Rank of elliptic curves associated to Brahmagupta quadrilaterals
Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 187-192
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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.
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
Affiliations des auteurs :
Farzali Izadi 1 ; Foad Khoshnam 2 ; Arman Shamsi Zargar 2
@article{10_4064_cm6556_12_2015,
author = {Farzali Izadi and Foad Khoshnam and Arman Shamsi Zargar},
title = {Rank of elliptic curves associated to {Brahmagupta} quadrilaterals},
journal = {Colloquium Mathematicum},
pages = {187--192},
publisher = {mathdoc},
volume = {143},
number = {2},
year = {2016},
doi = {10.4064/cm6556-12-2015},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm6556-12-2015/}
}
TY - JOUR AU - Farzali Izadi AU - Foad Khoshnam AU - Arman Shamsi Zargar TI - Rank of elliptic curves associated to Brahmagupta quadrilaterals JO - Colloquium Mathematicum PY - 2016 SP - 187 EP - 192 VL - 143 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/cm6556-12-2015/ DO - 10.4064/cm6556-12-2015 LA - en ID - 10_4064_cm6556_12_2015 ER -
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Farzali Izadi; Foad Khoshnam; Arman Shamsi Zargar. Rank of elliptic curves associated to Brahmagupta quadrilaterals. Colloquium Mathematicum, Tome 143 (2016) no. 2, pp. 187-192. doi: 10.4064/cm6556-12-2015
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