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:



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



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.)

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?

MD Academic Seminar

數學資料庫將於本月底舉行期待已久的 academic seminar！ 詳情如下：

日期：2009 年 12 月 27 日（星期日）
時間：下午 4 時 30 分至 5 時 30 分
地點：香港大學莊月明文娛中心 302 室
講者：樊偉堂先生（華盛頓大學數學系博士研究生）

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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.

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 converges. Also, for each positive integer k, it is known that (just to avoid confusion, allow me clarify here that jk means "j times k"). Prove that for all positive integers i.

2) S contains all elements such that for any , there exists a rational number (where p,q are positive integers) satisfying . Prove that S is uncountable.

香港大學「數趣漫話」講座

日期：2010 年 1 月 9 日（星期六）
時間：下午 2 時 30 分至 4 時
地點：香港大學明華綜合大樓 T1 演講廳
講者：李志光教授
講題：數學世界與武俠天地

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簡介：

數學世界中有抽象的科研題目，有孜孜不倦的學者，有勤奮的學生；武俠天地裏有神奇的絕世武功，有鋤強扶弱的俠客，有堅毅的徒底。數學與武俠，兩者仿似風馬牛不相及，細看又似有不少共通之處。

講者將以其多年從事數學科研教學之經驗，剖析數學世界與武俠天地之異同，聽眾可以輕鬆地瞭解數理學者的工作和生活。

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有關其他詳情，可瀏覽這裡。

培正數學邀請賽

數學資料庫協辦的「培正數學邀請賽 2010」將於 1 月 23 日（星期六）和 3 月 20 日（星期六）分別舉行初賽和決賽。學校報名的截止日期為 2009 年 12 月 5 日（星期五）。有興趣參賽的同學，可向就讀學校的數學老師查詢有關事宜。其他有關比賽的詳情，可瀏覽比賽網頁。