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

Abstract

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 $n=p_1{p}_2\cdots p_{t}q$ be a square-free integer, where each $p_i\equiv 1(\mathrm{mod}8)$ and $q\equiv 3(\mathrm{mod}8)$, 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 $\mathbb{Q}(\sqrt{-p_1{p}_2\cdots p_t})$, under the assumption that $n$ is a congruent number, using a modified Rédei matrix.

12 pages