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


Authors: Yong-Gao Chen and Ying Shi
Journal: Proc. Amer. Math. Soc. 136 (2008), 2351-2357
MSC (2000): Primary 11A25, 11A41, 37B99
DOI: https://doi.org/10.1090/S0002-9939-08-09207-1
Published electronically: March 4, 2008
MathSciNet review: 2390501
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Abstract: Let $ A_3$ be the set of all positive integers $ pqr$, where $ p,q,r$ are primes and possibly two, but not all three of them are equal. For any $ n=pqr\in A_{3} $, define a function $ w$ by $ w(n)=P(p+q)P(p+r)P(q+r),$ where $ P(m)$ is the largest prime factor of $ m$. 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 $ s$ and a positive integer $ t$ such that $ w^{s + t}(n)= w^{s}(n)$. We use ind$ (n)$ to denote the least such nonnegative integer $ s$. Wushi Goldring [Dynamics of the $ w$ 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 $ i$ such that $ w^{i}(n)\in\{20,98,63,75\}$, ind$ (n)\leqslant 4(\pi(P(n))-3)$, and conjectured that ind$ (n)$ can be arbitrarily large.

In this paper, it is proved that for any $ n\in A_{3}$ we have ind$ (n)=$ $ O((\log P(n))^2)$, and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.


References [Enhancements On Off] (What's this?)

  • 1. Wushi Goldring, Dynamics of the $ w$ function and primes, J. Number Theory 119 (2006), 86-98. MR 2228951 (2007a:11010)
  • 2. Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, to appear in Ann. Math.

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Additional Information

Yong-Gao Chen
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email: ygchen@njnu.edu.cn

Ying Shi
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-08-09207-1
Received by editor(s): October 6, 2006
Received by editor(s) in revised form: April 30, 2007
Published electronically: March 4, 2008
Additional Notes: The authors were supported by the National Natural Science Foundation of China, Grants No. 10471064 and 10771103.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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