Topics for an Undergraduate Expository Paper in Number Theory

Question

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.

Answer 1

• 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!

Answer 2

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 $p$.

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.

Answer 3

Just on top of my head, I can think of the following themes, all having in common that they can be approached with a minimal knowledge and that there exists a continuous path from them to current research (those with a star could be linked to real analysis):

Quadratic reciprocity*, higher reciprocity laws (starting with the biquadratic character of 2), sample cases of Fermat's last theorem, primality testing*, the last entry of Gauss diary, counting solutions of polynomial equations modulo various primes*, factorization of Fermat and Mersenne numbers, representing integers by quadratic forms, representing integers by sum of squares*, classifying integral quadratic forms, cyclotomy...

Generally speaking, I think perusing the table of content of Gauss' Disquisitiones is a very good source of inspiration for projects like that.

Answer 4

Here are some elementary topics that come to mind

• Continuous fractions and diophantine equations (linear and Pell equations). (Check Davenport's Higher arithmetic)
• Geometry of numbers, Minkowski's theorem and applications
• Quadratic reciprocity
• Any one of Khincine's Three pearls of number theory (Dover Publications)
• Transcendence of $e$ or $\pi$ (Felix Klein's description of these proofs is a gem.)
• Weyl's uniform distribution theorem.

Answer 5

• Lagrange's four square theorem
• High-speed computer multiplication using the FFT
• The zeta function and the Riemann hypothesis (i.e. to understand what they are and why they're related to prime numbers; they're not expected to prove the RH)
• Descriptions of extremely big numbers (Conway's chained-arrow notation and its recursive definition, and stuff like that.)
• Proof of Fermat's last theorem for exponent 4 (I think this is supposed to be pretty accessible)
• Hilbert's tenth problem
• Basics of error-correcting codes
• Heuristic estimates for difficult or unsolved problems like FLT, Goldbach, etc. (see e.g. http://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ )
• combinatorial identities
• The formula $1+2+\cdots+n=n(n-1)/2$ is well known. What are the corresponding formulas for sum of squares, cubes, etc.? This may be closer to high-school level.
• the Ulam spiral

etc.

Answer 6

Class numbers! $SL_2(\mathbb{Z})$ acts on integral binary quadratic forms, probably stick to the negative discriminant (positive definite) case. It is not too hard to write out a proof of a fundamental domain for the action of $SL_2(\mathbb{Z})$ (at least if the student has some proof to read, and then is expected to write it in his or her own words), from which the finiteness of the class number is immediate.

There's no reason that the students need to know the abstract definition of a group action here as everything is quite explicit.

The fun part is that you can then compute class numbers to your heart's content. For example, it is easy to compute that $h(-163) = 1$. Does that ever happen again? Nope! Gauss couldn't prove it. But the student who computes $h(-163)$ (and maybe $h(-167)$ and $h(-171)$, etc.) by hand is likely to appreciate why Gauss believed it.

Answer 7

Primes in quadratic field extensions of $Q$. If you have not done Gaussian primes, this is where they should start.

$L$-functions.

Answer 8

Several times, I've supervised a project on perfect numbers. For a start, the students can prove the classification theorem for even perfect numbers. They can also look up and state the known results on odd perfect numbers, and/or do stuff on multiplicative functions.

I don't know whether this is at the right level for the students you have. Maybe it's too elementary. But for the students I had, I thought it worked well.

Answer 9

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.