The main purpose of this paper is using the elementary methods to study several problems in [2], and four of them are solved

Wenji, Guan

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Scientia MagnaVol. 3 (2007), No. 4, 45-47Four problems related to thePseudo-Smarandache-Squarefree functionWenji Guan † ‡† Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China‡ Department of Mathematics, Weinan Teachers University, Weinan, Shaanxi, P.R.ChinaAbstract For any positive integer n, the Pseudo-Smarandache-Squarefree function Zw(n)is defined as the smallest positive integer m such that mn is divisible by n. That is,Zw(n) = min {m : m ∈ N, n | mn}. In reference [2], Felice Russo proposed many problemsand conjectures related to the Pseudo-Smarandache-Squarefree function Zw(n). The mainpurpose of this paper is using the elementary methods to study several problems in [2], andfour of them are solved.Keywords Pseudo-Smarandache-Squarefree function, equation, positive integer solution.§1. Introduction and resultsFor any positive integer n, the famous Pseudo-Smarandache-Squarefree function Zw(n) isdefined as the smallest positive integer m such that mn is divisible by n. That is,Zw(n) = min {m : m ∈ N, n | mn} .It is easy to see that if n > 1, thenZw(n) =∏p|np,where∏p|ndenotes the product over all different prime divisors of n (see reference [1]). From thisformula, we can easily get the value of Zw(n). For example, Zw(1) = 1, Zw(2) = 2, Zw(3) =3, Zw(4) = 2, Zw(5) = 5, Zw(6) = 6, Zw(7) = 7, Zw(8) = 2, Zw(9) = 3, Zw(10) = 10, · · · .In fact if n is a square-free number, then Zw(n) = n. In reference [2], Felice Russo studied theproperties of Zw(n), and proposed the following four problems:Problem 1. Find all the values of n such that Zw(n) = Zw(n+ 1) · Zw(n+ 2).Problem 2. Solve the equation Zw(n) · Zw(n+ 1) = Zw(n+ 2).Problem 3. Solve the equation Zw(n) · Zw(n+ 1) = Zw(n+ 2) · Zw(n+ 3).Problem 4. Find all the values of n such that S(n) = Zw(n), where S(n) is the Smaran-dache function.The main purpose of this paper is using the elementary methods to study these fourproblems, and solved them completely. That is, we shall prove the following conclusions:46 Wenji Guan No. 4Theorem 1. The following three equations have no positive integer solution.Zw(n) = Zw(n+ 1) · Zw(n+ 2); (1)Zw(n) · Zw(n+ 1) = Zw(n+ 2); (2)Zw(n) · Zw(n+ 1) = Zw(n+ 2) · Zw(n+ 3). (3)Theorem 2. There exist infinite positive integers n such that the equation S(n) = Zw(n),where S(n) is the Smarandache function defined by S(n) = min {k : k ∈ N, n | k!}.§2. Proof of the theoremsIn this section, we shall complete the proof of our theorems. First we prove that theequation Zw(n) = Zw(n+1) ·Zw(n+2) has no positive integer solution. It is clear that n = 1is not a solution of this equation. In fact if n = 1, then1 = Zw(1) 6= 2 · 3 = Zw(2) · Zw(3).If n > 1, suppose that the equation (1) has one positive integer solution n = n0, thenZw(n0) = Zw(n0 + 1) · Zw(n0 + 2).For any prime divisor p of n = n0, it is clear that p | Zw(n0).From Zw(n0) = Zw(n0 + 1) · Zw(n0 + 2)we deduce that p | Zw(n0 + 1) · Zw(n0 + 2).That is, p | Zw(n0 + 1) or p | Zw(n0 + 2).(a) If p | Zw(n0 + 1), then p | (n0 + 1), combining p | n0 and p | (n0 + 1) we getp | n0 + 1− n0 = 1, this is a contradiction.(b) If p | Zw(n0 + 2), then p | (n0 + 2), combining p | n0 and p | (n0 + 2) we deduce thatp | n0 + 2− n0 = 2, then we get n0 = p = 2.From equation (1) we have2 = Zw(2) 6= 3 · 2 = Zw(3) · Zw(4),It is not possible. So the equation (1) has no positive integer solution.Using the similar method as in proving problem 1, we find that the equation (2) andequation (3) also have no positive integer solution. This completes the proof of Theorem 1.In order to prove Theorem 2, we need some important properties of the Smarandachefunction S(n), which we mentioned as the following:Lemma 1. Let n = pα11 pα22 · · · pαkk is the prime powers factorization of n, thenS(n) = max1≤i≤k{S(pαii )}.Proof. See reference [3].Lemma 2. If p be a prime, then S(pk) ≤ kp; If k < p , then S(pk) = kp, where k is anypositive integer.Vol. 3 Four problems related to the Pseudo-Smarandache-Squarefree function 47Proof. See reference [4].Now we use these two simple lemmas to prove our Theorem 2. It’s clear that all primep are the solutions of the equation S(n) = Zw(n). So there are infinite positive integers nsatisfying the equation S(n) = Zw(n).Now we construct infinite composite numbers n satisfying the equation S(n) = Zw(n),let n = p1 · p2 · · · pk−1 · pαkk , where pi are the different primes, and pk > αk = p1p2 · · · · pk−1.This time, from the definition of S(n) and Zw(n) we have S(n) = p1 · p2 · · · pk−1 · pk andZw(n) = p1 · p2 · · · pk−1 · pk. So all composite numbers n = p1 · p2 · · · pk−1 · pαkk (where pi arethe different primes, and pk > αk = p1p2 · · · · pk−1 ) satisfying the equation S(n) = Zw(n).Note that k be any positive integer and there are infinite primes, so there are infinitecomposite numbers n satisfying the equation S(n) = Zw(n).This complete the proof of Theorem 2.References[1] Maohua Le, On the Pseudo-Smarandache-Squarefree function, Smarandache NotionsJournal, 13(2002), 229-236.[2] Felice Russo, A set of new Smarandache function, sequences and conjectures in numbertheory, Lupton USA, 2000.[3] BALACENOIUI, SELEACU V., History of the Smarandache function, SmarandacheNotions Journal, 1999, No. 10, 199-201.[4] F. Mark, M. Patrick, Bounding the Smarandache function, Smarandache Journal, 13(2002).[5] Ribenboim, P., The book of prime number records, New York, Springer-Verleg, 1989.[6] Birkhoff, G. D. and Vandiver, H. S., On the integral divisor of an − bn, Ann.of Math.,2(1904).[7] F.Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House,1993.[8] Pan Chengdong and Pan Chengbiao, Elementary Number Theory, Beijing UniversityPress, Beijing, 1992.[9] Yi Yuan and Kang Xiaoyu, Research On Smarandache Problems, High American Press,2006.

Four problems related to the Pseudo-Smarandache-Squarefree function

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