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Rational Points in Arithmetic Progressions on $y^2=x^n+k$

  Published:2011-03-31
 Printed: Mar 2012
  • Maciej Ulas,
    Jagiellonian University, Institute of Mathematics, 30 - 348 Kraków, Poland
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Abstract

Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$ for $f\in\mathbb{Z}[x]$ without multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\mathbb{Q})$ for $i=1,2,\dots, m$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,\dots, m$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\mathbb{Z}[t]$ with the property that on the elliptic curve $\mathcal{E}': y^2=x^3+k(t)$ (defined over the field $\mathbb{Q}(t)$) we can find four points in arithmetic progression that are independent in the group of all $\mathbb{Q}(t)$-rational points on the curve $\mathcal{E}'$. In particular this result generalizes earlier results of Lee and V\'{e}lez. We also show that if $n\in\mathbb{N}$ is odd, then there are infinitely many $k$'s with the property that on curves $y^2=x^n+k$ there are four rational points in arithmetic progressions. In the case when $n$ is even we can find infinitely many $k$'s such that on curves $y^2=x^n+k$ there are six rational points in arithmetic progression.
Keywords: arithmetic progressions, elliptic curves, rational points on hyperelliptic curves arithmetic progressions, elliptic curves, rational points on hyperelliptic curves
MSC Classifications: 11G05 show english descriptions Elliptic curves over global fields [See also 14H52] 11G05 - Elliptic curves over global fields [See also 14H52]
 

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