I change the pack parameter from "N" to "L", maybe the "L" could be enough to store the line number, while the "N" is not long enough.

Both "N" and "L" are 32-bit usigned integers, so that's not going to make a (useful) difference.  They only differ with respect to byte order:

• "N" — 32-bit unsigned integer in big-endian byte order
• "V" — 32-bit unsigned integer in little-endian byte order
• "L" — 32-bit unsigned integer in native byte order of the architecture perl is running on

A 32-bit unsigned int can hold values up to 4294967296 (4G). If you need to store larger values, you could use the "Q" template (64-bit), if your build of perl supports it.  Otherwise - or if you want to save space - you could just "add" another single byte ("C"), so you have 5 bytes / 40-bit in total — which would be able to handle indices of up to around 1 Terabyte:

my $i = 78187493530;

write_index($i);
print read_index();  # 78187493530

sub write_index {
    my $i = shift;
    open my $f, ">", "myindex" or die $!;
    my $pi = pack("CN", $i / 2**32, $i % 2**32);
    print $f $pi;    # writes 5 bytes
    close $f;
}

sub read_index {
    open my $f, "<", "myindex" or die $!;
    read $f, my $pi, 5;
    my ($C, $N) = unpack "CN", $pi;
    return $C * 2**32 + $N;
}
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This works even with 32-bit-int perls, because then numbers larger than 32-bit are handled as floats internally.