A new family of elliptic curves with positive ranks arising from the Heron triangles (original) (raw)

On elliptic curves via Heron triangles and Diophantine triples

‎In this article‎, ‎we construct families of elliptic curves arising from the Heron triangles and Diophantine triples with the Mordell-Weil torsion subgroup of BbbZ/2BbbZtimesBbbZ/2BbbZ\Bbb{Z}/2\Bbb{Z}\times \Bbb{Z}/2\Bbb{Z}BbbZ/2BbbZtimesBbbZ/2BbbZ‎. ‎These families have ranks at least 2 and 3‎, ‎respectively‎, ‎and contain particular examples with rank equal to 777.‎

A New Family Of Elliptic Curves With Positive Rank arising from Pythagorean Triples

arXiv (Cornell University), 2010

The aim of this paper is to introduce a new family of elliptic curves in the form of y 2 = x(x −a 2)(x −b 2) that have positive ranks. We first generate a list of pythagorean triples (a, b, c) and then construct this family of elliptic curves. It turn out that this new family have positive ranks and search for the upper bound for their ranks.

Some new families of positive-rank elliptic curves arising from Pythagorean triples

Notes on Number Theory and Discrete Mathematics, 2018

In the present paper, we introduce some new families of elliptic curves with positive rank arising from Pythagorean triples. We study elliptic curves of the form y 2 = x 3 − A 2 x + B 2 , where A, B ∈ {a, b, c} are two different numbers and (a, b, c) is a rational Pythagorean triple. First of all, we prove that if (a, b, c) is a primitive Pythagorean triple (PPT), then the rank of each family is positive. Furthermore, we construct subfamilies of rank at least 3 in each family but one with rank at least 2, and obtain elliptic curves of high rank in each family. Finally, we consider two other new families of elliptic curves of the forms y 2 = x(x − a 2)(x + c 2) and y 2 = x(x − b 2)(x + c 2), and prove that if (a, b, c) is a PPT, then the rank of each family is positive.

On a Family Of Elliptic Curves With Positive Rank arising from Pythagorean Triples

The aim of this paper is to introduce a new family of elliptic curves in the form of y2=x(x−a2)(x−b2)y^2=x(x-a^2)(x-b^2)y2=x(x−a2)(x−b2) that have positive ranks. We first generate a list of pythagorean triples (a,b,c)(a,b,c)(a,b,c) and then construct this family of elliptic curves. It turn out that this new family have positive ranks and search for the upper bound for their ranks.

A Family of Elliptic Curves With Rank geq5\geq5geq5

In this paper, we construct a family of elliptic curves with rank geq5\geq 5geq5. To do this, we use the Heron formula for a triple (A2,B2,C2)(A^2, B^2, C^2)(A2,B2,C2) which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers AAA, BBB, and CCC, along with the extra parameter DDD satisfy the quartic Diophantine equation A4+D4=2(B4+D4)A^4+D^4=2(B^4+D^4)A4+D4=2(B4+D4). keywords {Diophantine equation, elliptic curve and Heron formula}

A Family of Elliptic Curves With Rank ≥5

2015

In this paper, we construct a family of elliptic curves with rank ≥ 5. To do this, we use the Heron formula for a triple (A^2, B^2, C^2) which are not necessarily the three sides of a triangle. It turns out that as parameters of a family of elliptic curves, these three positive integers A, B, and C, along with the extra parameter D satisfy the quartic Diophantine equation A^4+D^4=2(B^4+D^4).

#A10 Integers 19 (2019) Elliptic Curves Arising from the Triangular Numbers

2019

We study the Legendre family of elliptic curves Et : y2 = x(x 1)(x t), parametrized by triangular numbers t = t(t + 1)/2. We prove that the rank of Et over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.