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


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 $n=14$.

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

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