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 Palindromic and p-gonal (Posted on 2014-01-05)
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 No Rating

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 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       deca1073823745  2 1000000000000010100000000000001     deca1637760195  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                      deca1073823745  4 1000000110000001                    deca1196692945  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`

Pentagonal: 51
Decagonal: 85

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

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