Writing $a\dot b$ every time would make group presentation theorems a night mare to read.

Writing x\y for "x divides y" seems much more horrible than "x|y". Maybe I'm just too used to the latter.

The hats-switching proof is not only mathematically correct but also goes to the essence of the matter. Hence it can also easily answer other related questions such as how many head-bumps occurred before all the ants fell of the rod.

@KConrad: Hmm, the fact that there is a polynomial whose values at the integers are primes only, even negative ones, still surprises me. Is there a reason that such a polynomial isn't very surprising? Also, why are irreducible polynomials more impressive than reducible ones? If a polynomial is reducible, it just seems much harder for its values to be primes...

I agree with Andrew L here. I too was disappointed at how little stuff was covered in this book.

The existence of a polynomial whose positive integer values are all primes is indeed surprising. I can't help to ask: what important consequences follow from that existence?

@Kevin: yes, I knew that and thanks for making it clear. I was just surprised to hear that Siegel had (as it turns out) another proof of the result.

Thanks. The papers listed in the answers to that question is the one I remember seeing. So this means that the Theorem above is a true theorem.

Is it true that the only Fibonacci numbers that are cubes are $0, \pm 1, \pm 8$? I seem to recall a recent paper proving this...

That's great to hear.

Straight from the author's website: May 2010. New version of Algebraic Groups, Lie Groups, and their Arithmetic Subgroups

I don't like this business of voting down stuff, especially when I do like the question.

Your first sentence on Rudin's book is very bad, unfair and very likely not true. Any serious college student who approaches analysis for the first time must know what a proof is, having seen it in Euclidean Geometry back in junior middle school. And I personally like Rudin's, read it when I was still in high school and found it clear, to-the-point, and with a good supply of excellent problems. Also, one cannot fault an author for giving slick proofs. I for one prefer slick proofs over tedious, drawn-out proofs (unless they're the correct conceptual ones).

After Sergey Ivanov's solution, I wonder how many discontinuity points a solution of $f(f(x)) = \cos(x)$ must have?