On Properly $θ$-Congruent Numbers Over Real Number Fields
Sajad Salami, Arman Shamsi Zargar
TL;DR
The paper resolves four questions about when being $(K,\theta)$-congruent implies being properly $(K,\theta)$-congruent for real number fields. It extends the equivalence to all real fields with degrees coprime to $6$ and to real cubic fields by showing torsion cannot grow beyond $2$-torsion on $E_{n,\theta}$, and by ruling out possible $3$- and higher-torsion via division-polynomial and Galois-structure analyses. It also addresses fields with degree divisible by $6$ (sextic fields) and analyzes exceptional $n$ values $\{1,2,3,6\}$ with explicit examples and rank-twist arguments. Collectively, these results clarify when θ-congruence over real fields guarantees infinite families of θ-triangle realizations and provide concrete instances where exceptions occur.
Abstract
The notion of $θ$-congruent numbers generalizes the classical congruent number problem. Recall that a positive integer $n$ is $θ$-congruent if it is the area of a rational triangle with an angle $θ$ whose cosine is rational. Das and Saikia [2] established criteria for numbers to be $θ$-congruent over certain real number fields and concluded their work by posing four open questions regarding the relationship between $θ$-congruent and properly $θ$-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 $6$.