The θ-Congruent Number Elliptic Curves via Fermat-type Algorithms

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. and its area is N root r(2) - s(2), where theta is an element of (0, pi), cos(theta) = s/r, and 0 <= vertical bar s vertical bar < 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 not equal 1, 2, 3,6 is a theta-congruent number if and only if rank of E-N(theta) (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) (Q). Then, based on the addition of two distinct points in E-N(theta) (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) (Q) with corresponding rational theta-triangles.