! I read another post on here that said "research is a must", and I am getting really down on myself. I am going to have my M.S. at the end of this semester and there's no more time for me to start a new project or learn the details of research in a new field. I will say my CS research is mathematical in nature in that I will come up with an idea, and try and both "prove" it mathematically as well as obtain experimental data to back this up by programming and then running simulations
Research is not a must for pure math PhD programs. While people do conduct mini projects, pure math classes get you asking the right kinds of questions. Still, as I said on the other thread, trying to understand a topic not as well-represented in the literature is a good exercise, to help you start getting used to synthesizing something.
You have more research experience than plenty of applicants. Your CS work is research-y enough - a lot of supposed 'pure math' research is graph theory this, graph theory that. Which is fine, but frankly not representative of what a lot of mainstream pure mathematics research in graduate school and beyond will be like.
Research IS kind of a must for many applied disciplines, where you cannot approximate grad level research by just paper and pencil work.