Dynamics of the 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 | References | Similar Articles | Additional Information

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 , define a function by where is the largest prime factor of . It is clear that if , then . For any , define , for . An element is semi-periodic if there exists a nonnegative integer and a positive integer such that . 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 is semi-periodic. He showed that there exists such that , ind, and conjectured that ind can be arbitrarily large.

In this paper, it is proved that for any we have ind , and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring's above conjecture.

**1.**Wushi Goldring,*Dynamics of the 𝑤 function and primes*, J. Number Theory**119**(2006), no. 1, 86–98. MR**2228951**, https://doi.org/10.1016/j.jnt.2005.10.010**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.