Required condition for a congruent number: pq with primes p ≡ 1 (mod 8) and q ≡ 3 (mod 8)

Abstract

In this paper, we establish a crucial requirement for a number of the form n, having two prime factors p and q such that $(p,q)\equiv (1,3)\left(\mathrm{mod}8\right)$, to qualify as a congruent number. Specifically, we present congruence relations modulo 16 for the 2-part of the class number of the imaginary quadratic field $Q\left(\sqrt{-2pq}\right)$ when n is congruent.