The $\text{thera}$-congruent numbers elliptic curves via a Fermat-type theorem

A positive integer $N$ is called a $\theta$-congruent number if there is a $\text{ta}$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $\theta$ and its area is $N\sqrt{r^2-s^2}$, where $\theta \in (0, \pi)$,$\cos(\theta)=s/r$,and$0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\text{ta}$-congruent number if and only if the elliptic curve $E_N^{\text{ta}}:y^2=x(x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\mathrm{\ne }1,2,3,6$ is a $\text{ta}$-congruent number if and only if rank of $E_N^{\text{ta}}(\text{Q})$ is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational $\theta$-triangle for a $\theta$-congruent number $N$ from given ones by generalizing the Fermat’s algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $\theta$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $E_N^\theta({\mathbb Q})$\mathrm{.}Then,basedontheadditionoftwodistinctpointsin$E_N^\theta({\mathbb Q})$,weprovideawaytofindnewrational$\ta$-trianglesforthe$\theta$-congruentnumber$N$ using given two distinct ones. Finally, we give an alternative proof for Fujiwara’s theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $E_N^{\theta }(\mathbb{Q})$ with corresponding rational $\theta$-triangles