A VARIANT OF THE CONGRUENT NUMBER PROBLEM

A positive integer n is called a theta-congruent number if there is a triangle with sides a , b and c for which the angle between a and b is equal to theta and its area is n root r(2) - s(2) , where 0< theta < pi , cos theta s/r s / r and 0 <= | s | < r are relatively prime integers. The case theta = pi/2 refers to the classical congruent numbers. It is known that the problem of classifying theta-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 +/-root 2 / 2.