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arXiv · 2308.14381

A variant of the congruent number problem

Jerome T. Dimabayao·Soma Purkait
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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 $0 \leq |s| < r$ 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 $\pm \sqrt{2}/2$.

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Jerome T. Dimabayao, Soma Purkait. 2023-08-28. A variant of the congruent number problem. https://arxiv.org/abs/2308.14381

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