Extremal Problems in Number Theory
Extremal Problems in Number Theory
复制标题
DOI：
10.1090/pspum/008/0174539
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
P. Erdos
中科院分区：
文献类型：
--
作者：
P. Erdos
我想用一个例子来说明我将在本文中研究的问题。用r）表示不超过n的整数的最大数目，其中没有k个整数形成算术级数。问题是确定或估计rk（n）的值。这个问题与数论的几个已知问题有关。如果对于每个K，r &）<（I-t）n/log n，如果n足够大，则素数定理意味着对于每个k，在算术级数中有k个素数。r）<n/2将意味着众所周知的货车der Waerden定理。关于rk（n）的第一篇论文是由Turzin和我[21]提出的。目前已知的r3（n）的最佳界是[1; 7]。
I would like to illustrate the problems which I shall investigate in this paper by an example. Denote by r) the maximum number of integers not exceeding n, no k of which form an arithmetic progression. The problem is to determine or estimate the value of rk(n). This problem is connected with several known questions of number theory. If r&) <(I-t)n/log n for every K, if n is sufficiently large, then the prime number theorem implies that for every k there are k primes in arithmetic progression. r) <n/2 would imply the well known theorem of Van der Waerden. The first paper on rk(n) is due to Turzin and myself [ 21. The best bounds for r3(n) presently known are [ 1;7 ]