A necessary condition for a congruent number of the form $8k+3$

A positive square-free integer is called a \textit{congruent number} if it arises as the area of a right triangle with rational side lengths. Let $n = p_1p_2 \cdots p_t q$ be a square-free integer, where each $p_i \equiv 1 \pmod{8}$ and $q \equiv 3 \pmod{8}$, with the $p_i$ and $q$ being distinct primes. In this article, we present a congruence relation modulo powers of 2 between the 2-part of the class numbers of $\mathbb{Q}(\sqrt{-n})$ and $\mathbb{Q}(\sqrt{-p_1p_2 \cdots p_t})$, under the assumption that $n$ is a congruent number, using a modified R\'edei matrix.