On the Inverse Problem Relative to 9 Dynamics of the w Function 0 0 2 Chaohua Jia n a J 4 Abstract. In this paper we shall study the inverse problem relative to dynamics 2 ofthew functionwhichisaspecialarithmeticfunctionandshallgetsomeresults. ] T N . h 1. Introduction t a m In 2006, Wushi Goldring [4] proposed some problems and conjectures [ on dynamics of the w function and gave some interesting results. Recently 1 Yong-Gao Chen and Ying Shi [2], [3] made further progress on these prob- v 4 lems. In this paperwe shall study the inverse problem relative to dynamics 3 8 of the w function. 3 . We begin by introducing some notations. Let P be the set of prime 1 0 numbers and P(n) denote the largest prime factor of integer n > 1. Write 9 0 v: C3 = {p1p2p3 : pi ∈ P(i = 1,2,3), pi 6= pj(i 6= j)}, i X B3 = {p1p2p3 : pi ∈ P(i = 1,2,3), p1 = p2 or p1 = p3 r a or p2 = p3, but not p1 =p2 = p3}, D = {p3 : p ∈ P}. 3 Then {p p p : p ∈ P(i = 1,2,3)} = C ∪B ∪D , 1 2 3 i 3 3 3 where no any two of C , B and D intersect. Let 3 3 3 A = C ∪B . 3 3 3 For n = p p p ∈ A , define the w function by 1 2 3 3 w(n) = P(p +p )P(p +p )P(p +p ) 1 2 1 3 2 3 1 and define w0(n)= n, wi(n)= w(wi−1(n)), i = 1, 2, ··· , which is reasonable according to Lemma 1 in section 2. Wushi Goldring [4] proved that for any n ∈ A , there exists i such 3 that wi(n) = 20. The smallest of such i is denoted by ind(n). He [4] proposed two conjectures on ind(n) (Conjectures 2.9 and 2.10) and gave the firstupperboundfor ind(n) which has been improved greatly by Yong- Gao Chen and Ying Shi [2] recently. Wushi Goldring [4] also asked the following inverse problems: 1. For n ∈A , can we find m ∈ A such that w(m) = n? 3 3 2. If so, how many such elements are there? 3. What form do they have? For n ∈ A , if there is m ∈ A such that w(m) = n, then we call m 3 3 a parent of n. If this m ∈ S ⊂ A , then we call it S-parent of n. Wushi 3 Goldring[4]provedthatthereareinfinitelymanyelementsofB whichhave 3 at least seven parents. He proposed the following conjecture (Conjecture 2.16 in [4]). Conjecture (Wushi Goldring). Every element of A (respectively 3 B ) has infinitely many C -parents (respectively B -parents). 3 3 3 Yong-Gao Chen and Ying Shi [3] proved that for any given positive integer k, there are infinitely many elements of B which have at least k 3 B -parents. On the other hand, they [3] proved that there are infinitely 3 many elements of B which have no B -parent. 3 3 In this paper we shall study parents of elements of C . It is obvious 3 that the element of C has no B -parent. We shall prove that there are 3 3 infinitely many elements of C which have enough C -parents. 3 3 Inthefollowing, p, p , p , p , q, r, r , r denoteprimenumbersandc , 1 2 3 1 2 1 c , ··· denote positive constants. The expression f ≪ g means f = O(g). 2 We suppose that x is sufficiently large throughout. Theorem 1. There exists an element r r q of C which satisfies 1 2 3 x12 logx < ri ≤ 2x21 logx(i = 1,2), q ≤ 4xandhasatleastc1logx4x different C -parents p p p with x < p ≤ 2x(i = 1,2,3). 3 1 2 3 i 2 We shall also prove that there are infinitely many elements of B which 3 have enough C -parents. 3 Theorem 2. There exists an element qr2 of B which satisfies q ≤ 3 4x, x21 logx < r ≤ 2x12 logx and has at least c2logx4x different C3-parents p p p with x < p ≤ 2x(i = 1,2,3). 1 2 3 i Moreover, we shall prove that there are infinitely many elements of B 3 that have enough B -parents, which is a quantitative improvement on the 3 result of Yong-Gao Chen and Ying Shi [3]. Theorem 3. There exists an element qr2 of B which satisfies x < q ≤ 3 2x, x12 logx < r ≤ 2x12 logx and has at least c3loxg212x different B3-parents pq2 with x < p ≤ 2x. 2. Lemmas Lemma 1. If n ∈A , then w(n) ∈ A . 3 3 This is Lemma 2.1 in [4]. Lemma 2. Let n (1 ≤ j ≤ Z) be distinct positive integers not exceed- j ing N and Z(N; r, a) denote the number of those n which are congruent j to a(modr). If X ≥ 2, then we have r Z 2 r Z(N; r, a)− ≪ (N +X2)Z. X X(cid:16) r(cid:17) r≤X a=1 This is Theorem 1 in [1], which is obtained by the large sieve method. Lemma 3. We have 1 2 x2 1− 1 ≪ . X X (cid:16) X r X (cid:17) logx x12 logx n2, then p|n ⇐⇒ P(n)= p. Then we have 1 X X X x

On the Inverse Problem Relative to Dynamics of the w Function

Similar On the Inverse Problem Relative to Dynamics of the w Function