New biggest prime. In class, we discussed why this might be important, since we've known since ancient times that the list of primes is infinite. One answer is the interesting collateral benefits of thinking deeply about numbers. The most important of these -- to our understanding of the universe -- is probably difficult to articulate. But sometimes it can be quite tangible, like when errors in Intel chips were discovered searching for primes. The tools we use (both mental and machine) are tested when we make these investigations.

Lewis Carroll's logic puzzles can be found in a number of places on the web, including the original text via Google books. But here's another source from a course at the University of Hawaii.

Theo found these notes online, from a course at the University of Iowa, which review and extend some of the theory of Gaussian integral approximation.

Keith Conrad at UConn does a terrific job of writing up supplementary notes on all sorts of topics, some of which are advanced topics in number theory. I often find my web searches for pedagogical materials leading back to his homepage. Here's another talking about the uses of differentiating under the integral sign. (It starts with a great quote from Feynman)

Here's a link to an old Mathematical Monthly article on the Dominated Convergence Theorem, giving a proof without using measure theory, and discussing limits under integral signs.

Terry Tao, a professor at UCLA and Fields Medalist, has a series of Java-based radio button quizzes on topics in real analysis. These are used for a course in analysis using the textbook by Rudin, which is a little more advanced than our course, but they still might be useful. They can be found at this repository.

One of our key themes for the year is that mathematics tells a story. It is not a collection of facts to be memorized. Nevertheless, it can be useful to have a good memory when doing mathematics. One strategy in the news lately is memory palaces. Apart from being a fun exercise, can you think of any situations in mathematics where a memory palace would be useful? Trigonometric identities? Long lists of hypotheses in theorems? Do we need them now that the internet is at our fingertips?

Theo reminded me of this nice set of notes, also by Terry Tao, on differential forms. It really conforms closely to the definitions in Hubbard and Hubbard, but I think you'll find the extra perspective useful.

Further Reading