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Rank of elliptic curves associated to Brahmagupta quadrilaterals

Volume 143 / 2016

Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar Colloquium Mathematicum 143 (2016), 187-192 MSC: Primary 11G05; Secondary 14H52. DOI: 10.4064/cm6556-12-2015 Published online: 21 December 2015

Abstract

We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals $(p^3,q^3,r^3,s^3)$ not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers $p,q,r,s$ along with the extra integers $u,v$ satisfy $u^6+v^6+p^6+q^6=2(r^6+s^6)$, $uv=pq$, which, by previous work, has infinitely many integer solutions.

Authors

  • Farzali IzadiDepartment of Mathematics
    Faculty of Science
    Urmia University
    Urmia 165-57153, Iran
    e-mail
  • Foad KhoshnamDepartment of Pure Mathematics
    Faculty of Basic Science
    Azarbaijan Shahid Madani University
    Tabriz 53751-71379, Iran
    e-mail
  • Arman Shamsi ZargarDepartment of Pure Mathematics
    Faculty of Basic Science
    Azarbaijan Shahid Madani University
    Tabriz 53751-71379, Iran
    e-mail

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