A. S. Zargar

activity

20152025

collaborators

17 papers

math.NT2025

On Properly $θ$ -Congruent Numbers Over Real Number Fields

Sajad Salami, Arman Shamsi Zargar

The notion of $θ$ -congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $θ$ -congruent if it is the area of a rational triangle…

math.NT2025

The splitting fields and Generators of Shioda's elliptic surfaces $y^{2} = x^{3} + t^{m} + 1$ (I)

Sajad Salami, Arman Shamsi Zargar

The splitting field of an elliptic surface $E$ defined over $Q (t)$ is the smallest subfield $K$ of $C$ such that ${\mathcal E}({\mathbb C}(t))\c…

math.NT2025

Integer triangles with a rational ratio of circumcircle radius to excircle radius

Lorenz Halbeisen, Norbert Hungerbühler, Arman Shamsi Zargar

We consider the problem of finding integer triangles with $R / r$ a positive rational, where $R$ and $r$ are the radii of the circumcircle and an excircle, respectively. We show that…

math.NT2024

Partitions into Triples with Equal Products and Families of Elliptic Curves

Ahmed El Amine Youmbai, Arman Shamsi Zargar, Maksym Voznyy

Let $S_{l} (M, N)$ denote a set of $ℓ$ triples of positive integers having the same sum $M$ and the same product $N$ . For each $2 \leq ℓ \leq 4$ we establish a connection between a…

math.GM2024

The diophantine equation $x^{4} + y^{4} = z^{4} + w^{4}$

Ajai Choudhry, Arman Shamsi Zargar

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^{4} + y^{4} = z^{4} + w^{4}$ have…

math.NT2022

An octic diophantine equation and related families of elliptic curves

Ajai Choudhry, Arman Shamsi Zargar

We obtain two parametric solutions of the diophantine equation $ϕ (x_{1}, x_{2}, x_{3}) = ϕ (y_{1}, y_{2}, y_{3})$ where $ϕ (x_{1}, x_{2}, x_{3})$ is the octic form defined by $ϕ(x_1, x_2, x_3)=x_1^8+ x_…

math.NT2022

Generators and splitting fields of certain elliptic K3 surfaces

Sajad Salami, Arman Shamsi Zargar

Let $k \subset C$ be a number field and $E$ be an elliptic curve defined over $k (t)$ , the rational function field of the projective line $P_{k}^{1}$ , is…

math.NT2021

Pairs of equiperimeter and equiareal triangles whose sides are perfect squares

Ajai Choudhry, Arman Shamsi Zargar

In this paper we consider the problem of finding pairs of triangles whose sides are perfect squares of integers, and which have a common perimeter and common area. We find two such…

math.NT2020

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

Sajad Salami, Arman Shamsi Zargar

A positive integer $N$ is called a $θ$ -congruent number if there is a $\ta$-triangle $(a, b, c)$ with rational sides for which the angle between $a$ and $b$ is equal to $θ$ and its a…

math.NT2020

Rational $θ$ -parallelogram envelopes via $θ$ -congruent elliptic curves

Sajad Salami, Arman Shamsi Zargar

We introduce a new generalization of $θ$ -congruent numbers by defining the notion of rational $θ$ -parallelogram envelope for a positive integer $n$ , where $θ \in (0, π)$ is an angle…

math.NT2019

A sextic diophantine chain and a related Mordell curve

Ajai Choudhry, Arman Shamsi Zargar

In this paper we obtain parametric as well as numerical solutions of the sextic diophantine chain $ϕ (x_{1}, y_{1}, z_{1}) = ϕ (x_{2}, y_{2}, z_{2}) = ϕ (x_{3}, y_{3}, z_{3}) = k$ where $ϕ (x, y, z)$ …

math.NT2018

Evaluation of Gaussian hypergeometric series using Huff's models of elliptic curves

Mohammad Sadek, Nermine El-Sissi, Arman Shamsi Zargar +1

A Huff curve over a field $K$ is an elliptic curve defined by the equation $a x (y^{2} - 1) = b y (x^{2} - 1)$ where $a, b \in K$ are such that $a^{2} \neq = b^{2}$ . In a similar fashion, a general Huff c…

math.NT2017

On the Diophantine equations $z^{2} = f (x)^{2} \pm g (y)^{2}$ concerning Laurent polynomials

Yong Zhang, Arman Shamsi Zargar

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^{2} = f (x)^{2} \pm g (y)^{2}$ for some simple…

math.NT2016

On the Diophantine equations $z^{2} = f (x)^{2} \pm f (y)^{2}$ involving Laurent polynomials

Yong Zhang, Arman Shamsi Zargar

By the theory of elliptic curves, we study the nontrivial rational parametric solutions and rational solutions of the Diophantine equations $z^{2} = f (x)^{2} \pm f (y)^{2}$ for some simple…

math.NT2016

A parametrised family of Mordell curves

Ajai Choudhry, Arman Shamsi Zargar

An elliptic curve defined by an equation of the type $y^{2} = x^{3} + d$ is called a Mordell curve. We obtain a parametrised family of Mordell curves whose rank, in general, is at least th…

math.NT2016

On Parametric Spaces of Bicentric Quadrilaterals

Farzali Izadi, Foad Khoshnam, Allan J. MacLeod +1

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both a circumcircle passing through the four vertices and an incircle having the four sides as t…

math.NT2015

Rank of elliptic curves associated to the Brahmagupta quadrilaterals

Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar

In this paper, we construct a family of elliptic curves of rank at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals $(p^{3}, q^{3}, r^{3}, q^{3})$ no…

On Properly θ-Congruent Numbers Over Real Number Fields

The splitting fields and Generators of Shioda's elliptic surfaces y2=x3+tm+1 (I)

Integer triangles with a rational ratio of circumcircle radius to excircle radius

Partitions into Triples with Equal Products and Families of Elliptic Curves

The diophantine equation x4+y4=z4+w4

An octic diophantine equation and related families of elliptic curves

Generators and splitting fields of certain elliptic K3 surfaces

Pairs of equiperimeter and equiareal triangles whose sides are perfect squares

The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem

Rational θ-parallelogram envelopes via θ-congruent elliptic curves

A sextic diophantine chain and a related Mordell curve

Evaluation of Gaussian hypergeometric series using Huff's models of elliptic curves

On the Diophantine equations z2=f(x)2±g(y)2 concerning Laurent polynomials

On the Diophantine equations z2=f(x)2±f(y)2 involving Laurent polynomials

A parametrised family of Mordell curves

On Parametric Spaces of Bicentric Quadrilaterals

Rank of elliptic curves associated to the Brahmagupta quadrilaterals

On Properly $θ$ -Congruent Numbers Over Real Number Fields

The splitting fields and Generators of Shioda's elliptic surfaces $y^{2} = x^{3} + t^{m} + 1$ (I)

The diophantine equation $x^{4} + y^{4} = z^{4} + w^{4}$

Rational $θ$ -parallelogram envelopes via $θ$ -congruent elliptic curves

On the Diophantine equations $z^{2} = f (x)^{2} \pm g (y)^{2}$ concerning Laurent polynomials

On the Diophantine equations $z^{2} = f (x)^{2} \pm f (y)^{2}$ involving Laurent polynomials