Integral points on elliptic curves associated with generalized twin primes

Volume 121 / 2020

Tomasz Jędrzejak Banach Center Publications 121 (2020), 71-78 MSC: 11G05, 11G50. DOI: 10.4064/bc121-7

Abstract

This article is a continuation of our previous paper [Bull. Pol. Acad. Sci. Math. 67 (2019)] concerning elliptic curves $E_{p,m}:y^{2}=x(x-2^{m})(x+p)$, where $p$ and $p+2^{m}$ are primes. There we proved inter alia that $E_{p,1}$ has at most two non-torsion integral points, and $E_{p,2}$ has no such points. Now by using completely different methods, namely an analysis of local height functions, we try to get upper bounds for the number of integral points and for the number of multiples of such points on our curves for any $m$. In particular, we show that no even multiples of an integral point on $E_{p,m}$ are also integral, and if $E_{p,m}$ has rank 1 and $p\equiv 3\; ({\rm mod}\; 4)$ then there are at most twelve non-torsion integral points in the union of the non-identity component and the certain subset of the identity component.

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