A search for "localtime" gives you some info on the function :
$now_string = localtime;  # e.g., "Thu Oct 13 04:54:34 1994"

I suspect you need to open the file for writing ?

open ( FILE, ">>input.txt" ) or die ( "input.txt: $!" );
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So the main problem i see here is your string "This is the local time :" is not included any where in the code, so how would it know to print "This is the time : $now_string" to the file? Try this :

my $variable = localtime;
my $words = "This is the time : $localtime";

open (FH">>input.txt") or die "$!\n";

while (<FH>) {

print FH $words;

}
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