@ElenaParent, your son sounds brilliant. He is clearly super intelligent and ready for further study in math at the graduate level.
But is he really only 16? Is there anyone with whom he can socialize? Sure, he can talk math. But I worry about his finding balance in life. He needs more than math to be a fully rounded adult. Does he have friends? Does he do any activities outside of math where he is around others his age? It is highly unlikely that other PhD students who are 22 years old and older are going to chat with him outside of class beyond saying “hi” in class and talking about the weather. He will have nothing in common with them.
Maybe a Master’s would be a good way for him to continue to grow as a mathematician while at the same time putting his doctoral studies on hold for another year or two to allow him to mature. Sure, he will likely get into some of those PhD programs…but oh, I worry for him.