#!/usr/bin/perl
use strict;
use warnings;
use WWW::Mechanize ;
my $mech = WWW::Mechanize->new();
 
foreach my $line (100982..106146) {
    next if 0.99 > rand ;
    print "Now processing $line \n" ;
    my $get_file = "http://securities.stanford.edu/filings-case.html?i
+d=".$line;
    my $filename = "file_".$line ;
        open OUT, ">$filename" or die $!;
        print "file $filename \n";
        $mech->get($get_file) ;
[download]

As a side note, as long as you are using double quotes, your code will probably read better if you interpolate your variables, a la:

my $get_file = "http://securities.stanford.edu/filings-case.html?id=$l
+ine";
[download]

You might also consider using a three-argument open and indirect filehandle, particularly since you don't seem to be closing OUT prior to opening the next file.

Also, if you are trawling everything off someone's server, it's considered polite to put a sleep in there. And you should check that you aren't violating terms of service.