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Palindromic and p-gonal (Posted on 2014-01-05) Difficulty: 3 of 5
Determine the minimum value of a pentagonal number each of whose binary, quaternary and hexadecimal representations is a palindrome. What is the smallest decagonal number with this property?

Any solution must have more than one digit in any given base. So, trivial solutions like (0)base 2, (2)base 4 or, (B)base 16 are not allowed.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 1 of 1

DECLARE FUNCTION isPalinb# (n#, b#)
DECLARE FUNCTION isPalin# (s$)
DEFDBL A-Z

OPEN "palinpgn.txt" FOR OUTPUT AS #2

FOR n = 1 TO 99999
  pent = (3 * n * n - n) / 2: deca = 4 * n * n - 3 * n
  IF isPalinb(pent, 2) THEN PRINT #2, " pent"
  IF isPalinb(pent, 4) THEN PRINT #2, " pent"
  IF isPalinb(pent, 16) THEN PRINT #2, " pent"
  IF isPalinb(deca, 2) THEN PRINT #2, " deca"
  IF isPalinb(deca, 4) THEN PRINT #2, " deca"
  IF isPalinb(deca, 16) THEN PRINT #2, " deca"
NEXT

CLOSE

FUNCTION isPalin (s$)
 good = 1

 FOR i = 1 TO LEN(s$) / 2
   IF MID$(s$, i, 1) <> MID$(s$, LEN(s$) + 1 - i, 1) THEN good = 0: EXIT FOR
 NEXT
 isPalin = good
END FUNCTION

FUNCTION isPalinb (n, b)
  s$ = ""
  q = n
  WHILE q > 0
    r = q MOD b
    q = q \ b
    IF b < 10 THEN
     s$ = LTRIM$(STR$(r)) + s$
    ELSE
     s$ = MID$("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ", r + 1, 1) + s$
    END IF
  WEND
  IF LEN(s$) > 1 AND isPalin(s$) THEN PRINT #2, USING "########## ## \" + SPACE$(33) + "\"; n; b; s$; : isPalinb = 1:  ELSE isPalinb = 0
  EXIT FUNCTION
END FUNCTION


After the output of the above is sorted appropriately (on pentagonal vs decagonal, base and the number itself), the results are:

number in |base| representation                  |type
 decimal  |   /                                  |
        27  2 11011                               deca
        85  2 1010101                             deca
       297  2 100101001                           deca
      1105  2 10001010001                         deca
      2047  2 11111111111                         deca
      4257  2 1000010100001                       deca
     16705  2 100000101000001                     deca
     27307  2 110101010101011                     deca
     66177  2 10000001010000001                   deca
    121975  2 11101110001110111                   deca
    263425  2 1000000010100000001                 deca
   1051137  2 100000000101000000001               deca
   4199425  2 10000000001010000000001             deca
   7735351  2 11101100000100000110111             deca
  16787457  2 1000000000010100000000001           deca
  67129345  2 100000000000101000000000001         deca
 268476417  2 10000000000001010000000000001       deca
1073823745  2 1000000000000010100000000000001     deca
1637760195  2 1100001100111100011110011000011     deca
        10  4 22                                  deca
        85  4 1111                                deca
       855  4 31113                               deca
      1105  4 101101                              deca
     16705  4 10011001                            deca
    227767  4 313212313                           deca
    263425  4 1000110001                          deca
   2223826  4 20132323102                         deca
   4199425  4 100001100001                        deca
  39278422  4 2111311131112                       deca
  58449847  4 3132331332313                       deca
  67129345  4 10000011000001                      deca
 153245830  4 21020211202012                      deca
1073823745  4 1000000110000001                    deca
1196692945  4 1013111001113101                    deca
        85 16 55                                  deca
      2232 16 8B8                                 deca
      2425 16 979                                 deca
     18292 16 4774                                deca
    564376 16 89C98                               deca
    755595 16 B878B                               deca
    787212 16 C030C                               deca
 144366232 16 89ADA98                             deca
 156982105 16 95B5B59                             deca
         5  2 101                                 pent
        51  2 110011                              pent
    348245  2 1010101000001010101                 pent
    465095  2 1110001100011000111                 pent
   1407957  2 101010111101111010101               pent
  15483447  2 111011000100001000110111            pent
 400730365  2 10111111000101010100011111101       pent
         5  4 11                                  pent
        51  4 303                                 pent
       425  4 12221                               pent
       477  4 13131                               pent
     48510  4 23311332                            pent
    134850  4 200323002                           pent
    348245  4 1111001111                          pent
        51 16 33                                  pent
      8177 16 1FF1                                pent
     15555 16 3CC3                                pent
     98945 16 18281                               pent
    227955 16 37A73                               pent
    326900 16 4FCF4                               pent
    348245 16 55055                               pent
    399126 16 61716                               pent
    545112 16 85158                               pent
    818812 16 C7E7C                               pent
   2145026 16 20BB02                              pent
   5767301 16 580085                              pent
  30116801 16 1CB8BC1                             pent
  36885042 16 232D232                             pent
  61532835 16 3AAEAA3                             pent
  75526276 16 4807084                             pent
  90905445 16 56B1B65                             pent
 252480527 16 F0C8C0F                             pent
 261868447 16 F9BCB9F                             pent


Answers to the specific questions:

Pentagonal: 51
Decagonal: 85


  Posted by Charlie on 2014-01-05 17:47:49
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