On simultaneous arithmetic progressions on elliptic curves. (English) Zbl 1215.11056

García-Selfa, Irene; Tornero, José M.

Exp. Math. 15, No. 4, 471-478 (2006).

From the text: In this paper we study elliptic curves that have a number of points whose coordinates are in arithmetic progression. We first motivate this Diophantine problem; then we prove some results, provide a number of interesting examples, and finally point out open questions that focus on the most interesting aspects of the problem for us.
Especially, we have managed to prove the following results:
Theorem 1.3. Given an elliptic curve with an x-arithmetic progression (x-a.p.), there exists an algorithm that decides whether the curve also has an simultaneous arithmetic progression (s.a.p.) with the given x-a.p. as support.
Theorem 1.4. There are integers \(n\), and examples of curves with s.a.p. of length \(n\), that do not contain any s.a.p. of length \(n-1\).
Theorem 1.5. There are no elliptic curves defined over \(\mathbb Q\) with s.a.p. of length 7. There are only finitely many nonisomorphic curves defined over \(\mathbb Q\) with s.a.p. of length 6.

MSC:

11G05	Elliptic curves over global fields
11B25	Arithmetic progressions

Keywords:

Elliptic curves; arithmetic progressions