本课程的主要内容是影响深远的三个著名数学问题，即费马大定理（350年历史，1994年被证明），黎曼猜想（150年历史，迄今仍不知真假的天下第一难题，数学研究的最前沿）和庞加莱猜想（100年历史，2003年被证明）。 费马大定理1637年由法国律师、著名业余数学家费马提出，1994年被英国数学家怀尔斯所证明。被誉为二十世纪最伟大的数学成果。 黎曼假设是德国著名数学家黎曼1859年提出的至今不知真假的天下第一的数学难题。黎曼假设与素数定理等价. 庞加莱猜想是法国著名数学家、物理学家、天体力学家、哲学家1904年提出的有关宇宙形状的数学猜想，2004年被俄罗斯数学家佩雷尔曼所证明。佩雷尔曼拒绝了菲尔兹奖(数学的诺贝尔奖)和100万美元。 丘成桐：如果把庞加莱猜想比作黄河、长江，那么哥德巴赫猜想只能算是一条比较小、但很漂亮的河。 通过本课程的学习，学生在知识层面可以初步懂得有关素数的若干定理和猜想的意义与原理，及其若干精彩证明的思想和方法；在逻辑层面可以了解数学理论形成的内在规律，以此感悟如何从具体的实际问题凝练出抽象的普遍理论。更重要的是，通过了解数学精英的奋斗历程，学生或多或少会在思维或思想或人生层面有所感悟。 The course consists of three parts, each focusing on one of the most famous mathematical problems, Fermat’s Last Theorem(about 350 years ago and proved in 1994), Riemann’s Hypothesis(about 150 years ago and still not proved or disproved) and Poincare Conjecture(about 110 years ago and proved in 2004). The Fermat’s Last Theorem was stated by the most famous nonprofessional French mathematician Fermat that the equation x^n+y^n=z^n has no solutions of positive integers for all n>2. This was not proved for over 350 yeas until 1994 when the British mathematician Wiles gave a proof known as the most important achievement of mathematics during the whole 20th century. The Riemann’s hypothesis was conjectured by the famous German mathematician Riemann in 1859 that the so-called Riemann theta function 1+2^s+3^s+...+n^s+... has all its nontrivial zeros on the line Re(s)=0.5. This hypothesis is regarded as the number 1 question in mathematics and is still unknown whether it is true or false. The Poincare conjecture was stated as a question in 1904 by the famous French mathematician, physist and philosopher Poincare as he studied the topics of the shape of our universe. This conjecture was proved in 2004 by Russian mathematician Perelman, who refused the highest mathematical medal-Fields medal and an award of one million dollars for the first solver of one of seven million dollars problems, two among which are Poincare conjecture and Riemann’s hypothesis. The most famous mathematician S.T.Yau said that if Poincare conjecture were the Yellow river, the Goldbach conjecture is only a small brook, a beautiful brook. This course aims to make students to know some theorems about prime numbers and several concise proofs and bright ideas; to understand the logic of the formation of mathematical theories and to apply mathematics to the real world; most importantly, to get a feeling of the struggling history of mathematical heroes and to make themselves a wonderful struggling lives.