NONCONGRUENT NUMBERS OF THE FORM $2pq$ WITH $pq\equiv 5\ (\textbf {mod } 8)$

Abstract Let p and q be two primes with $(p,q)\equiv (1,5)$ or $(7,3) \pmod 8$ . Lagrange [‘Nombres congruents et courbes elliptiques’, Séminaire Delange-Pisot-Poitou. Théorie des Nombres 16 (1) (1974–1975), Article no. 16] and Qin [‘Congruent numbers, quadratic forms and $K_2$ ’, Math. Ann. 383 (3–4) (2022), 1647–1686] showed that if $(\frac {q}{p})=-1$ , then $2pq$ is not a congruent number. By using Qin’s method, we prove that if $(p,q)\equiv (1,5) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-pq)\not \equiv p-1 \pmod {16}$ , then $2pq$ is not a congruent number; if $(p,q)\equiv (7,3) \pmod 8$ and $(\frac {q}{p})=1$ with $h(-2pq)\not \equiv p+1 \pmod {16}$ , then $2pq$ is not a congruent number. Here, $h(-d)$ denotes the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-d})$ .