We study the numbers of the form 2PQ to determine if they qualify as congruent numbers in connection to the 8-rank of the class groups of the associated imaginary quadratic fields, where P and Q are products of distinct primes \(p_i\) ’s and \(q_j\) ’s respectively with certain conditions. We focus on the case that in contrast to the previous developments on the case . Additionally, we establish a necessary condition for a congruent number of the form \(P= p_{1}p_{2} \cdots p_{t}\) with certain conditions; this depends on the parity of the number of \(p_i\) ’s, where the 2-part of the class number of the imaginary quadratic field \(\mathbb {Q}(\sqrt{-p_i})\) satisfies particular linear congruence relations modulo 8. Furthermore, we provide quantitative lower bounds on the number of non-congruent numbers of the form 2pq (and P, respectively) with certain conditions respectively, where p and q are primes satisfying \((p, q) \equiv (3, 7)\pmod {8}\) . Equivalently, we obtain a lower bound on the number of the corresponding congruent number elliptic curves \(E_{2pq}\) (and \(E_{P}\) , respectively) with Mordell-Weil rank zero, whose 2-primary part of the Shafarevich-Tate groups are isomorphic to \((\mathbb {Z}/2\mathbb {Z})^{2}\) .
