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

TL;DR

This paper establishes a congruence relation involving class numbers of quadratic fields, providing a necessary condition for certain congruent numbers of the form 8k+3.

Contribution

It introduces a new congruence relation modulo powers of 2 between class numbers of specific quadratic fields, assuming the number is congruent.

Findings

Derived a congruence relation between class numbers of quadratic fields.

Identified a necessary condition for a number of the form 8k+3 to be congruent.

Utilized a modified Rédéi matrix in the analysis.

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_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.