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 significant unique digits when expressed 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 significant unique digits when expressed in base N, for each of base 2 through 12. Display each number in the base N for which it was calculated.

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

Julia

<lang julia>using Combinatorics

const num = "0123456789abcdef" hasallin(n, numerals, b) = (s = string(n, base=b); all(x -> occursin(x, s), numerals))

function squaresearch(bas)

   basenumerals = [c for c in num[1:bas]]
   highest = parse(Int, "10" * num[3:bas], base=bas)
   lowest = Int(trunc(sqrt(highest)))
   for n in lowest:highest
       nsquared = n * n
       if hasallin(nsquared, basenumerals, bas)
           return nsquared
       end
   end
   throw("failed to find num for base $bas")

end

println("Base Root N") for b in 2:16

   n = squaresearch(b)
   println(lpad(b, 3), lpad(string(Int(trunc(sqrt(n))), base=b), 10), "  ", string(n, base=b))

end

</lang>

Output:

Base     Root   N
  2        10  100
  3        22  2101
  4        33  3201
  5       243  132304
  6       523  452013
  7      1431  2450361
  8      3344  13675420
  9     11642  136802574
 10     32043  1026753849
 11    111453  1240a536789
 12    3966b9  124a7b538609

Perl 6

Works with: Rakudo version 2019.03

<lang perl6># Only search perfect squares that have at least N digits;

1. 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