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 Powers often in other bases. (Posted on 2013-04-07)
Create a table of powers of 10 in binary starting with 101 = 10102 then create a similar table in base 5 starting with 101 = 205.

If you look at the lengths of the numbers in the two tables combined, prove there is exactly one each of length 2, 3, 4...

 No Solution Yet Submitted by Jer No Rating

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 Tables: no proof except for 2,3,4,...,67 | Comment 1 of 3

This does not provide a general proof, but obviously proves for lengths 2,3,4,...,67.

`base 10        base 2                                       base 510            1010                                         20100           1100100                                      4001000          1111101000                                   1300010000         10011100010000                               310000100000        11000011010100000                            112000001000000       11110100001001000000                         22400000010000000      100110001001011010000000                     10030000000100000000     101111101011110000100000000                  2011000000001000000000    111011100110101100101000000000               402200000000010000000000   1001010100000010111110010000000000           130440000000000100000000000  1011101001000011101101110100000000000        31143000000000001000000000000 1110100011010100101001010001000000000000     112341000000000000`
`Lengths:                                                            length                                         representational length   missed by                                              base 2   base 5       base 5                                              10                                                4         2         1100                                               7         31000                                              10        5         410000                                             14        6100000                                            17        8         71000000                                           20        910000000                                          24        11        10100000000                                         27        121000000000                                        30        1310000000000                                       34        15        14100000000000                                      37        161000000000000                                     40        18        1710000000000000                                    44        19100000000000000                                   47        21        201000000000000000                                  50        2210000000000000000                                 54        23100000000000000000                                57        25        241000000000000000000                               60        2610000000000000000000                              64        28        27100000000000000000000                             67        291000000000000000000000                            70        31        3010000000000000000000000                           74        32100000000000000000000000                          77        331000000000000000000000000                         80        35        3410000000000000000000000000                        84        36100000000000000000000000000                       87        38        371000000000000000000000000000                      90        3910000000000000000000000000000                     94        41        40100000000000000000000000000000                    97        421000000000000000000000000000000                   100       4310000000000000000000000000000000                  103       45        44100000000000000000000000000000000                 107       461000000000000000000000000000000000                110       48        4710000000000000000000000000000000000               113       49100000000000000000000000000000000000              117       51        501000000000000000000000000000000000000             120       5210000000000000000000000000000000000000            123       53100000000000000000000000000000000000000           127       55        541000000000000000000000000000000000000000          130       5610000000000000000000000000000000000000000         133       58        57100000000000000000000000000000000000000000        137       591000000000000000000000000000000000000000000       140       61        6010000000000000000000000000000000000000000000      143       62100000000000000000000000000000000000000000000     147       631000000000000000000000000000000000000000000000    150       65        6410000000000000000000000000000000000000000000000   153       66100000000000000000000000000000000000000000000000  157       68        67`

So up through 67, the lengths missed by the base-5 representations match those found in the base-2 representations.

10   cls
20
30   Pw=10
40   while Pw<1000000000000000000000000000000000000000000000000
45     Pl=L:L=len(fnCv\$(Pw,5))
50     print Pw;tab(50);len(fnCv\$(Pw,2));tab(60);L;
55     if L>Pl+2 then print "*";
56     if L>Pl+1 then print Pl+1:else print
60     Pw=Pw*10
70   wend
80   end
90   fnCv\$(N,B)
100     R=N:V\$=""
110     while R>0
120       Q=int(R/B):R=R-Q*B
130       V\$=cutspc(str(R))+V\$
140       R=Q
150     wend
160   return(V\$)

 Posted by Charlie on 2013-04-07 12:18:07

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