Integral distances from (two) given lattice points

Abstract

We completely characterize pairs of lattice points $P_1\ne P_2$ in the plane with the property that there are infinitely many lattice points $Q$ whose distance from both $P_1$ and $P_2$ is integral. In particular, we show that it suffices that $P_2-P_1\ne (\pm 1,\pm 2),(\pm 2,\pm 1)$, and we show that $|P_1-P_2|>\sqrt{20}$ suffices for having infinitely many such $Q$ outside any finite union of lines. We use only elementary arguments, the crucial ingredient being a theorem of Gauss which does not appear to be often invoked. We further include related remarks (and open questions), also for (rational and integral) distances from an arbitrary prescribed finite set of lattice points.