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Gaussian integer
A complex number of the form $a+bi$, where $a,b\in\mathbb{Z}$, is called a Gaussian integer.
It is easy to see that the set $S$ of all Gaussian integers is a subring of $\mathbb{C}$; specifically, $S$ is the smallest subring containing $\{1,i\}$, whence $S=\mathbb{Z}[i]$.
$\mathbb{Z}[i]$ is a Euclidean ring, hence a principal ring, hence a unique factorization domain.
There are four units (i.e. invertible elements) in the ring $\mathbb{Z}[i]$, namely $\pm 1$ and $\pm i$. Up to multiplication by units, the primes in $\mathbb{Z}[i]$ are

ordinary prime numbers $\equiv 3\mod 4$

elements of the form $a\pm bi$ where $a^{2}+b^{2}$ is an ordinary prime $\equiv 1\mod 4$ (see Thue’s lemma)

the element $1+i$.
Using the ring of Gaussian integers, it is not hard to show, for example, that the Diophantine equation $x^{2}+1=y^{3}$ has no solutions $(x,y)\in\mathbb{Z}\times\mathbb{Z}$ except $(0,1)$.
Mathematics Subject Classification
11R04 no label found5500 no label found55U05 no label found32M10 no label found32C11 no label found1402 no label found1800 no label found Forums
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Gaussian integers
A lot more could be said about the Gaussian integers. For instance, they are a subring of the complex numbers, their graph on the complex plane forms a lattice of "squares", and they are a Euclidean domain.