A necessary condition for \(2pq\) to be a congruent number

Abstract

In this article, we derive a necessary condition for a number of the form n = \(2pq\), where p and q are distinct prime numbers satisfying \((p, q) \equiv (1, 5) \pmod{8}\), to be a congruent number. Specifically, we obtain congruence relations involving the 2-part of the class number of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-pq})\), which must hold whenever n is congruent.