Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.1

A Note on Arithmetic Progressions on Quartic Elliptic Curves

Maciej Ulas
Jagiellonian University
Institute of Mathematics
Reymonta 4
30-059 Kraków
Poland

Abstract: G. Campbell described a technique for producing infinite families of quartic elliptic curves containing a length-9 arithmetic progression. He also gave an example of a quartic elliptic curve containing a length-12 arithmetic progression. In this note we give a construction of an infinite family of quartics on which there is an arithmetic progression of length 10. Then we show that there exists an infinite family of quartics containing a sequence of length 12.

Received November 11 2004; revised version received May 21 2005. Published in Journal of Integer Sequences May 24 2005.