- #1

- 104

- 0

## Homework Statement

Show that (1+xy)(1+zy)(1+zx) is a perfect square iff (1 + xy), (1+yz) , and (1+zx) are perfect squares.

## Homework Equations

## The Attempt at a Solution

I initially tried to solve it like this

let 1 + xy = s

^{2}, 1 + yz = t

^{2}, 1 + zx = u

^{2}. I substituted the new variables and got

(1+xy)(1+zy)(1+zx) = s

^{2}t

^{2}u

^{2}- s

^{2}t

^{2}+t

^{4}-t

^{2}+s

^{2}

I tried to get the RHS of the equation to be in some square form, but couldn't. I thought that introducing new variables to the problem would add to the complexity of the problem, but I thought that adding new variables would give me a new way to solve the problem also.

So .. any thoughts?