On Properly $\theta$-Congruent Numbers Over Real Number Fields

TL;DR

This paper resolves open questions about properly $ heta$-congruent numbers over various real number fields, extending previous criteria and removing technical assumptions, with detailed analysis for specific cases.

Contribution

It provides complete answers to open questions on $ heta$-congruent numbers, removing assumptions and extending results to new classes of real number fields.

Findings

Criteria for $ heta$-congruent numbers over real cubic and degree 6 fields

Removal of technical assumptions in previous results

Analysis of special cases $n=1, 2, 3, 6$

Abstract

The notion of $\theta$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $\theta$-congruent if it is the area of a rational triangle with an angle $\theta$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $\theta$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $\theta$-congruent and properly $\theta$-congruent numbers. In this work, we provide complete answers to those questions. Indeed, we remove a technical assumption from their result on fields with degrees coprime to $6$, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree~$6$, and examine the exceptional cases $n=1, 2, 3$ and $6$.