14-term Arithmetic Progressions on Quartic Elliptic Curves

Let P4(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 x1, x2,..., xn, with P4(xi) a rational square for 1 ≤ i ≤ 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. 1