ARITHMETIC PROGRESSIONS ON CONIC SECTIONS

Abstract

The set {1, 25, 49} is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set {(1, 1), (5, 25), (7, 49)} as a 3-term collection of rational points on the parabola y = x2 whose y-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections [Formula: see text] with respect to a linear rational map [Formula: see text]. We explain how this construction is related to rational points on the universal elliptic curve Y2 + 4XY + 4kY = X3 + kX2 classifying those curves possessing a rational 4-torsion point.