## 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.]