Arithmetic Progressions on Conics.

Abstract

In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of three term progressions on the unit hyperbola, as well as conics ax² + cy² = 1 containing arithmetic progressions as long as 8 terms.