Quantitative Bounds on the Number of Non-Congruent Numbers of the form 2PQ

Abstract

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 pi’s and qj’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=p1p2⋯pt with certain conditions; this depends on the parity of the number of pi’s, where the 2-part of the class number of the imaginary quadratic field Q(-pi) 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)≡(3,7)(mod8). Equivalently, we obtain a lower bound on the number of the corresponding congruent number elliptic curves E2pq (and EP, respectively) with Mordell-Weil rank zero, whose 2-primary part of the Shafarevich-Tate groups are isomorphic to (Z/2Z)2. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.