DYNAMICS OF THE w FUNCTION AND THE GREEN-TAO THEOREM ON ARITHMETIC PROGRESSIONS IN THE PRIMES

Let A3 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 ∈ A3, 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 ∈ A3, then w(n) ∈ A3. For any n ∈ A3, define w0(n) = n, wi(n) = w(wi−1(n)) for i = 1, 2, . . . . An element n ∈ A3 is semi-periodic if there exists a nonnegative integer s and a positive integer t such that ws+t(n) = ws(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 ∈ A3 is semi-periodic. He showed that there exists i such that wi(n) ∈ {20, 98, 63, 75}, ind(n) 4(π(P (n))− 3), and conjectured that ind(n) can be arbitrarily large. In this paper, it is proved that for any n ∈ A3 we have ind(n) = O((logP (n))2), and the Green-Tao Theorem on arithmetic progressions in the primes is employed to confirm Goldring’s above conjecture.