The $${\theta }$$-Congruent Number Elliptic Curves via Fermat-type Algorithms

Abstract

A positive integer N is called a $$\theta $$ -congruent number if there is a $${\theta }$$ -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 $$0 \le |s|<r$$ are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235–241, 1997) that N is a $${\theta }$$ -congruent number if and only if the elliptic curve $$E_N^{\theta }: y^2=x (x+(r+s)N)(x-(r-s)N)$$ has a point of order greater than 2 in its group of rational points. Moreover, a natural number $$N\ne 1,2,3,6$$ is a $${\theta }$$ -congruent number if and only if rank of $$E_N^{\theta }({{\mathbb {Q}}})$$ is greater than zero. In this paper, we answer positively to a question concerning with 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 $$E_N^{\theta }({{\mathbb {Q}}})$$ . Then, based on the addition of two distinct points in $$E_N^{\theta }({{\mathbb {Q}}})$$ , we provide a way to find new rational $${\theta }$$ -triangles for the $${\theta }$$ -congruent number N using given two distinct ones. Finally, we give an alternative proof for the 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.