FROM BINARY QUADRATIC FORMS TO POTENTIAL FAMILIES OF CONGRUENT NUMBERS

We investigate specialized families of integers arising from the classical binary quadratic forms both in general settings and under Pythagorean constraints. For each in these families, we consider the associated elliptic curve and compute its local and global root numbers. According to the Birch and Swinnerton-Dyer conjecture, values of with are potential congruent numbers. Our initial analysis shows that the families and produce integers , with and , whereas the Pythagorean families and appear much poorer in this respect. To overcome this limitation, we introduce quadratic twists (with small square-free ) which permits a systematic "rescreening". The rescue effect recovers many additional candidates, balancing the distribution of potential congruent numbers across all families.