This post stated an easy Number Theory question regarding the number of solutions of the Diophantine equation \(x(x+z)=y^2\).
I decided to make the question a bit more interesting:
Characterize all positive integer solutions to \(x(x+z)=y^2\) given that \(2y-2x\ge z-1\).
Indeed, the following is also true:
Given that \(x,y,z\) are positive integers that satisfy both \(x(x+z)=y^2\) and \(2y-2x\ge z-1\), then it follows that \(2y-2x=z-1\).
My solution is elementary, but well-hidden. I await your solutions.