Given are non-negative integer variables $x$, $y$ and $z$. I am trying to deduce the absolute difference between a certain value of $C=x^2+y^2+z^2$ and the very next smallest increase in $C$ possible.
I'd like to do this so I can (dis)prove the following:
- Whether small absolute differences occur less frequently at higher values of $C$
- Whether larger absolute differences start appearing at higher values of $C$
Perhaps I am missing something obvious but can a formula be deduced that (dis)proves this?