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 5 Digit Number II (Posted on 2011-01-03)
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 No Rating

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 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.rootbase 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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