Sorry, I don't understand really what your input and desired output is. Thus I can't also not follow what your code wants to do. Can you give us an example of your input and a description what good numbers are.
However, of course, I can rewrite your code to shorten it and make it more Perlish :-) I assume, that the data comes from a file.

use Tie::CSV_File;
tie my @array, "Tie::CSV_File", $filename, WHITESPACE_SEPARATED;
for my $i (1 .. $#array) {
    push @good_nums, 
       $array[$i-1]->[0] * $array[$i]->[1] *
       $array[$i-1]->[1] * $array[$i]->[0];
    # Did you really want to compute 4x a multiplication ?!
}
print @good_nums;
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Well, I still don't know what's going on :-(
If the data doesn't come from a file, you could also achieve quite the same, all you have to do is to nest the data:

@array = map {[split]} @array;
# instead of the first two lines of my snippet
[download]

Greetings,
Janek