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5 Digit Number II (Posted on 2011-01-03) Difficulty: 4 of 5
In continuation of 5 Digit Number, let us define a 5-digit non leading zero base N (N > 3) positive integer x as a split number whenever, 3*x is a perfect square and, when the digits of x are split, the first number is double the second one.

How many split numbers are there whenever 11 ≤ N ≤ 36. What are the respective minimum and maximum values?

(Splitting a base-N 5-digit number into two numbers means 12345 into 1 and 2345 or, 123 and 45.)

No Solution Yet Submitted by K Sengupta    
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Solution computer solution | Comment 2 of 4 |

    5       cls:print "----------"
    6       kill "5digii.txt"
    7       open "5digii.txt" for output as #2
   10       for N=11 to 36
   20         Flag=1:Flag2=0
   30         for D4=0 to N-1
   40          for D5=0 to N-1
   50            V2=D4*N+D5
   60            V1=2*V2
   70            if V1>=N^2 then
   80             :D3=V1 @ N:D2=V1\N
   90             :D1=D2\N:D2=D2 @ N
  100             :V=V1*N^2+V2
  110             :Sq=3*V
  120             :Sr=int(sqrt(Sq)+0.5)
  130             :if Sr*Sr=Sq then
  135               :if Flag then
  136                 :print "base";N:Flag=0:Flag2=1
  137                 :print #2,"base";N:Flag=0:Flag2=1
  139               :endif
  140               :print D1;D2;D3;D4;D5,V;V1;V2
  141               :print #2,D1;D2;D3;D4;D5,V;V1;V2,3*V;sqrt(3*V)
  200          next D5
  210         next D4
  220         if Flag2 then print #2,
  230       next N

finds the following. Each solution row contains the five digits of the base-N number, the decimal representation of the number, the decimal representation of the first part of the split (first 3 base-n digits), the decimal representation of the remainder of the number, three times the full value of the number in decimal, and that number's square root in decimal.

                 ---- decimal ----------------
    digits       whole  pt 1 pt 2  3x    sq.root
base 11
 1  0  7  5  9   15552  128  64   46656  216.0
 1  3  8  7  4   19683  162  81   59049  243.0
 1  7  2  9  1   24300  200  100   72900  270.0
base 12 
 1  0  6  6  3   21675  150  75   65025  255.0
 1  6  0  9  0   31212  216  108   93636  306.0
base 13 
 1  4  5  8  9   38307  226  113   114921  339.0
base 14 
 1  4  10  9  5   51483  262  131   154449  393.0
base 16 
 1  5  6  10  11   87723  342  171   263169  513.0
base 17 
 1  5  12  11  6   111747  386  193   335241  579.0
base 19 
 1  6  7  12  13   174243  482  241   522729  723.0
base 20 
 1  6  14  13  7   213867  534  267   641601  801.0
base 22 
 1  7  8  14  15   312987  646  323   938961  969.0
base 23 
 1  7  16  15  8   373827  706  353   1121481  1059.0
base 25 
 1  8  9  16  17   521667  834  417   1565001  1251.0
base 26 
 1  8  18  17  9   610203  902  451   1830609  1353.0
base 28 
 1  9  10  18  19   820587  1046  523   2461761  1569.0
base 29 
 1  9  20  19  10   944163  1122  561   2832489  1683.0
base 31 
 1  10  11  20  21   1232643  1282  641   3697929  1923.0
base 32 
 1  10  22  21  11   1399467  1366  683   4198401  2049.0
base 34 
 1  11  12  22  23   1783323  1542  771   5349969  2313.0
base 35 
 1  11  24  23  12   2002467  1634  817   6007401  2451.0

a total of 21 solutions within the given range of bases.


  Posted by Charlie on 2011-01-03 16:55:37
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