Logistics

I'll want you to turn in a 5-page (or so) paper on your topic. The papers will be due on Thursday, 4/24.

Suggested topics

These are just suggestions. Let me know if you have an idea of your own, and also let me know either way what topic you're going to cover, so I can suggest references and ideas. Please tell me asap.

Irrationality and transcendence proofs: Give proofs that pi and e are irrational/transcendental numbers. (The latter is harder than the former!)

Hurwitz's theorem and the golden ratio: Give a proof of Hurwitz's theorem on approximations to irrational numbers, and discuss how it can be strengthened. Explain what this has to do with the golden ratio!

Continued fractions: Give proofs of the following results:
(1) If |P/Q - r| < 1/2Q^2, then P/Q is a convergent in the continued fraction expansion for r.
(2) If r is a quadratic surd, then its continued fraction expansion is periodic.

Cyclic unit groups: Give a proof that the unit group of Z/nZ is cyclic if and only if n is an odd prime power, twice an odd prime power, or 4. Half of this proof is tangentially related to something called Hensel's lemma. Explain Hensel's lemma. (NB: These two topics are fairly unrelated, but I'm not sure either one of them is enough on its own; hence the merger.)

Fermat's last theorem for regular primes: Explain what a regular prime is. Explain how Fermat's last theorem follows for regular primes. How many regular primes are there? (NB: This probably requires a pretty decent algebra background, much more than the other projects.)

Densities: In class we have occasionally mentioned a measure of the size of an infinite set of integers, e.g. "about half" of all integers are even. Make this rigorous by defining the density of a set of integers. Find, with proof, the densities of various sets of interest (e.g. the set of primes, the set of square-free integers, the set of all integers whose leftmost digit is 8...)