On extremal sets without coprimes
On extremal sets without coprimes
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DOI：
10.4064/aa-66-1-89-99
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发表时间：
1994
期刊：
影响因子：
0.7
通讯作者：
R. Ahlswede;L. Khachatrian
中科院分区：
文献类型：
--
作者：
R. Ahlswede;L. Khachatrian
1.定义、问题的表述和说明。我们使用以下符号：Z表示所有整数的集合，N表示正整数的集合，并且P = {p1，p2，. . .} = {2，3，5，. . .}表示所有素数的集合。我们设（1.1）Qk = ki =1 pi。对于两个数u，v ∈ N，我们写（u，v）= 1，如果u和v是互素的。我们特别感兴趣的是集合（1.2）N s = {u ∈ N：（u，Q s−1）= 1}和（1.3）N s（n）= N s <$[1，n]，（n，k，s）作为最大整数r，其中an（1.4）An <$N s（n），|A n| = r，存在A n中没有k + 1个数是互素的。当然，这个集合没有k + 1个互素。s = 1的情况，其中我们有N1（n）= [1，n]，是特别感兴趣的。猜想1.看来，这一猜想的Erd Ekesos首次出现在打印在他的论文[6]的1962年。
1. Definitions, formulation of problems and conjectures. We use the following notations: Z denotes the set of all integers, N denotes the set of positive integers, and P = {p 1 , p 2 ,. . .} = {2, 3, 5,. . .} denotes the set of all primes. We set (1.1) Q k = k i=1 p i. For two numbers u, v ∈ N we write (u, v) = 1 if u and v are coprimes. We are particularly interested in the sets (1.2) N s = {u ∈ N : (u, Q s−1) = 1} and (1.3) N s (n) = N s ∩ [1, n], (n, k, s) as the largest integer r for which an (1.4) A n ⊂ N s (n), |A n | = r , exists with no k + 1 numbers in A n being coprimes. Certainly the set does not have k + 1 coprimes. The case s = 1, in which we have N 1 (n) = [1, n], is of particular interest. Conjecture 1. It seems that this conjecture of Erd˝ os appeared for the first time in print in his paper [6] of 1962.