How should I start Undergraduate Research in Mathematics?
While it is possible for a highly competent mathematician to dole out a doable problem for an undergraduate to solve over the summer, empirical data has proven otherwise - i.e., it's rather hard for an undergrad to prove anything original in number theory if only given a few weeks during the summer.

For an REU in NT, it might be more realistic to expect to read some interesting topics in number theory or perform some numerical analysis. Take it as a bonus if you obtain any original result.

Number theory is known to be a very difficult topic to get into. Before you decide to commit, take courses in complex analysis and abstract algebra. They are crucial if you want to read more advanced texts in number theory.

Good luck!