Integral distances from (two) given lattice points
Umberto M. Zannier · L’Enseignement Mathématique · 2022
We completely characterize pairs of lattice points P_1 eq 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 eq (\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.