Equidistant Pairs on Two Lattice Lines

Jin Akiyama, Midori Kobayashi, Gisaku Nakamura, and Chie Nara
School of Administration and Informatics, University of Shizuoka,
Shizuoka 422-8526, Japan

Abstract

Consider the xy-coordinate plane. A point A(a,b) is called a lattice point if a,b are integers. If two lattice points A(a,b) and B(c,d) have the same distance from the origin O(0,0), i.e., a² +b²= c² +d², we say that A and B form an equidistant pair. The line passing through two lattice points is called a lattice line. We consider two lattice lines, and equidistant pairs consisting of one point from each line.
Are there two lattice lines on which there are infinitely many equidistant pairs? And are there two lattice lines on which there are no equidistant pairs? Reflecting a lattice line about y=x, y=-x, x=0 or y=0, or rotating it by around the origin, or any combination of these operations, maps it to another lattice line with the same distances from the origin. We exclude these cases as trivial.
In this paper, we show that there are two lattice lines on which there are infinitely many equidistant pairs, and we also show similar results for finitely many equidistant pairs or no equidistant pairs.