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Integer solution to exponential diophantine equation

  1. Sep 28, 2011 #1
    Hey everyone!

    I was recently scribbling on paper, and after a series of ideas, I got stuck with a problem. That is, can I find out if there exists some integers A and B such that

    [itex]C=2^{A}3^{B}[/itex]

    For some integer C?

    For an arbitrary C, how do I know whether some [itex]A, B \in \textbf{Z}[/itex] exist?

    Cheers for reading!
    Adrian
     
  2. jcsd
  3. Sep 28, 2011 #2
    Hi, Adrian,
    it shouldn't be harder than testing if the number is divisible by 2 or by 3; and, in that case, if you are interested in the actual values of A and B, just divide and iterate. Unless you mean really big numbers.
     
  4. Sep 28, 2011 #3
    Code (Text):

    Solving with a computer:

    Factor C  givin a list of it's prime factors and their occurences;
    if there i9s a factor > 3 then 'No, C is not of the required form'
    else 'Yes, A and B are the number, the factor 2 (resp 3) occurs

    Solving with paper and your head:

    Set A and B to zero
    Loop2:
    If C is even replace C by C / 2 andd add 1 to A
       loop until C is odd
    Loop3:
    If C is multiple of 3 (add the digits modulo 3)
       replace C by C / 3 andd add 1 to B
       loop until C is not a multiple of 3
    Test: it the remaining C is one, then 'Yes' else 'No'
     
     
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