2025/12/18 by Salami, Sajad, Zargar, Arman Shamsi
#11R16 #FOS: Mathematics #Number Theory (math.NT) #primary 11G05 #secondary 11R21
paper · doi:10.48550/arxiv.2512.16597
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.