Rank of elliptic curves associated to Brahmagupta quadrilaterals

Farzali Izadi¹ ; Foad Khoshnam² ; Arman Shamsi Zargar²

¹ Department of Mathematics Faculty of Science Urmia University Urmia 165-57153, Iran
² Department of Pure Mathematics Faculty of Basic Science Azarbaijan Shahid Madani University Tabriz 53751-71379, Iran

Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 187-192.

Résumé

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.

Détail

DOI : 10.4064/cm6556-12-2015

Keywords: construct family elliptic curves six parameters arising system diophantine equations whose rank least five brahmagupta formula area cyclic quadrilaterals necessarily representing genuine geometric objects turns out parameters curves integers s along extra integers satisfy which previous work has infinitely many integer solutions