# Arithmetics in Gaussian Integers $$\mathbb Z[i]$$

We have already seen that the norm of element $$a+bi\in\mathbb{Z}[i]$$ ($$a,b\in\mathbb{Z}$$) is $$N(a+bi)=a^2+b^2$$ and the units are $$\pm1$$ and $$\pm i$$. Therefore, all divisors of a prime element $$\pi\in\mathbb{Z}[i]$$ are $$\pm1,\pm i,\pm\pi,\pm i\pi$$.

Theorem 6

The Fundamental Theorem of Arithmetic (FTA) holds in the set of Gaussian integers $$\mathbb{Z}[i]$$.

The following proposition describes all prime elements in the set of Gaussian integers.

Theorem 7

An element $$x\in\mathbb{Z}[i]$$ is prime if and only if $$N(x)$$ is a prime or $$|x|$$ is a prime integer of the form $$4k-1$$, $$k\in\mathbb{N}$$.

Problem 3

Solve the equation $$x^5-1=y^2$$ in integers.

Problem 4

Suppose that $$x,y$$ and $$z$$ are natural numbers satisfying $$xy=z^2+1$$. Prove that there exist integers $$a,b,c,d$$ such that $$x=a^2+b^2$$, $$y=c^2+d^2$$ and $$z=ac+bd$$.

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