Some variants of the congruent number problem. I (Q2772944)

Let \(\theta\) be a real number with \(0<\theta <\pi\). A natural number \(n\) is said to be \(\theta\)-congruent if there exists a triangle with rational sides and an angle \(\theta\) whose area is \(n\sqrt{r^2-s^2}\), where \(\cos \theta= s/r\) with \(r,s\in\mathbb{Z}\), \(\text{gcd} (r,s)=1\), and \(r>0\). In this paper, among other results, the author proves the following. For a prime number \(p\), (i) if \(p\equiv 7,11,13\pmod {24}\), then \(p\) is not \(2\pi/3\)-congruent, and (ii) if \(p\equiv-1\pmod {24}\) then \(p,2p\), and \(3p\) are \(2\pi/3\)-congruent.