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Pandigital and Pretty Powerful II (Posted on 2009-05-06) Difficulty: 3 of 5
Find all possible positive integer(s) N, such that the decimal representation of the sum of two distinct perfect powers of N consists of non leading zeroes and contains each of the digits from 0 to 9 exactly once.

See The Solution Submitted by K Sengupta    
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Solution basic computer solution Comment 2 of 2 |

DEFDBL A-Z
nlim = SQR(9876543210#)
lnlim = LOG(nlim)
FOR n = 2 TO nlim
 plim = 2 * lnlim / LOG(n)
 FOR p1 = 2 TO plim
  FOR p2 = p1 + 1 TO plim
    tot = INT(n ^ p1 + n ^ p2 + .5)
    t$ = LTRIM$(STR$(tot))
    IF LEN(t$) = 10 THEN
      good = 1
      FOR i = 1 TO 9
        IF INSTR(i + 1, t$, MID$(t$, i, 1)) > 0 THEN good = 0: EXIT FOR
      NEXT
      IF good THEN PRINT n; p1; p2, n ^ p1; n ^ p2, tot
    END IF
  NEXT
 NEXT
NEXT
 
finds

   n  pwr1 pwr2      n^pwr1     n^pwr2        n^pwr1 + n^pwr2
  264   3    4     18399744  4857532416         4875932160
 2016   2    3      4064256  8193540096         8197604352

If it hadn't been required that the powers be distinct, there would have been many more solutions, one of which would have two 4th powers, and the rest of the equal powers as 2. In fact 264 is the one that has two 4th powers.

  264  3  4     18399744  4857532416      4875932160
  264  4  4   4857532416  4857532416      9715064832
 2016  2  3      4064256  8193540096      8197604352
22887  2  2    523814769   523814769      1047629538
23124  2  2    534719376   534719376      1069438752
24957  2  2    622851849   622851849      1245703698
25941  2  2    672935481   672935481      1345870962
26409  2  2    697435281   697435281      1394870562
26733  2  2    714653289   714653289      1429306578
27276  2  2    743980176   743980176      1487960352
29685  2  2    881199225   881199225      1762398450
31389  2  2    985269321   985269321      1970538642
35367  2  2   1250824689  1250824689      2501649378
39036  2  2   1523809296  1523809296      3047618592
39147  2  2   1532487609  1532487609      3064975218
39432  2  2   1554882624  1554882624      3109765248
39702  2  2   1576248804  1576248804      3152497608
40293  2  2   1623525849  1623525849      3247051698
41997  2  2   1763748009  1763748009      3527496018
42843  2  2   1835522649  1835522649      3671045298
43059  2  2   1854077481  1854077481      3708154962
44922  2  2   2017986084  2017986084      4035972168
45258  2  2   2048286564  2048286564      4096573128
45624  2  2   2081549376  2081549376      4163098752
46464  2  2   2158903296  2158903296      4317806592
49059  2  2   2406785481  2406785481      4813570962
50889  2  2   2589690321  2589690321      5179380642
53568  2  2   2869530624  2869530624      5739061248
54354  2  2   2954357316  2954357316      5908714632
57321  2  2   3285697041  3285697041      6571394082
59268  2  2   3512695824  3512695824      7025391648
59727  2  2   3567314529  3567314529      7134629058
60984  2  2   3719048256  3719048256      7438096512
61098  2  2   3732965604  3732965604      7465931208
61611  2  2   3795915321  3795915321      7591830642
61866  2  2   3827401956  3827401956      7654803912
62634  2  2   3923017956  3923017956      7846035912
65436  2  2   4281870096  4281870096      8563740192
68823  2  2   4736605329  4736605329      9473210658
68982  2  2   4758516324  4758516324      9517032648
69087  2  2   4773013569  4773013569      9546027138
69696  2  2   4857532416  4857532416      9715064832
69732  2  2   4862551824  4862551824      9725103648
69798  2  2   4871760804  4871760804      9743521608

  Posted by Charlie on 2009-05-06 13:10:35
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