Dynamics of the <IMG WIDTH="21" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="/proc/2008-136-07/S0002-9939-08-09207-1/gif-title0/img1.gif" ALT="$ w$"> function and the Green-Tao theorem on arithmetic progressions in the primes期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Dynamics of the function and the Green-Tao theorem on arithmetic progressions in the primes

Authors:	Yong-Gao Chen Ying Shi

Institution:	Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China ; Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China

Abstract:	Let be the set of all positive integers , where are primes and possibly two, but not all three of them are equal. For any $n=pqr\in A_{3}$ , define a function by where is the largest prime factor of . It is clear that if $n=pqr\in A_{3}$ , then $w(n) \in A_3$ . For any $n\in A_{3}$ , define $w^{0}(n)=n$ , $w^{i}(n)=w(w^{i-1}(n))$ for $i=1,~2,~\ldots$ . An element $n\in A_{3}$ is semi-periodic if there exists a nonnegative integer and a positive integer such that $w^{s + t}(n)= w^{s}(n)$ . We use ind to denote the least such nonnegative integer . Wushi Goldring Dynamics of the function and primes, J. Number Theory 119(2006), 86-98] proved that any element $n\in A_{3}$ is semi-periodic. He showed that there exists such that $w^{i}(n)\in\{20,98,63,75\}$ , ind $(n)\leqslant 4(\pi(P(n))-3)$ , and conjectured that ind can be arbitrarily large. In this paper, it is proved that for any $n\in A_{3}$ we have ind $O((\log P(n))^2)$ , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

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