TITLE: Topics for an Undergraduate Expository Paper in Number Theory
QUESTION [11 upvotes]: I am teaching an undergraduate course in number theory and am looking for topics that students could take on to write an expository paper (~10 pages). No new results are expected of them. Many of the students have an undergraduate course in abstract algebra and a course in real analysis (but few have any complex analysis background).
The only topics that I have come up with are
1.) Elliptic curves
2.) Cryptography
Of course these are related but I think these could be two projects. Other topics (such as the prime number theorem) seem too difficult to me. What other good projects are there? In particular are there good projects based on analysis? References would be greatly appreciated, including references for the two projects above.

REPLY [2 votes]: Arithmetic functions -- there are some nice, very accessible results about arithmetic functions and some questions about these (notably the sum of divisors, $\sigma$) date back to the Greeks (perfect numbers, amicable pairs, etc.)  There is also a nice algebra of arithmetic functions using the operation of convolution.  Many basic number-theoretic ideas (RSA!) have arithmetic functions hiding in the background.  
Example of arithmetic functions include the Euler totient $\phi$-function, $\sigma(n)=$ sum of divisors of $n$, $\tau(n)=$ number of prime divisors of $n$ and the Moebius function $\mu$.  
I once had a small team of undergraduates (who had no higher mathematics in their backgrounds) create their own arithmetic functions and analyze questions about iterates of the functions.