How about Fibonacci numbers? Many of their mod $p$ properties can be easily deduced from arithmetic of finite fields. (e.g. $p|F_{p\pm 1}$, where the sign is determined by mod 5 class of $p$).

And similarly for Pascal's triangle. They have beautiful patterns mod $p$.

![picture from Wikipedia of Sierpinski's triangle][1]

Pell's equation and continued fractions are also beautiful.

If the students know holomorphic functions, then Dirichlet's theorem on arithmetic progressions is a cool topic.


  [1]: https://upload.wikimedia.org/wikipedia/commons/thumb/4/45/Sierpinski_triangle.svg/220px-Sierpinski_triangle.svg.png