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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