Integral distances from (two) lattice points

TL;DR

This paper characterizes pairs of lattice points in the plane for which infinitely many lattice points have integral distances to both, using elementary methods and a classical theorem of Gauss.

Contribution

It provides a complete characterization of such pairs, identifying specific conditions on their difference vector for the existence of infinitely many lattice points with integral distances.

Findings

Pairs with difference not equal to (b1 1,b1 2) or (b1 2,b1 1) have infinitely many points with integral distances.

If the distance between points exceeds c7 20, infinitely many such points exist outside finite line unions.

Elementary arguments and Gauss's theorem are used to establish these results.

Abstract

{\it .}We completely characterize pairs of lattice points $P_1\neq P_2$ 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 $P_2-P_1\neq (\pm 1,\pm 2), (\pm 2,\pm 1)$, 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 applied. We further include related remarks (and open questions), also for distances from an arbitrary prescribed finite set of lattice points % $P_1,\ldots ,P_r$. }