A variant of the congruent number problem

Abstract

A positive integer $n$ is called a $\theta$-congruent number if there is a triangle with sides $a,b$ and $c$ for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $n\sqrt{r^2-s^2}$, where $0<\theta <\pi$, $\cos \theta =s/r$ and $0\le |s|<r$ are relatively prime integers. The case $\theta =\pi /2$ refers to the classical congruent numbers. It is known that the problem of classifying $\theta$-congruent numbers is related to the existence of rational points on the elliptic curve $y^2=x(x+(r+s)n)(x-(r-s)n)$. In this paper, we deal with a variant of the congruent number problem where the cosine of a fixed angle is $\pm \sqrt{2}/2$.

20 pages