# Thread: Prove that the expression is a perfect square

1. ## 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.
the question asks that if
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

2. ## Re: Prove that the expression is a perfect square

The question does not contain a, b and c please clarify?

3. ## Re: Prove that the expression is a perfect square

Sorry, wasn't sure which notation I was going to use

4. ## 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!