It is shown that any primitive integral Apollonian circle packing captures a fraction of the prime numbers. Basically the method consists in applying the circle method, considering the curvatures produced by a well-chosen family of binary quadratic forms.
Cite
@article{arxiv.1105.5127,
title = {Integral Apollonian circle packings and prime curvatures},
author = {Jean Bourgain},
journal= {arXiv preprint arXiv:1105.5127},
year = {2012}
}