Computing GCDs over the Gaussian Integers

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I wrote up a very simple Perl script for computing GCDs over the Gaussian integers. It comes with no guarantee or proof of correctness. One can sketchily argue something like this however:

Claim: \mathbb{Z}[i] is a Euclidean domain, and hence a PID, and hence a UFD.


Consider the norm function \nu(a + bi) = a^2 + b^2. We write the standard norm on the complex plane as ||a+bi||_{\mathbb{C}} = \sqrt{a^2 + b^2}. We show that for all x \neq 0 and y we have,  y = qx + r where \nu(r) = 0 or \nu(r) < \nu(x). Consider x and y as points on the complex plane. Since x \neq 0 we have that y/x is another point in the complex plane. Consider \mathbb{Z}[i] as the points with integer coordinates in \mathbb{C}. For any point \zeta \in \mathbb{C} we can find a \zeta' \in \mathbb{Z}[i] such that ||\zeta - \zeta'||_{\mathbb{C}} \leq \sqrt{2}/2 by taking the \zeta' to be the nearest lattice point to \zeta. Since any point on the plane is contained in a square whose vertices are lattice points and whose side lengths are one, there must be a lattice point nearer than half the diagonal of the square. We then take z to be the nearest lattice point to y/x. We then have y = zx + (y - zx). We compute the norm of \nu(y - zx). We have:  \sqrt{\nu(y - zx)} = ||y - zx||_{\mathbb{C}} = ||x||_{\mathbb{C}} \cdot ||y/x - z||_{\mathbb{C}} \leq ||x||_{\mathbb{C}} \frac{\sqrt{2}}{2} < ||x||_{\mathbb{C}} = \sqrt{\nu(x)} It follows that \nu(y - zy) < \nu(x). We obtain that \mathbb{Z}[i] is a Euclidean domain. Thus \mathbb{Z}[i] is a PID and hence a UFD by results given in class.



#!/usr/bin/perl
use Math::Complex;

## A Quick hack for computing GCDs of Gaussian integers.

$z2 = 857 + i;
$z1 = 255;


sub gcd {
	# the Euclidean algorithm

	my $x = $_[0];
	my $y = $_[1];
	
	if ($x * $y == 0) {
		print "Done!\n";
	} else {
		$q = &approx($x/$y); 
		$r = $x - $q*$y;
		print "($x) = ($q)($y) + ($r)\n";

		&gcd($y,$r);
	}
}

sub approx {
	# find the nearest Gaussian integer to a point on the complex plane

	my $z = $_[0];

	my $x = int(Re($z));
	my $y = int(Im($z));

	if (abs($z - (($x+1) + i*$y) ) < 1/sqrt(2)) {
		return ($x+1) + i*$y;
	} elsif (abs($z - (($x) + i*($y+1)) ) < 1/sqrt(2)) {
		return $x + i*($y+1);
	} elsif (abs($z - (($x+1) + i*($y+1)) ) < 1/sqrt(2)) {
		return ($x+1) + i*($y+1);
	} else {	
		return $x + i*$y;
	}	

	
}

&gcd($z1, $z2);