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RE question regarding base expansion

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  • Jose Ramón Brox
    ... From: Gordon Chalmers Given a number and a base expansion in base x, N=a x^2 + b x + c , with c known, is there a
    Message 1 of 2 , Oct 11, 2005
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      ----- Original Message -----
      From: "Gordon Chalmers" <gordon_as_number@...>
      Given a number and a base expansion in base x,

      N=a x^2 + b x + c ,

      with c known, is there a (convenient) algorithm to
      determine x? or a or b?

      -------------------------------------------------

      If the exist, you have several solutions, since your number can have 3 digits in several
      bases x. For example, using the notation 'number)base' :

      100)10 = 121)9 = 144)8 = 202)7 = 244)6 = 400)5 (note that if you have a known c, you
      still have several solutions, like 100)10 = 400)5 )

      (nice curios: 10^2)10 = 11^2)9 = 12^2)8 )

      Can you specify your question more? When you said "or a or b" did you mean "fixing x"? If
      so, then for a fixed x you have a single terna (a,b,c) with a,b,c<x IF IT EXISTS (that is,
      if N can be expressed as a number with 3 digits in base x) and the algorithm for getting
      (a,b,c) is based on division.

      Regards. Jose Brox
    • Jose Ramón Brox
      ... From: Jose Ramón Brox If they exist, you have several solutions, since your number can have 3 digits in several bases x. [...] ...
      Message 2 of 2 , Oct 11, 2005
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        ----- Original Message -----
        From: "Jose Ramón Brox" <ambroxius@...>

        If they exist, you have several solutions, since your number can have 3 digits in several
        bases x. [...]

        ------------------------------------

        There is always at least one base in which N>2 has 3 digits:

        if x = N+1 then N = N)x has 1 digit
        if x = N then N = 10)x has 2 digits
        if x = Floor(Sqrt(N)) then N)x has 3 digits, since
        x^2 = [Floor(Sqrt(N))]^2 < Sqrt(N)^2 = N
        and (Floor(Sqrt(N)))^2 >= (Sqrt(N)-1)^2 = N - 2Sqrt(N) +1 -->
        --> 2*x^2 >= 2N - 4Sqrt(N) +1 > N (if N>2)

        So we have that x^2 < N < 2*x^2 and then, if x>2, N has 3 digits in base x =
        Floor(Sqrt(N)).

        Examples:

        N = 1532 --> x = 39 --> N)x = 10B
        N = 54489 --> x = 233 --> N)x = 10(200) (I don't have a proper symbol for the digit
        (200) in base 233)
        N = 9999 --> x = 99 --> N)x = 120

        So here you have a procedure to find ONE way of expressing N as a number of 3 digits, with
        a = 1: take x = Floor(Sqrt(N)), then compute R = mod(N/x^2) and compare it with x; if R <
        x, then your number N is 10R)x. If R >= x, then divide again to get S = mod(R/x) and
        finally your number is 1RS)x.

        Regards. Jose Brox
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