A VARIANT OF THE CONGRUENT NUMBER PROBLEM

抄録

A positive integer n is called a 𝜃-congruent number if there is a triangle with sides a, b and c for which the angle between a and b is equal to 𝜃 and its area is n√r ² − s ², where 0 < 𝜃 < 𝜋, cos 𝜃 = s/r and 0 ≤ |s| < r are relatively prime integers. The case 𝜃 = 𝜋/2 refers to the classical congruent numbers. It is known that the problem of classifying 𝜃-congruent numbers is related to the existence of rational points on the elliptic curve y ² = x (x + (r + s) n )( x − (r − s )n ). In this paper, we deal with a variant of the congruent number problem where the cosine of a fixed angle is ±√2/2.