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arXiv · 2604.23450

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

Shamik Das·Sudipa Mondal
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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.

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Shamik Das, Sudipa Mondal. 2026-04-25. A necessary condition for a congruent number of the form $8k+3$. https://doi.org/10.1016/j.jalgebra.2026.04.009

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