Look more closely at the docs with regards to $month

$mon is the month itself, in the range 0..11 with 0 indicating January and 11 indicating December

So your code should look like this

my ($day, $month, $year) = (localtime)[3..5];
$year += 1900;
## increase $month to get desired output
$month++;
my $date = "$year-$month-$day";

print "$date\n";

$date = localtime;
print "$date\n";

__output__

2004-5-17
Mon May 17 10:41:08 2004
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