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\title{Gaussian Integers}
\author{Manuel Eberl}
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\begin{abstract}
The Gaussian integers are the subring $\mathbb{Z}[i]$ of the complex numbers, i.\,e.\ the ring of all complex numbers with integral real and imaginary part. This article provides a definition of this ring as well as proofs of various basic properties, such as that they form a Euclidean ring and a full classification of their primes. An executable (albeit not very efficient) factorisation algorithm is also provided.

Lastly, this Gaussian integer formalisation is used in two short applications:
\begin{enumerate}
\item The characterisation of all positive integers that can be written as sums of two squares
\item Euclid's formula for primitive Pythagorean triples
\end{enumerate}
While elementary proofs for both of these are already available in the AFP, the theory of Gaussian integers provides more concise proofs and a more high-level view.
\end{abstract}

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