ON PROPERLY $\theta $ -CONGRUENT NUMBERS OVER REAL NUMBER FIELDS

Abstract The notion of $\theta $ -congruent numbers generalises the classical congruent number problem. 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 [‘On $\theta $ -congruent numbers over real number fields’, Bull. Aust. Math. Soc. 103 (2) (2021), 218–229] 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 six, provide a definitive answer for real cubic fields without congruence restrictions, extend the analysis to fields of degree six and examine the exceptional cases $n=1, 2, 3$ and $6$ .