Question 101
Find all values of the positive integer $\kappa$ satisfying the following property:
There do not exist positive integers $\alpha, \beta$ such that the number
$$
A(\kappa, \alpha, \beta) = \frac{\alpha + \beta}{\alpha^2 + \kappa^2 \beta^2 - \kappa^2 \alpha \beta}
$$
is a composite positive integer.
There do not exist positive integers $\alpha, \beta$ such that the number
$$
A(\kappa, \alpha, \beta) = \frac{\alpha + \beta}{\alpha^2 + \kappa^2 \beta^2 - \kappa^2 \alpha \beta}
$$
is a composite positive integer.