Why Can't d Be a Perfect Square in Pell Equations?

[cragar]

Homework Statement

A pell equation is an equation $x^2-dy^2=1$ 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
$x^2-d^2y^2=1$ z=dy then $x^2-z^2=1=(x+z)(x-z)=1$
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 won't work so d can't be a perfect square. Does this work?

 

[tiny-tim]

hi cragar!

yes, that's ok

(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)