Rational points in arithmetic progression on $y^2=x^n+k$

Abstract

Let $C$ be a hyperelliptic curve given by the equation $y^2=f(x)$, where $f\in\Z[x]$ and $f$ hasn't multiple roots. We say that points $P_{i}=(x_{i}, y_{i})\in C(\Q)$ for $i=1,2,..., n$ are in arithmetic progression if the numbers $x_{i}$ for $i=1,2,..., n$ are in arithmetic progression. In this paper we show that there exists a polynomial $k\in\Z[t]$ with such a property that on the elliptic curve $\cal{E}: y^2=x^3+k(t)$ (defined over the field $\Q(t)$) we can find four points in arithmetic progression which are independent in the group of all $\Q(t)$-rational points on the curve $\cal{E}$. In particular this result generalizes some earlier results of Lee and V\'{e}lez from \cite{LeeVel}. We also show that if $n\in\N$ is odd then there are infinitely many $k$'s with such a property that on the 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 the curves $y^2=x^n+k$ there are six rational points in arithmetic progression.