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From Rosetta Code

First perfect square in base N with N unique digits[edit]

The most time consuming ist evaluating the used digits.

static bool allInQS(BigInteger b) {
        BigInteger re; int c = ic; hs.Clear(); hs.UnionWith(o); while (b > bllim) {
            b = BigInteger.DivRem(b, Base, out re);
            hs.Add((byte)re); c += 1; if (c > hs.Count) return false;
        } return true;
    }

I think, using a combination of biginteger and machine size integers can improve this. First calculate the power of base, that fits into a machine integer and than use this maximised base for Bigint calculationes.

  b = BigInteger.DivRem(b, BaseToPower, out re);
       for(i = 1; i<power, i++)
// extract the count= power digits

If you need more speed, create for every base a seperat function. The compiler will use mul and shift instead of div, if the divisor is a constant.

// think of UInt64 as machine size of integer.
static bool allInQSBase28(BigInteger b) {
        BigInteger re; int c = ic; hs.Clear(); hs.UnionWith(o); while (b > bllim) {
            b = BigInteger.DivRem(b, (28**13)6502111422497947648‬, out re);
            for(i = 1; i<13, i++){
              div = re \ 28;
              mod = re - div*28;
              hs.Add((byte)mod);}
....
        } return true;
    }