A ¡§lattice point¡¨ has integer coordinates. Then, A = (m, n) is a lattice point if both

Question

m and n are integers. Let¡¦s call a point P = (x, y) ¡§generic¡¦¡¦ if all the distances from P
to lattice points are different.
With some algebraic work, I checked that the point S = ( , ) is generic.
However, the point T = (0, £k) is not generic because it is equally distant from the lattice
points (1, 0) and (-1, 0).
„½ Is there some generic point with rational coordinates?
That is, if Q = (r, s) for rational numbers r and s, must there exist two lattice points
equidistant from Q ?
As a first step, show that R = ( , ) is not generic. (Find lattice points A, B equidistant from R.)
Can you use those ideas to answer the general question?

Answer 1

But how ???