Journal of Integer Sequences, Vol. 6 (2003), Article 03.1.3

A Note on Arithmetic Progressions on Elliptic Curves

Garikai Campbell
Department of Mathematics and Statistics
Swarthmore College
Swarthmore, PA 19081
USA

Abstract: Andrew Bremner (Experiment. Math. 8 (1999), 409-413) has described a technique for producing infinite families of elliptic curves containing length 7 and length 8 arithmetic progressions. This note describes another way to produce infinite families of elliptic curves containing length 7 and length 8 arithmetic progressions. We illustrate how the technique articulated here gives an easy way to produce an elliptic curve containing a length 12 progression and an infinite family of elliptic curves containing a length 9 progression, with the caveat that these curves are not in Weierstrass form.

Received February 5, 2003; revised version received February 7, 2003. Published in Journal of Integer Sequences February 25, 2003.