Question

What is the largest integer n for which a circle with center can pass through exactly n with integer coordinates?

Answer 1

The general equation for a circle is given by:

$x^2+y^2-2ax-2by-r^2=0$

(a,b) is the center coordinates and r is the radius.

For the given circle, we have:

$x^2+y^2-2\sqrt[]{2}x-2\sqrt[]{2}y=r^2$

SInce it is a 2th degree equation for each variable and th, the largest integer n for which this circle can pass through exactly n integer coordinates is 2.