# Prove that the expression is a perfect square

• Sep 21st 2013, 10:14 PM
pikachu26134
Prove that the expression is a perfect square
Hello,
Please excuse my inability to use latex, and hence the poor format of the equation.
we are given an expression with x,y,z positive integers.
z= x + y/x - 1/y then prove a is a perfect square.
i figure that y/x = 1/y in order for the whole thing to be an integer but I'm not sure that this is sufficient to then go on and prove it. (Like couldn't y/x - 1/y = 1 for y>x ?)
thank you
• Sep 21st 2013, 10:21 PM
ibdutt
Re: Prove that the expression is a perfect square
The question does not contain a, b and c please clarify?
• Sep 21st 2013, 10:23 PM
pikachu26134
Re: Prove that the expression is a perfect square
Sorry, wasn't sure which notation I was going to use
• Oct 2nd 2013, 09:25 PM
timetime44
Re: Prove that the expression is a perfect square
hi pikachu, here's a hint:

y/x - 1/y = (y^2 - x)/xy which must be integer as z, x is integers
=> x divides (
y^2 - x) & y divides (y^2 - x)
=> x
divides y^2 & y divides x

let y^2= ax & x=by

from here just substitute y^2= ax into (y^2 - x)/xy and aim to prove a=b=1 and so x= y^2
And thus z= y^2

Good luck!