Rational points in arithmetic progression on y²=xⁿ+k

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Let C be a hyperelliptic curve given by the equation y²=f(x), where f∈[x] and f hasn't multiple roots. We say that points Pᵢ=(xᵢ, yᵢ)∈ C() for i=1,2,..., n are in arithmetic progression if the numbers xᵢ for i=1,2,..., n are in arithmetic progression. In this paper we show that there exists a polynomial k∈[t] with such a property that on the elliptic curve E: y²=x³+k(t) (defined over the field (t)) we can find four points in arithmetic progression which are independent in the group of all (t)-rational points on the curve E. In particular this result generalizes some earlier results of Lee and Vélez from LeeVel. We also show that if n∈ is odd then there are infinitely many k's with such a property that on the curves y²=xⁿ+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 y²=xⁿ+k there are six rational points in arithmetic progression.