Heron triangles and a family of elliptic curves with rank zero

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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