Each of X, Y and Z is a nonzero integer such that:

Z = X + Y/X – 1/Y is an integer.

Is Z always a perfect square?
Give reasons for your answer.

Say y=1.

Then z=x+1/x-1; x can be 1, when z= (1+1-1)=1, which is a square, or x can be -1, when z=-1+-1-1 = -3, which is not a square.

It's always worth checking the proposition, before embarking on a proof!

(Incidentally, if y=2, x can take the value -4, when z is -5, while if  y=-2, x can take the value -4, when z is -3.)

The answer might be different if x,y,z were required to be positive integers...

Edited on October 11, 2015, 1:52 pm

Posted by broll on 2015-10-11 13:42:14