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Subtract The Exponents, Get 3

  1. Jul 31, 2007 #1
    Problem

    Determine all possible non negative integer pairs (x, y) satisfying this equation:

    2^x – 5^y = 3


    My Attempt:

    If x =0, then 5^y = -2, which is a contradiction.

    If x =1, then 5^y =-1, which is a contradiction.

    If x = 2, then 5^y = 1, so that y = 0

    If x>=3, then we observe that:

    3^y = 3(Mod 8), so that y must be odd

    Let us substitute y = 2s+1, where s is a positive integer. …….(*)

    Again, if y – y, then 2^x = 4, giving: x = 2

    If y =1, 2^x = 8, giving x = 3

    For y>=3, substituting y = 2s+ 1 in terms of (*), we obtain:

    2^x - 5^(2s+1) = 3
    Or, 2^x = 3 (Mod 5)
    or, x = 4t+3, where s is a non negative integer.

    So we have:

    2^(4t+3) – 5^(2s+1) = 3
    Or, 8*(16^t) - 5*(25^s) = 3

    For t =1, we obtain s =1, so that: (x, y) = (7, 3)

    Hence, so far we have obtained (x,y) = (7, 3); (3,1) and (2, 0) as valid solutions to the problem.

    **** I am unable to proceed any further, and accordingly, I am looking for a methodology giving any further valid solution(s) or any procedure conclusively proving that no further solutions can exist for the given problem.
     
  2. jcsd
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