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

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.


14G05 Rational points
11B83 Special sequences and polynomials
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