Go to a good math library, pick up a few books (perhaps even randomly) in number theory and start reading them until you find one book which you really like. I am (almost) sure that you will ask yourself plenty of questions when studying deeply some interesting mathematics. At the beginning, many of these questions are already solved (and perhaps not hard) but questions help advancing. At some point, some of your questions will have no known answer. With some luck, one of these questions is interesting and solvable. (Books can of course be replaced by attendance of lectures, discussions with collegues etc.)

The point is that one needs often some external input and good books can provide this.

A last piece of advice: Do not chose the crowdest corners (e.g. around Riemann hypothesis), at least at the beginning, if you want to make some contributions.