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2^x prove x is even

  1. Mar 15, 2012 #1
    1. The problem statement, all variables and given/known data
    i want to prove that x is even in this equation

    2^x+3^y=z^2 (1)

    2. Relevant equations



    3. The attempt at a solution

    what i have so far is

    (1) is congruent to mod3
    2^x=z^2(mod3)
    when z=1 and x=2 then
    4=1(mod3)
    so therefore x is even?

    have i proved that x is even, or am i missing anything?
     
  2. jcsd
  3. Mar 15, 2012 #2

    Office_Shredder

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    You proved that if x=2 that it is even perhaps. There are two questions that should immediately come to mind. What can z squared be mod three be as z varies, and what can two to the x look like mod three as x varies? Calculating some small examples of each should suggest a general formula
     
  4. Mar 15, 2012 #3
    so z^2 =1 (mod3) unless z=3m for any integer m
    with 2^x
    im not sure whether
    2=1(mod3) or 2=2(mod3)
    assuming 2=2(mod 3)
    when x is odd
    then 2^(2k+1)=2(mod3)
    and when x is even
    2^x=1(mod 3)

    so im not too sure where to go from here
     
  5. Mar 15, 2012 #4

    Office_Shredder

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    So if x is odd, does 2^x and z^2 ever agree mod 3
     
  6. Mar 15, 2012 #5
    if i was right with assuming 2^(2k+1)=2(mod3)
    then no when x is odd it doesnt agree with z^2
     
  7. Mar 15, 2012 #6
    oh so does that mean that x has to be even, is that how i prove it?
     
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