Here is a question that has puzzled me for a long time. I have figured out how to do the question a and b(i).
(a) An ellipse, F, is centred on the origin. There are two distinct points inside F with position vectors $\vec x$ and $\vec y$, such that $\vec y = \vec x + \begin{pmatrix}2a\\ 2b\end{pmatrix}$, where a and b are integers. Explain why the point with position vector $-\vec x$ is inside F and show that the point with positive vector $\begin{pmatrix}a \\ b\end{pmatrix}$ is also inside F.
For question (a), I used the property of symmetry and convexity of ellipse.
(b) The diagram shows an ellipse E centred on the origin. Also shown are points with integer coordinates and $2\times 2$ squares. enter image description here Consider E to be split into its intersections with the various $2\times2$ squares. In this particular case there are four such intersections and each of these parts can be translated into the region $0\leq x,y \leq 2$, as shown.enter image description here All the parts of this particular ellipse fit inside the $2\times 2$ square without overlapping and therefore, in this case, E has area less than 4.
(i) Use part (a) to prove that any ellipse centred on the origin with area at least 4 has a point with integer coordinates, other than the origin, inside the ellipse.
For question b(i), we can transform vectors $\begin{pmatrix}x \\ y\end{pmatrix}$ into the $2\times 2$ squares by a translation of $\begin{pmatrix}2\lfloor{x/2} \rfloor\\ 2\lfloor y/2 \rfloor\end{pmatrix}$. As the ellipse has an area larger than 4, there must exist some image points overlapping in the $2\times 2$ square. Hence, with (a), the difference between these 2 vectors $\begin{pmatrix}\lfloor{x_1/2} \rfloor-\lfloor{x_2/2} \rfloor\\ \lfloor y_1/2 \rfloor -\lfloor{y_2/2} \rfloor\end{pmatrix}$ must be a lattice point inside E.
(ii) Let $p$ and $u$ be positive integers and let C be the circle centred on the origin and with radius $2\sqrt{\frac{p}{\pi}}$ . Explain why there is a point with coordinates $(mp-nu,n)$, where m and n are integers which are not both zero, inside C.
(c) The positive integer $p$ and $u$ are such that $u^2+1$ is an integer multiple of $p$. Prove that there are integers $x$ and $y$ such that $x^2+y^2=p$.
I can see that question b(ii) and c are trying to use some number theory idea such as gcd or Bezout's identity. However, I am stucked and do not know how to apply results from a and b(i).
If it is possible, please give me some hints to proceed on b(ii) or c. Thank you!
