Equidistant Pairs on Two Lattice Lines

概要

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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), 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 l, m and equidistant pairs A, B with A ∈ l and B ∈ m. 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. Next we show some examples of two quadratic curves, two cubic curves, and two curves of the fourth degree, on which there are infinitely many equidistant pairs.
東海大学の論文