The integer sides of triangles with a prime side opposite to a pi/3 angle

Abstract

Consider the non-equilateral triangles with integer sides, one of which is prime and lies opposite a pi/3 angle. Given a positive integer it is easy to predict whether or not it can occur as the length of one of the two remaining sides of any such triangle. We will establish an asymptotic upper bound and conditionally an asymptotic lower bound for the rate of occurrence of such integers. Because it is natural to separate them, we prove results independently for odd and even integers. We expect the correct size to be a double logarithmic factor smaller than the upper bound, and a small logarithmic factor larger than the lower bound. We further expect that the results can be replicated for a wider range of triangles.