you can escape the special chars stored in the variable $constant using \Q and \E. See also perlre.

     if($_ =~ m/^(\Q$constant\E)/) {
       $line_count++;
     }
[download]

Since $constant is user specified you could simply call quotemeta on the user input before using it. Then you won't need \Q and \E in your regex. E.g.:

my $searchterm = quotemeta $user_input;
[download]

You are currently only counting lines which start with that pattern. You wrote that you wanted to count all occurrences of the pattern. If you want to match all the occurrences of $constant per line you could change your code to:

     while ($_ =~ m/\Q$constant\E/gc) {
       $count++; 
     }
[download]

Other thoughts:

• use lexical filehandles (open my $FILE ... instead of open FILE)
• $constant doesn't seem like a fitting variable name to me. How about $searchterm?
• in case you only open the file to look for $constant: don't loop through the file if $constant is empty/undef. You currently have the check _within_ the while (<FILE>) loop. So even if you don't have a valid searchterm you are still looping through the entire file.

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