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¡¶ÄϾ©Ê¦´óѧ±¨£¨×ÔÈ»¿Æѧ°æ£©¡·[ISSN:1001-4616/CN:32-1239/N]

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2013Äê04ÆÚ

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1-

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Title:

The Erdös-Tur¨¢n Conjecture and Related Topics

Author(s):

Chen Yonggao

School of Mathematical Sciences and Institute of Mathematics,Nanjing Normal University,Nanjing 210023,China

Keywords:

Erdö; s-Tur¨¢n conjecture; representation function; additive basis

PACS:

O156.1

DOI:

-

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¡ÊA.If R(A,n)¡Ý1 for all integers n¡Ý0,then A is called a basis of N.The well known Erdös-Tur¨¢n 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-Tur¨¢n conjecture.We also prove that the Erdös-Tur¨¢n 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 ¦Á can be uniquely(neglecting the order)represented as a product of two elements of A.Finally,seven open questions are posed for further research.

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Memo

Memo:

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Common functions

Last Update: 2013-12-30