NONCONGRUENT NUMBERS OF THE FORM 2 p q $2pq$ 2 p q WITH p q ≡ 5 ( mod 8 ) $pq\equiv 5\ (\textbf {mod } 8)$ p q identical to 5 left parenthesis mod 8 right parenthesis

Abstract Let p and q be two primes with ( p , q ) ≡ ( 1 , 5 ) $(p,q)\equiv (1,5)$ left parenthesis p comma q right parenthesis identical to left parenthesis 1 comma 5 right parenthesis or ( 7 , 3 ) ( mod 8 ) $(7,3) \pmod 8$ left parenthesis 7 comma 3 right parenthesis left parenthesis mod 8 right parenthesis . 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 $K_2$ upper K 2 ’, Math. Ann. 383 (3–4) (2022), 1647–1686] showed that if ( q p ) = − 1 $(\frac {q}{p})=-1$ left parenthesis StartFraction q Over p EndFraction right parenthesis equals negative 1 , then 2 p q $2pq$ 2 p q is not a congruent number. By using Qin’s method, we prove that if ( p , q ) ≡ ( 1 , 5 ) ( mod 8 ) $(p,q)\equiv (1,5) \pmod 8$ left parenthesis p comma q right parenthesis identical to left parenthesis 1 comma 5 right parenthesis left parenthesis mod 8 right parenthesis and