integer points on an ellipse

Question

I have the equation $1=x^2-xy+y^2 = \frac 1 4 (x+y)^2 + \frac 3 4 (x-y)^2$ where I am looking for integer solutions $x,y \in Z$. When you draw this ellipse it is quite obvious that the integer points are $(1,1),(-1,-1),(1,0),(-1,0),(0,1),(0,-1)$, but is there a simple way to prove that theser are in fact all the integer points on the ellipse?

Answer 1

$$ 4 (x^2 - xy+y^2) = 4 x^2 - 4 x y + y^2 + 3 y^2 = (2x-y)^2 + 3 y^2 $$ $$ x^2 - xy + y^2 \geq (3/4) y^2. $$ $$ x^2 - xy + y^2 \geq (3/4) x^2. $$