 Primary Gaussian integer

An odd Gaussian integer is primary when it satisfies the displayed congruence. The four units have distinct images in $(\mathbb{Z}[i]/(1+i{)}^3{)}^{\times }$, a group of order four; hence every odd ideal has a unique primary generator. Indeed $v_{1+i}(-2)=2$ and $v_{1+i}(i-1)=v_{1+i}(-i-1)=1$.