[1]陈永高.Erdös-Turán猜想与相关的问题[J].南京师大学报(自然科学版),2013,36(04):1.
　Chen Yonggao.The Erdös-Turán Conjecture and Related Topics[J].Journal of Nanjing Normal University(Natural Science Edition),2013,36(04):1.
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Erdös-Turán猜想与相关的问题
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《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

卷:
第36卷
期数:
2013年04期
页码:
1
栏目:
出版日期:
2013-12-31

文章信息/Info

Title:
The Erdös-Turán Conjecture and Related Topics
作者:
陈永高
南京师范大学数学科学学院与数学研究所,江苏 南京 210023
Author(s):
Chen Yonggao
School of Mathematical Sciences and Institute of Mathematics,Nanjing Normal University,Nanjing 210023,China
关键词:
Erdö; s-Turán猜想; 表示函数; 加法基
Keywords:
Erdö; s-Turán conjecture; representation function; additive basis
分类号:
O156.1
文献标志码:
A
摘要:
设N是所有非负整数所成的集合,对于AN,用R(A,n)表示方程n=a+b,a,b∈A的解数.著名的Erdös-Turán猜想为:如果对所有非负整数n,总有R(A,n)≥1,则R(A,n)一定无界.本文简单介绍了Erdös-Turán猜想的进展,同时证明了Erdös-Turán猜想在有理数域上关于加法和乘法(乘法不含零)均不成立.如关于乘法,本文证明了如下结论:存在非零有理数集的一个子集A,使得每个非零有理数均可以唯一地(不考虑次序)表成A中两个数的乘积.最后,本文提出了7个未解决的问题供进一步研究.
Abstract:
Let N be the set of all nonnegative integers.For any subset A of N,let R(A,n)be the number of solutions to the equation n=a+b,a,b∈A.If R(A,n)≥1 for all integers n≥0,then A is called a basis of N.The well known Erdös-Turán conjecture says that if A is a basis of N,then R(A,n)cannot be bounded.In this paper,we give a brief review on the Erdös-Turán conjecture.We also prove that the Erdös-Turán conjecture is false in rational number field on both addition and multiplication.For example,the following result is proved:There exists a subset A of nonzero rational numbers such that every nonzero rational number α can be uniquely(neglecting the order)represented as a product of two elements of A.Finally,seven open questions are posed for further research.

参考文献/References:

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备注/Memo

备注/Memo:
收稿日期:2013-09-20.
基金项目:国家自然科学基金(11371195).
通讯联系人:陈永高,特聘教授,博士生导师,研究方向:数论.E-mail:ygchen@njnu.edu.cn
更新日期/Last Update: 2013-12-30