Sequences of consecutive squares on quartic elliptic curves. (English) Zbl 1415.14011

Kamel, Mohamed; Sadek, Mohammad

Funct. Approximatio, Comment. Math. 60, No. 2, 245-252 (2019).

Let \(C:y^2=f(x)\), where \(f\in\mathbb{Q}[x]\) is symmetric quartic polynomial, i.e., \(f(x)=f(-x)\). The authors of the paper under review are interested in finding examples of curves \(C\) with long sequence of rational points \((x_{i},y_{i})\) such that \(x_{i}=(t+i)^{2}\) for \(i=0,1,\dots m\). The question is one of the possible variations on the problem considered by Bremner, Campbell, Ulas and others related to finding long progressions of arithmetic nature on elliptic and hyperelliptic curves. The main result of the paper says that there are infinitely many curves \(C\) (parameterized by rational parameter \(t\)) such that there is sequence of six rational points \((x_{i}, y_{i})\in C(\mathbb{Q})\) with \(x_{1},\dots,x_{6}\) being consecutive squares. This result is proved by various elementary methods.

Reviewer: Maciej Ulas (Kraków)

MSC:

14G05	Rational points
11B83	Special sequences and polynomials

Keywords:

elliptic curves; rational points; sequences of consecutive squares

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