报告人：孙智伟教授

       南京大学数学系

报告题目：REFINING LAGRANGE’S FOUR-SQUARE THEOREM

报告时间：2016年05月12日下午16:30

报告地点：海韵实验楼105

学院联系人:祝辉林助理教授

报告摘要：Lagrange’s four-square theorem asserts that any natural number can be written as x 2 +y 2 +z 2 +w 2 with x,y,z,w nonnegative integers. This classical result is very famous and it has several different proofs. The speaker recently found that Lagrange’s theorem can be further refined by requiring additionally that P(x,y,z,w) is a square, where P(x,y,z,w) is a suitable polynomial with integer coefficients. We have proved few results along this line, for example, P(x,y,z,w) = x − y is okay for our purpose. Moreover,we have formulated lots of surprising conjectures on this topic; for example, we conjecture that one may take P(x,y,z,w) = x + 3y + 5z. A quite mysterious conjecture of the speaker asserts that any natural number can be written as x 2 + y 2 + z 2 + w 2 with x,y,z,w nonnegative integers such that (10w+5x) 2 +(12y+36z) 2 is a square. This reveals a surprising connection between Lagrange’s theorem and Phthagorean triples. In this talk we will tell the story of such discoveries as well as related new results on partitions of integers motivated by our refinements of Lagrange’s theorem.

报告人简介：孙智伟，1965年10月生。现为南京大学数学系教授、博士生导师，数学系数学与应用数学专业主任, 其研究方向为数论与组合。他获得过教育部首届青年教师奖（2000）,国家杰出青年科学基金（2005-2008）与国务院政府特殊津贴(2011)。他是著名数论期刊《International Journal of Number Theory（国际数论杂志）》的编委。曾多次应邀去美国、欧洲、香港、台湾等地访问讲学或担任客座教授。他在数论与组合领域有许多创新成果, 迄今已在国外著名数学期刊《Trans. Amer. Math. Soc.（美国数学会汇刊）》等SCI杂志上发表了一百多篇学术论文。其工作被一些著名数学家（如Fields奖获得者T. Tao与著名组合学家N. Alon）在专著或论文中所引用。他提出了许多数学猜想（见他的网页 http://math.nju.edu.cn/~zwsun/），并愿意悬赏给第一个解决这些猜想或者否定这些猜想的人。

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