There is a student in US. He has completed 4 hours of high school and is applying for college. As part of the application procedure, he has to submit a research paper in Mathematics. He is a very good student. He accelerated math and enrolled for Differential Equation, while still being in high school, although differential equations is a college level course. The colleges to which he is applying want to see a Math Research Paper. I am not from US. So I do not know what kind of paper will be good. Could you suggest some good topics? And also, any idea how long should the paper be (i.e. how many pages?) Please give any other details you think appropriate.