GEOMETRIC PROGRESSIONS ON ELLIPTIC CURVES.

In this paper, we look at long geometric progressions on different model of elliptic curves, namely Weierstrass curves, Edwards and twisted Edwards curves, Huff curves and general quartics curves. By a geometric progression on an elliptic curve, we mean the existence of rational points on the curve whose -coordinate (or -coordinate) are in geometric progression. We find infinite families of twisted Edwards curves and Huff curves with geometric progressions of length 5, an infinite family of Weierstrass curves with 8 term progressions, as well as infinite families of quartic curves containing 10-term geometric progressions.

Arithmetic Progressions on Conics.

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 -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.