# Investigating number of equivalence classes in the Gaussian integers formed by adding integers.

I'm trying to find how many equivalence classes on the Gaussian integers can be formed just by adding integers (as part of a wider consideration on how many there are altogether).

Let $$\gamma \in \mathbb{Z}[i]$$

Consider $$\mathbb{Z}[i]_{mod\gamma}$$.

The set is not infinite so there exists $$z \in \mathbb{Z}$$ such that $$\gamma + z$$ loops back to the equivalence class of $$[0]_\gamma$$. I'm trying to find the lowest such $$z$$.

So I considered $$\gamma = a + bi \Rightarrow \gamma + z = a + z + bi$$.

If $$\gamma + z \in [0]_\gamma$$ then $$\frac{(a+z+bi)(a-bi)}{(a+bi)(a-bi)} = \frac{(a^2 + b^2 + za) -zbi}{a^2+b^2} \in \mathbb{Z}[i]$$

Since I'm only investigating adding integers for now I considered:

$$\frac{a^2 + b^2 + za}{a^2 + b^2} \in \mathbb{Z}$$

This implies $$a^2 + b^2 + za = (a^2+b^2)k$$ for some $$k \in \mathbb{Z}$$

Rearranging: $$za = (a^2+b^2)k - a^2 + b^2 = (a^2+b^2)(k-1)$$.

So $$za = (a^2+b^2)l$$ for some $$l \in \mathbb{Z}$$

So $$z = (a^2 + b^2)\frac{l}{a}$$. But this is only true if l is a multiple of a.

So it follows $$z = (a^2+b^2)m$$ for some $$m \in \mathbb{Z}$$.

In other words, the integers which wrap back around to the equivalence class of $$[0]_\gamma$$ when added to $$\gamma$$ are the multiples of the norm $$N(\gamma)$$.

Therefore, there should be precisely $$N(\gamma)$$ equivalence classes that can be made just by adding integers.

The problem is I know (because it was told to me) that the number of equivalence classes is $$N(\gamma)$$, which means I must have gone wrong somewhere. My method implies that there are more than $$N(\gamma)$$ equivalence classes since I haven't even begun considering adding Gaussian integers.

Where did I go wrong?

• What makes you so sure that adding Gaussian integers creates any new equivalence classes? Commented Feb 9, 2023 at 17:10
• What is $\mathbb{Z}[i]_{\bmod \gamma}$? Commented Feb 9, 2023 at 17:13
• What makes you believe that "The set is not infinite", and that "there exists z∈Z [$\ne0$] such that γ+z loops back to the equivalence class of [0]γ"? Commented Feb 9, 2023 at 17:22

Show that if $$\gcd(a,b)=1$$ then $$(a+ib)\Bbb{Z}[i]\cap \Bbb{Z} = (a^2+b^2)\Bbb{Z}$$: if $$(a+ib)(c+id)= k$$ then $$c-id=\frac{c^2+d^2}{k}(a+ib)$$, $$\frac{c^2+d^2}{k}$$ must be an integer from which $$c+id$$ is a multiple of $$a-ib$$ and $$k$$ is a multiple of $$a^2+b^2$$.
From there $$\Bbb{Z}[i]/(a+ib)$$ is of characteristic $$g ((a/g)^2+(b/g)^2)$$ where $$g=\gcd(a,b)$$.