CONGRUENT NUMBERS AND LOWER BOUNDS ON CLASS NUMBERS OF REAL QUADRATIC FIELDS

Abstract

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields Q(root t(4) - n(2)) as t varies over positive integers for a congruent number n. Furthermore, we provide lower bounds on class numbers of Richaud-Degert type real quadratic fields of the form Q(root n(2)k(4) - 1) for positive integers k and congruent numbers n whose elliptic curves have algebraic rank greater than 2.