Rational points in arithmetic progression on $y^2=x^n k$ - Mathematics > Number TheoryReport as inadecuate




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Abstract: Let $C$ be a hyperelliptic curve given by the equation $y^2=fx$, where$f\in\Zx$ 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 thenumbers $x {i}$ for $i=1,2,

., n$ are in arithmetic progression.In this paper we show that there exists a polynomial $k\in\Zt$ with such aproperty that on the elliptic curve $\cal{E}: y^2=x^3+kt$ defined over thefield $\Qt$ we can find four points in arithmetic progression which areindependent in the group of all $\Qt$-rational points on the curve $\cal{E}$.In particular this result generalizes some earlier results of Lee and V\-{e}lezfrom \cite{LeeVel}. We also show that if $n\in\N$ is odd then there areinfinitely many $k$-s with such a property that on the curves $y^2=x^n+k$ thereare four rational points in arithmetic progressions. In the case when $n$ iseven 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.



Author: Maciej Ulas

Source: https://arxiv.org/



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