Some comments:

opendir(DIRE, "/home/lenn287/public_html/diary");
my @files_exist = reverse sort  map { $_ =~ s/^(\d+)\.html$/$1/; $_} 
     grep { /^\w+.\.html$/} readdir DIRE;
closedir DIRE;
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Here your first regexp checks for (\w) and the second matches digits, so if you have a file called "bleh.html" it will pass the first grep but it will produce an empty entry on the display. Change the \w+ into \d+.

open my $FH,'<', '/home/lenn287/public_html/diary/'.$file_name.".html"
+;
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You don't check if your open succeeded or not.

for($first_file; $first_file<=$last_file;$first_file++) {
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Since you're coding Perl:

for my $f ($first_file .. $last_file) {
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On another note, since you want to search within the files, you should either create an index of some sorts, instead of opening each file and searching, or use a database to store the contents and then you can use fulltext search on them.

just a couple of cents

--
Leviathan.