## 2009年12月28日 星期一

### Polynomials and topology

Recently my friend is working on a problem in topology, and out of his work, it appears that the there is a special pattern in the coefficients of the following polynomial, when m,n are relatively prime:

$\frac{(x^{mn}-1)(x-1)}{(x^m-1)(x^n-1)} (x^{4n}-x^{2n}+1)$

It appears that if one expands this polynomial out, collects terms and arranges them in decreasing powers of x, then the non-zero coefficients are all either 1 and -1, and they appear to alternate as the power decreases. (e.g. when m=4, n=3, the polynomial is

$x^{18}-x^{17}+x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x+1$

It is not known whether this pattern really exists. But I thought this is cute and may be of interest to some of you. Does any of you have any idea about how to prove/disprove it?

(The case of interest in topology is when m > 3n, but it looks like this pattern persists as long as m,n are relatively prime.)

## 2009年12月24日 星期四

### Elementary number theory

Someone say that 167588402882520529579353108764873470755823697 is the smallest positive integer k such that all digit of 1989k are the same.

Do you agree?

## 2009年12月19日 星期六

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Probability from a gambler's viewpoint --- A taste of Martingale Theory

Suppose you keep flipping a fair coin until 10 heads occur consecutively. How many times of flipping do you need on average?

We will solve this and other related problems as an application of the Optional Stopping Theorem. The basic notions and properties of discrete martingales will be introduced in an informal manner, with emphasis on the intuitive ideas.

Prerequisite: Basic probability in secondary school level.

## 2009年12月11日 星期五

### Two Analysis Problems

Recently I heard two problems in analysis, both I think are interesting, and they do not require too deep knowledge in analysis, which is the kind of questions I like most. Share here.

1) Suppose $\sum_{j=1}^\infty |a_j|$ converges. Also, for each positive integer k, it is known that $\sum_{j=1}^\infty a_{jk} = 0$ (just to avoid confusion, allow me clarify here that jk means "j times k"). Prove that $a_i=0$ for all positive integers i.

2) S contains all elements $x\in[0,1]$ such that for any $\epsilon>0$, there exists a rational number $\frac{p}{q}$ (where p,q are positive integers) satisfying $\left| x - \frac{p}{q}\right| < \frac{\epsilon}{q^3}$. Prove that S is uncountable.