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

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.