A variant of the congruent number problem (Q6654967)

Let \(n\) be a given positive integer. We say that \(n\) is \(\theta\)-congruent number if there is a triangle with sides \(a, b, c\) for which the angle between \(a\) and \(b\) is equal to \(\theta\) and its area is equal to \(n\sqrt{r^2-s^2}, 0<|s|<r\) and \(0<\theta<\pi\) with \(\cos \theta=r/s\). The case when \(\theta=\pi/2\) is related to congruent numbers and has a long and rich history. There are many other variants (with essentially the same name) of the notion of \(\theta\)-congruent numbers. In the paper the the authors study the case when \(\theta=\pi/4\) or \(\theta=3\pi/4\). In these cases \(\cos \theta=\pm \sqrt{2}/2\). If \(\theta =\pi/4\) they ask about the existence of positive rational \(a, b, c\) such that \(a, b\sqrt{2}, c\) are sides of the triangle and the angle \(\pi/4\) is opposite to the side \(c\). For given \(n\), the problem is equivalent with the study of rational points on the elliptic curve \(E_{n}:\; y^2=x^3+2nx^2-n^2x\). In case of \(\theta=3\pi/4\) the corresponding elliptic curve takes the form \(E_{n}:\; y^2=x^3-2nx^2-n^2x\). There are many results proved in the paper. In particular, the existence of \(\pi/4\)-congruent numbers with arbitrarily many prime factors is obtained, as well as the existence of infinitely many numbers which are not \(\pi/4\)-congruent. Moreover, in this context, some applications of Birch-Swinerton-Dyer conjectures are obtained. Similar results were obtained in the case when \(\theta=3\pi/4\).