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.
Best Answer
The transcendence of $2^{\sqrt{2}}$ and $e^{\pi}$: Gel'fond's proof. (Assuming some basic complex analysis).
Nathanson's problem: show that $3^n \nmid 5^n-2$ for $n > 1$. (This involves the $p$-adic analog of the above topic).
More elementary (see also Yuhao Huang's answer above): the determination of $F_p \mod{p}$ for the Fibonacci sequence (i.e., periodic modulo $5$), as a consequence of the congruence $2\cos{(px)} \equiv 2\cos^p(x) \mod{p\mathbb{Z}[\cos{x}]}$ and the formula $(1+\sqrt{5})/4 = \cos{(2\pi/10)}$. This I think is a good, algebraic point of entry into quadratic reciprocity (obviously, it is equivalent to the splitting law for $\mathbb{Q}(\sqrt{5})$). A key point of course is to explain that $1/p$ is not a sum of roots of 1 (or, if one prefers, that is not an albgebraic integer), so that the congruence may be exploited appropriately. Interpretation as a "Fermat's little theorem" for $\mathbb{Q}(\sqrt{5})$.
Related: Exhibit a formula showing that $\sqrt{N} \in \mathbb{Q}\big( e^{2\pi i/4N} \big)$, and perhaps use this to conclude that the residue of $N^{\frac{p-1}{2}} \mod{p}$ only depends on the residue of $p \mod{4N}$.
$\mathbb{Q}$ has no unramified extensions.
There is always a prime between $n$ and $2n$: Erdos' elementary proof.
The Wolstenholme-Jacobsthal congruence $\binom{np}{mp} \equiv \binom{n}{m} \mod{p^3}$ using the "Stirling formula" for the $p$-adic $\Gamma$-function. Or combinatorial proofs of such congruences. (Or indeed, any other congruence from A. Granville's "Arithmetic Properties of Binomial Coefficients.")
Completely elementary: Zsygmondy's theorem and applications. (Here is one: find all integer solutions of $a^n = b^n + c^k$ subject to $|c| \leq n$).
If $a^n - 1 \mid b^n - 1$ for all $n > 0$, then $b = a^j$. If $a4^n + b6^n + c9^n$ is a perfect square for each $n$, then $(a,b,c) = (r^2,2rs,s^2)$. Solve, and generalize both!