A variant of the congruent number problem

TL;DR

This paper explores a variant of the congruent number problem focusing on triangles with a fixed angle whose cosine is ±√2/2, linking the problem to rational points on specific elliptic curves.

Contribution

It introduces a new variant of the congruent number problem with a fixed cosine value and analyzes its connection to elliptic curves.

Findings

Relation between the variant and rational points on elliptic curves

Characterization of $ heta$-congruent numbers for $ heta$ with cosine ±√2/2

Extension of classical congruent number problem to specific fixed angles

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 $\pm \sqrt{2}/2$.