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

Abstract

A positive integer N is called a θ-congruent number if there is a θ-triangle (a, b, c) with rational sides for which the angle between a and b is equal to θ and its area is N √ r 2 − s 2 , where θ ∈ (0, π), cos(θ) = s/r , and 0 ≤ |s| < r are coprime integers. It is attributed to Fujiwara (Number Theory, de Gruyter, pp 235-241, 1997) that N is a θ-congruent number if and only if the elliptic curve E θ N : 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 = 1, 2, 3, 6 is a θ-congruent number if and only if rank of E θ N (Q) is greater than zero. In this paper, we answer positively to a question concerning with the existence of methods to create new rational θ-triangle for a θ-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 θ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in E θ N (Q). Then, based on the addition of two distinct points in E θ N (Q), we provide a way to find new rational θ-triangles for the θ-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 θ N (Q) with corresponding rational θ-triangles.