Congruent numbers and lower bounds on class numbers of real quadratic fields

We give effective lower bounds on caliber numbers of the parametric family of real quadratic fields \mathbb {Q}(\sqrt {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 \mathbb {Q}(\sqrt {n^2k^4-1}) for positive integers k and congruent numbers n whose elliptic curves have algebraic rank greater than 2.