And, to make it work as you were expecting, use Math::BigInt

use 5.016;
use Test::More qw(no_plan);
use Math::BigInt;

for (51..66) {
    # substr to remove 0b prefix
    my $s = substr(Math::BigInt->new("2")->bpow($_)->bdec->as_bin, 2);
    # Or: Math::BigInt->new("1")->blsft($_)->bdec->as_bin
    is length($s), $_, "1 << $_ -1";
}
__END__
ok 1 - 1 << 51 -1
ok 2 - 1 << 52 -1
ok 3 - 1 << 53 -1
ok 4 - 1 << 54 -1
ok 5 - 1 << 55 -1
ok 6 - 1 << 56 -1
ok 7 - 1 << 57 -1
ok 8 - 1 << 58 -1
ok 9 - 1 << 59 -1
ok 10 - 1 << 60 -1
ok 11 - 1 << 61 -1
ok 12 - 1 << 62 -1
ok 13 - 1 << 63 -1
ok 14 - 1 << 64 -1
ok 15 - 1 << 65 -1
ok 16 - 1 << 66 -1
1..16
[download]

Or Math::BigNum if one has a large number of calculations and needs a bit more speed.

use 5.016;
use Test::More qw(no_plan);
use Math::BigNum;

for (51..66) {
    # no need to substr to remove 0b prefix
    my $s = Math::BigNum->new(2)->bpow($_)->bdec->as_bin;
    is length($s), $_, "1 << $_ -1";
}
__END__
ok 1 - 1 << 51 -1
ok 2 - 1 << 52 -1
ok 3 - 1 << 53 -1
ok 4 - 1 << 54 -1
ok 5 - 1 << 55 -1
ok 6 - 1 << 56 -1
ok 7 - 1 << 57 -1
ok 8 - 1 << 58 -1
ok 9 - 1 << 59 -1
ok 10 - 1 << 60 -1
ok 11 - 1 << 61 -1
ok 12 - 1 << 62 -1
ok 13 - 1 << 63 -1
ok 14 - 1 << 64 -1
ok 15 - 1 << 65 -1
ok 16 - 1 << 66 -1
1..16
[download]

Perl 5.24.1 64-bit, for reference:

use warnings;
use strict;

for(51..66){
    my $s = sprintf("%b", 2**$_-1);
    print "$_:" . length($s) . "\n";
}
[download]

Output (* denotation is added after the fact to represent where it starts to "break").:

51:51
52:52
53:53
54:54
55:55
56:56
57:57
58:58
59:59
60:60
61:61
62:63 *
63:64 *
64:64 *
65:64 *
66:64 *
[download]

Try using 1 << 56 instead, ** works on floats, where precision (nb of significant digits) is lowered in favour of a wider range of values (very small or very big numbers are possible)

Depends on the system: if you've got a 32-bit IV (perl -V:ivsize or perl -MConfig -le 'print $Config{ivsize}' will show 4 bytes for 32-bit, 8 bytes for 64-bit), then even that won't work, because it will overrun the 32-bit integer at 1<<32 - 1. And even with 64 bits, it will stop working above 64:

require v5.16;
use Test::More qw(no_plan);

for(51..66)
 {my $s = sprintf("%b", 1 << $_ -1);
  is length($s), $_, "1 << $_ -1";
 }
__END__
ok 1 - (1 << 51) -1
ok 2 - (1 << 52) -1
ok 3 - (1 << 53) -1
ok 4 - (1 << 54) -1
ok 5 - (1 << 55) -1
ok 6 - (1 << 56) -1
ok 7 - (1 << 57) -1
ok 8 - (1 << 58) -1
ok 9 - (1 << 59) -1
ok 10 - (1 << 60) -1
ok 11 - (1 << 61) -1
ok 12 - (1 << 62) -1
ok 13 - (1 << 63) -1
ok 14 - (1 << 64) -1
not ok 15 - (1 << 65) -1
#   Failed test '(1 << 65) -1'
#   at - line 6.
#          got: '64'
#     expected: '65'
not ok 16 - (1 << 66) -1
#   Failed test '(1 << 66) -1'
#   at - line 6.
#          got: '64'
#     expected: '66'
1..16
# Looks like you failed 2 tests of 16.
[download]

So, philiprbrenan, "is this something to worry about?" depends on what you want to do. It's doing exactly what's expected for a double-float NV. And my example code does what's expected for a 64-bit integer. But as to whether that's behavior to worry about depends on what you're trying to do with your 2**$_n-1.

Double-precision floating point numbers have 53 bits of precision. 2**54-1 requires 54 bits of precision to represent, so it can't be represented exactly by a double-precision floating point number.