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pro vyhledávání: '"Zargar, Arman Shamsi"'
Let $S_l(M,N)$ denote a set of $\ell$ triples of positive integers whose parts have the same sum $M$ and the same product $N$. For each $2\leq\ell\leq 4$ we establish a connection between the set $S_l(M,N)$ and a family of elliptic curves. When $\ell
Externí odkaz:
http://arxiv.org/abs/2408.13867
A pair $(a, b)$ of positive integers is a pythagorean pair if $a^2 + b^2$ is a square. A pythagorean pair $(a, b)$ is called a pythapotent pair of degree $h$ if there is another pythagorean pair $(k,l)$, which is not a multiple of $(a,b)$, such that
Externí odkaz:
http://arxiv.org/abs/2405.12989
Autor:
Choudhry, Ajai, Zargar, Arman Shamsi
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 been published. All these methods yield parametric solutions in terms
Externí odkaz:
http://arxiv.org/abs/2403.19694
Autor:
Choudhry, Ajai, Zargar, Arman Shamsi
We obtain two parametric solutions of the diophantine equation $\phi(x_1, x_2, x_3)=\phi(y_1, y_2, y_3)$ where $\phi(x_1, x_2, x_3)$ is the octic form defined by $\phi(x_1, x_2, x_3)=x_1^8+ x_2^8 + x_3^8 - 2x_1^4x_2^4 - 2x_1^4x_3^4 - 2x_2^4x_3^4$. Th
Externí odkaz:
http://arxiv.org/abs/2206.14084
Autor:
Salami, Sajad, Zargar, Arman Shamsi
Let $k \subset {\mathbb C}$ be a number field and ${\mathcal E}$ be an elliptic curve that is isomorphic to the generic fiber of an elliptic surface defined over the rational function field $k(t)$ of the projective line ${\mathbb P}^1_k$. The set ${\
Externí odkaz:
http://arxiv.org/abs/2206.05372
We give new parametrisations of elliptic curves in Weierstrass normal form $y^2=x^3+ax^2+bx$ with torsion groups $\mathbb{Z}/10\mathbb{Z}$ and $\mathbb{Z}/12\mathbb{Z}$ over $\mathbb{Q}$, and with $\mathbb{Z}/14\mathbb{Z}$ and $\mathbb{Z}/16\mathbb{Z
Externí odkaz:
http://arxiv.org/abs/2106.06861
Autor:
Choudhry, Ajai, Zargar, Arman Shamsi
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 pairs of triangles, and prove that there exist infinitely many pairs
Externí odkaz:
http://arxiv.org/abs/2104.06270
Autor:
Salami, Sajad, Zargar, Arman Shamsi
We introduce a new generalization of $\theta$-congruent numbers by defining the notion of rational $\theta$-parallelogram envelope for a positive integer $n$, where $\theta \in (0, \pi)$ is an angle with rational cosine. Then, we study more closely s
Externí odkaz:
http://arxiv.org/abs/2012.13471
Autor:
Salami, Sajad, Zargar, Arman Shamsi
A positive integer $N$ is called a $\theta$-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 $\theta$ and its area is $N \sqrt{r^2-s^2}$, where $\theta \in (0, \pi)$, $\co
Externí odkaz:
http://arxiv.org/abs/2012.13451
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