After seeing the difference in outputs between the OP code and

liverpole's version, and then actually looking up the references at wikipedia,

~~I realized that there were other problems besides execution speed: way too many spots were being drawn in black, which meant way too many numbers were being categorized as primes~~ *I felt compelled to figure out who was wrong; I have updated my earlier reply accordingly.*
I also noticed that the OP code actually draws all the points (from 1 to $max), both the supposed primes and the supposed non-primes. But you really only need to draw one set, and just leave the other set as "background color".

Then, I thought it would be helpful to get a sense of what took longest -- computing or drawing -- so I decided to refactor the code as follows:

- run through all the desired numeric range before doing any drawning, keeping track of the x/y coords as we go;
- each time we find a prime number, store that as both a hash key and as an item in an ascending array; use the hash value to store the x/y coords where the dot should be drawn for that prime;
- once all the primes have been found, set up the Tk::Canvas, then run through the sorted list of prime values (which are hash keys), and draw a dot for each one.

As it turns out, the computing takes about twice as long as the drawing (4 sec vs. 2 sec for a max number of 150,000, which pretty well fills the given grid); but bear in mind the compute step loops over all integers in the given range, while the drawing step only loops over the primes.

And luckily, the resulting canvas looks a lot more like the pictures I saw over at wikipedia (it still might not be entirely correct -- but you could try dumping a portion of the %primecoord hash structure to make sure the values are as intended).

`#!/usr/local/bin/perl
use strict;
use Tk;
die "Usage: $0 max_number\n" # use the command-line, Luke!
if ( @ARGV != 1 or $ARGV[0] !~ /^\d+/ );
my $lastnumber = shift;
my ( $mincoord, $midcoord, $maxcoord ) = ( 50, 250, 450 );
plot_grid( compute_primes( $lastnumber ));
MainLoop;
{ # closure for building prime number hash structure
my @sorted_primes = ( 1 ); # primes in ascending order
my %primecoord = ( 1 => [$midcoord, $midcoord] );
# HoA: keys are primes, values are x/y coords in grid
sub ck_prime {
my ( $num, $x, $y ) = @_;
my $end = int( sqrt( $num ));
my $keep = 1;
for my $prime ( @sorted_primes ) {
if ( 0 == $num % $prime ) {
$keep = 0;
last;
}
elsif ( $prime > $end ) {
last;
}
}
if ( $keep ) {
$primecoord{$num} = [ $x, $y ];
push @sorted_primes, $num;
}
}
sub compute_primes
{
# list of directions for incrementing x, y coordinates:
my %vector = ( 0 => [ 1, 0 ], # rightward
1 => [ 0, 1 ], # downward
2 => [ -1, 0 ], # leftward
3 => [ 0, -1 ], # upward
);
my $x = my $y = 250; # starting position
my $direction = 0; # changes when we've gone $pathlen steps
my $traveled = 0; # no. of points drawn in current direction
my $pathlen = 1; # increments on every second direction chang
+e
my $changes = 0; # no. of dir. changes since last $pathlen in
+crement
my $number = 1;
my $bgn = time;
while ( ++$number <= $lastnumber )
{
$x += $vector{$direction}[0];
$y += $vector{$direction}[1];
last if ( $x < $mincoord or $x > $maxcoord );
if ( ++$traveled == $pathlen ) {
$direction = ++$direction % 4;
$traveled = 0;
if ( ++$changes % 2 == 0 ) {
$changes = 0;
$pathlen++;
}
}
ck_prime( $number, $x, $y );
}
my $end = time;
warn "Checked $lastnumber integers in ", $end - $bgn, " sec\n"
+;
return( \@sorted_primes, \%primecoord );
}
}
sub plot_grid
{
my ( $primes, $coords ) = @_;
my $mw = MainWindow->new;
my $c = $mw->Canvas(-width => $maxcoord+50, -height => $maxcoord+5
+0,
-background => 'white' )->pack;
$c->createLine( $mincoord, $midcoord, $maxcoord, $midcoord );
$c->createText( $mincoord-30, $midcoord, -fill => 'blue', -text =>
+ 'X');
$c->createLine( $midcoord, $mincoord, $midcoord, $maxcoord );
$c->createText( $midcoord, $mincoord-30, -fill => 'blue', -text =>
+ 'Y');
my $bgn = time;
for my $prime ( @$primes ) {
my ( $x, $y ) = @{$$coords{$prime}};
$c->createText( $x, $y, -fill => 'black', -text => "\xb7" );
}
my $end = time;
warn "Plotted ", scalar @$primes, " primes in ", $end - $bgn, " se
+c\n";
}
`

Note that I used "\xb7" (centered dot) as the character -- seemed like "." (period) was off-center.

(updated to correct the comments about the "%vector" hash, to reflect the fact that coords in Tk::Canvas put 0,0 at the upper left corner)

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