Rank of elliptic curves associated to Brahmagupta quadrilaterals

Farzali Izadi, Foad Khoshnam, and Arman Shamsi Zargar. "Rank of elliptic curves associated to Brahmagupta quadrilaterals." Colloquium Mathematicae 143.2 (2016): 187-192. <http://eudml.org/doc/284096>.

@article{FarzaliIzadi2016,
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 (p³,q³,r³,s³) 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 u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.},
author = {Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar},
journal = {Colloquium Mathematicae},
keywords = {brahmagupta formula; Heron formula; quadrilaterals; Diophantine equation; elliptic curves; rank of elliptic curves},
language = {eng},
number = {2},
pages = {187-192},
title = {Rank of elliptic curves associated to Brahmagupta quadrilaterals},
url = {http://eudml.org/doc/284096},
volume = {143},
year = {2016},
}

TY - JOUR
AU - Farzali Izadi
AU - Foad Khoshnam
AU - Arman Shamsi Zargar
TI - Rank of elliptic curves associated to Brahmagupta quadrilaterals
JO - Colloquium Mathematicae
PY - 2016
VL - 143
IS - 2
SP - 187
EP - 192
AB - 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 (p³,q³,r³,s³) 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 u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions.
LA - eng
KW - brahmagupta formula; Heron formula; quadrilaterals; Diophantine equation; elliptic curves; rank of elliptic curves
UR - http://eudml.org/doc/284096
ER -