Move the push down outside the y loop, use another variable like y0 to hold the minimum value

#!/usr/bin/env perl
use strict;
use warnings;

my @primes = (2,3,5,7,11,13,17) ; 
my @start  = (0..209);
my @results =();

foreach my $x (@primes){
  my $y0=0;
  my @best = @start;
  my $count = scalar @best;
  print "\nprime $x $count\n";
  foreach my $y (0..$x-1){
    my @new = grep { ($_+$y) % $x } @best;
    my $new = scalar @new;
    print " ${y}mod${x} = $new\n";
    if ($new < @start){
      @start = @new;
      $y0 = $y;
    }
  }
  push @results, "[$y0,$x]";
}

print "$_\n" for @start;
print join(',',@results),"\n";
[download]