Explicit criteria for 3p or 6p to be non-θ-congruent numbers

The θ-congruent number problem, introduced by Fujiwara, is an extended notion of the congruent number problem. We find the explicit criteria for numbers of the form 3p or 6p to qualify as θ-congruent numbers, where p is a prime and θ=π3 or 2π3. Our criteria are obtained in connection to the divisibility of the class numbers of the associated imaginary quadratic fields by some powers of 2, where p≡1or13(mod24). Furthermore, we establish quantitative lower bounds on the number of non-θ-congruent numbers of the form 3p and 6p separately, where p≡1or13(mod24). We use the method of 2-descent on the corresponding elliptic curve to obtain our results.