Having much of your grade determined by a writing project is very unusual in a mathematics class. To do well on the final project, you must write something that is of very high quality. It does not have to include any original research, but must (1) show that you completely understand what you are writing, and (2) can write clear correct succinct mathematics with no typos.
You are strongly encouraged to start thinking about your project right now! Flip through the course textbook, think about number theory you've heard about, etc., and come with an idea, and come talk to me about it or email me.
So that you have a sense of how the projects will be graded, you will have two chances to turn in your project before it is due for a preliminary grading. I will grade the early version, but this grade will in no way influence your actual course grade. The only relation, is that your final project grade will be at least the preliminary grade. For example, suppose you turn in a draft of your project two weeks before your final project is due, and I look at it and give it a grade of 71% (out of 100%). Then your final project will receive a grade of at least 71%, but that preliminary 71% will in no way count against you.
All projects that receive a grade of at least 85% will be placed on the course website for others to look at in the future. (The grade will not be listed.)