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arXiv · 2105.03957

Heron triangles and a family of elliptic curves with rank zero

Vinodkumar Ghale·Shamik Das·Debopam Chakraborty
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Abstract

Given any positive integer $n$, it is well-known that there always exists a triangle with rational sides $a,b$ and $c$ such that the area of the triangle is $n$. For a given prime $p \not \equiv 1$ modulo $8$ such that $p^{2}+1=2q$ for a prime $q$, we look into the possibility of the existence of the triangles with rational sides with $p$ as the area and $\frac{1}{p}$ as $\tan \frac{\theta}{2}$ for one of the angles $\theta$. We also discuss the relation of such triangles with the solutions of certain Diophantine equations.

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Vinodkumar Ghale, Shamik Das, Debopam Chakraborty. 2021-05-09. Heron triangles and a family of elliptic curves with rank zero. https://doi.org/10.1007/s10998-022-00491-5

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