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Question about pell equation

  1. Mar 1, 2014 #1
    1. The problem statement, all variables and given/known data
    A pell equation is an equation [itex] x^2-dy^2=1 [/itex] where d is a positive integer that is not a perfect square. Can you figure out why we do not want d to be a perfect square?
    3. The attempt at a solution
    if d was a perfect square then we would have
    [itex] x^2-d^2y^2=1 [/itex] z=dy then [itex] x^2-z^2=1=(x+z)(x-z)=1 [/itex]
    x>z for this to work so if x>z then x-z is at least 1 and then x+z would be bigger than 1
    so (x+z)(x-z)>1 so this wont work so d cant be a perfect square. Does this work?
     
  2. jcsd
  3. Mar 1, 2014 #2

    tiny-tim

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    hi cragar! :wink:

    yes, that's ok :smile:

    (though i'd be inclined to say that the only divisors of 1 are 1 and 1,
    or -1 and -1, so x+z = x-z, so z = 0 and x = 1)
     
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