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Integral points on elliptic curves $y^{2}=x(x-2^{m}) (x+p)$

Volume 67 / 2019

Tomasz Jędrzejak, Małgorzata Wieczorek Bulletin Polish Acad. Sci. Math. 67 (2019), 53-67 MSC: Primary 11G05; Secondary 11D25, 11D45. DOI: 10.4064/ba8152-1-2019 Published online: 28 March 2019

Abstract

We provide a description of the integral points on elliptic curves $y^{2}=x(x- 2^{m}) \times (x+p)$, where $p$ and $p+2^{m}$ are primes. In particular, we show that for $m=2$ such a curve has no nontorsion integral point, and for $m=1$ it has at most one such point (with $y \gt 0$). Our proofs rely upon numerical computations and a variety of results on quartic and other diophantine equations, combined with an elementary analysis.

Authors

  • Tomasz JędrzejakInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail
  • Małgorzata WieczorekInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail

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