First perfect square in base n with n unique digits

Find the first perfect square in a given base N that has at least N digits and exactly N unique digits; display the number in base N.

E.G. In base 10, the first perfect square with at least 10 unique digits is 1026753849 (32043²).

You may use analytical methods to reduce the search space, but the code must do a search. Do not use magic numbers or just feed the code the answer to verify it is correct.

Task

Find and display here, on this page, the first perfect square in base N, with N unique digits, expressed in base N, for each of base 2 through 12.

(optional) Do the same for bases 13 through 16.

Perl 6

Works with: Rakudo version 2019.03

<lang perl6># Only search square numbers that have at least N digits;

smaller could not possibly match.

sub first-square (Int $n) {

   my $start = (($n - 1)/2).exp($n).floor || 1;
   my $sq = ($start .. *).map( *² ).hyper.first: *.base($n).comb.Bag.elems >= $n;
   sprintf "Base %2d: %10s² == %s", $n, $sq.sqrt.base($n), $sq.base($n);

}

say "First perfect square with N unique digits in base N: "; say .&first-square for flat

  2 .. 12, # required
 13 .. 16  # optional

</lang>

Output:

First perfect square with N unique digits in base N:
Base  2:         10² == 100
Base  3:         22² == 2101
Base  4:         33² == 3201
Base  5:        243² == 132304
Base  6:        523² == 452013
Base  7:       1431² == 2450361
Base  8:       3344² == 13675420
Base  9:      11642² == 136802574
Base 10:      32043² == 1026753849
Base 11:     111453² == 1240A536789
Base 12:     3966B9² == 124A7B538609
Base 13:    3828943² == 10254773CA86B9
Base 14:    3A9DB7C² == 10269B8C57D3A4
Base 15:   1012B857² == 102597BACE836D4
Base 16:   404A9D9B² == 1025648CFEA37BD9