I would like to find the number of all points with integer coordinates ($(x,y,z): x,y$ and $z$ integer) in an ellipsioid
$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} < P $
Is it possible to find an equation for the number of them, and how?
I encountered this problem while solving a physics task, and it quite a bit stalled my progress. Also, I find it an interesting problem.
My idea was taking the plane segments at each integer z (yielding an ellipse), and count the integer points in the segments.The problem is, even for a 2D ellipse you get an ugly equation:
$\frac{x^2}{a^2}+\frac{y^2}{b^2} < 1 $
So for a given $x=k_x$ coordinate
$y^2 = b^2(1-\frac{b^2}{a^2} k_{x}^2)$
So the number of integer y points is
$n_y = 2[b\sqrt{1-\frac{b^2}{a^2} k_{x}^2}]$
([ ] marks the integer part.)
(The physical problem is the following: I should find the number of possible energy states for a particle enclosed in a 3d box, that are smaller tha n a given energy. Everything is parametric. The coordinates are the quantum numbers, a, b and c are the box's sides, and P is related to the maximum energy.)
