Tannersfriend, just for clairfication, protoytpes are not required for patent. A mathematical concept, demonstrated "in theory" that others can follow and understand is sufficient tp make patent application.

I have added this quote, for would be inventors, from the US Attorney General's office:

"United States patent laws do not require you to have a prototype in order to apply for a patent, all that is required is that you be able to describe the invention so that others could both make and use it. So while you do need to have some kind of identifiable embodiment you can start by proving your concept on paper.