Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.2

14-term Arithmetic Progressions on Quartic Elliptic Curves


Allan J. MacLeod
Department of Mathematics and Statistics
University of Paisley
High St.
Paisley, Scotland PA1 2BE
UK

Abstract:

Let $P_4(x)$ be a rational quartic polynomial which is not the square of a quadratic. Both Campbell and Ulas considered the problem of finding an rational arithmetic progression $x_1,x_2,\ldots,x_n$, with $P_4(x_i)$ a rational square for $1 \le i \le n$. They found examples with $n=10$ and $n=12$. By simplifying Ulas' approach, we can derive more general parametric solutions for $n=10$, which give a large number of examples with $n=12$ and a few with $n=14$.

Full version:  pdf,    dvi,    ps,    latex    

Received October 25 2005; revised version received November 18 2005. Published in Journal of Integer Sequences November 18 2005.

Return to Journal of Integer Sequences home page