Question about pell equation

  • Thread starter cragar
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  • #1
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Homework Statement


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?

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?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
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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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