The smallest P can be is 1

If N is 3, the largest Q^N can be is BA987654320 ^ (1/3) which is (decimal) 9032.262

My algorithm is to make a list of all cubes (or whatever power) from decimal 1^3 to 9032^3 converted to base 12 (or whatever the largest Q can be raised to whatever power), then separate that into a dictionary by the length of the base 12 representation.  Then I only need to test cubes of length 2 against cubes of length 10; length 3 vs length 9, etc.

The dictionary is reconstructed for each power.

I tested powers from 2 to 21.  At power 22, 2^22 is already 7 digits in base 12.

I only found solutions in powers 2 (12 solutions) and 3 (3 solutions).  

-------  Base 2 ----------

11 109065 2

B 53149 2

A1 237B805469 

11 159918 2

B 78666 2

A1 4B58729630 

33 42955 2

29 20A37 2

769 435B208A1 

33 66319 2

29 32467 2

769 A28B50341 

66 13343 2

56 787B 2

2630 4B759A81 

99 10120 2

83 5A34 2

5809 2A36B714 

99 15059 2

83 886B 2

5809 63B427A1 

198 5852 2

146 3478 2

1A830 B576294 

242 5797 2

182 3431 2

29A84 B307561 

275 4202 2

1AB 2522 2

37921 5AB6084 

473 4686 2

335 2866 2

A9581 742B630 

484 3949 2

344 2351 2

B3694 52807A1 

-------  Base 3 ----------

6 1083 3

6 763 3

160 2B5497A83 

14 437 3

12 305 3

1708 23B469A5 

18 587 3

16 40B 3

3460 578A192B

--------------------

def b12(n):

    return base2base(n,10,12)

def b10(n):

    return base2base(n,12,10)

for power in range(2,22):

    maxQ = int(b10('BA987654320') ** (1/power))

    pqPowers = [b12(i**power) for i in range(1,maxQ + 1)]

    pqLengthDict = {}

    keys = pqLengthDict.keys()

    for p in pqPowers:

        length = len(p)

        if length not in pqLengthDict:

            pqLengthDict[length] = [p]

        else:

            pqLengthDict[length].append(p)

    

    for pLength in range(1,7):

        qLength = 12 - pLength

        if pLength in keys and qLength in keys:

            for pPow in pqLengthDict[pLength]:

                for qPow in pqLengthDict[qLength]:

                    pqConcat = pPow + qPow

                    if len(pqConcat) == 12 and len(set(pqConcat)) == 12 :

                        power10P = round(b10(pPow)**(1/power))

                        power10Q = round(b10(qPow)**(1/power))

                        print (power10P, power10Q, power )

                        print (b12(power10P), b12(power10Q), power )

                        print(pPow,qPow, ' ')