ there are only $\pm 1\pm 2i$ and $\pm 2 \pm i$
The answer is in fact $(1+i)(2+i)$ so I can see that guessing does work eventually.
How do I remove more cases efficiently?
In the case of $1 + i$ all this numbers are associates (i.e. they differ by an invertible multiple). So it is enough to check only one of them.
In the other case there are two groups of associates, so you just need to check divisibility by $1\pm 2i$.