## 2008年11月27日 星期四

### Solve it in ten minutes!

My friend heard a problem from his friend:

Find a positive integer solution to 4024x + 4178y + 4609z = 3757057.

Yes, this is a not-so-interesting question if you have a computer and you know how to write a programme (just do some brute-force calculation).

Also, if you requires an integer solution (so you may have non-positive integers as x, y or z), it is still easy to do it by hand quickly. (Do you know how?)

However, assume what you have is a sheet of paper and a pen, can you find a positive solution in ten minutes?

## 2008年11月16日 星期日

### Analytic function on a disk

Suppose f is a holomorphic function on the unit disk that is continuous up to the boundary. If f vanishes on an arc of the boundary circle, show that it is identically zero.

This is a standard question from complex analysis. I just learn today that there is a really cute solution to it. You may want to think about it before continuing.

The classical method is to use Schwartz reflection principle and argue that you can analytically continue the function outside the disk a little bit; then the continued function vanishes on a segment, so it must vanish identically.

The cute solution that I was referring to is the following: take (finitely many) copies of f, rotate each of them suitably and multiply the rotated functions altogether. Then the product is going to vanish identically on the boundary of the unit disk, and of course the product is holomorphic inside the disk. Hence the product is identically zero, and thus f is identically zero. (Just argue that all derivatives of f vanishes at the origin by differentiating the product.)

## 2008年11月13日 星期四

### Factorial

Today I went to a talk by Manjul Bhargava, and he stated some interesting facts in number theory that has to do with the factorial:

1. If $a_0,a_1,\dots,a_n$ are integers, then $\prod_{i is a multiple of $0!1!...n!$.

2. Suppose that f is a primitive polynomial with integer coefficients and let k be its degree. Let Let d(f) be the gcd of all f(a) as a runs through all the integers. Then d(f) divides k!.

3. The number of polynomial maps $f:\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ is $\prod_{k=0}^{\infty}\frac{n}{\gcd(n,k!)}$.

4. A function $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous if and only if it has the form $f(x)=\sum_{n=0}^{\infty}\frac{c_n}{n!}x(x-1)\dots(x-n+1)$ where $c_n\to 0$ as $n \to \infty$.

In fact in the talk he gave a far reaching generalization of the factorial functions. For each compact subset of the p-adic rationals (e.g. the p-adic integers), he defined a factorial function adapted to that set such that the above seemingly unrelated facts about the factorial goes through. It's amazing to see how he could generalize things that are so well known, and give non-trivial results that fits in so many settings.

[I just realize that his original article can be found here.]