The main purpose of this paper is to study the solvability of some equations involving the pseudo Smarandache function Z(n) and the Smarandache reciprocal function Sc(n), and propose some interesting conjectures

Li, Ling

Wenpeng, Zhang

English

Wenpeng Zhang

Ling Li

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Two problems related to the Smarandache function

ZENODO

Scientia MagnaVol. 4 (2008), No. 2, 1-3Two problems related to theSmarandache functionWenpeng Zhang† and Ling Li† ‡† Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China‡ Basic Department, Shaanxi Polytechnic Institute, Xianyang, Shaanxi, P.R.ChinaAbstract For any positive integer n, the famous pseudo Smarandache function Z(n)is defined as the smallest positive integer m such that n |m(m+ 1)2. That is, Z(n) =min{m : n |m(m+ 1)2, n ∈ N}. The Smarandache reciprocal function Sc(n) is defined asSc(n) = max {m : y | n! for all 1 ≤ y ≤ m, and m+ 1 † n!}. That is, Sc(n) is the largestpositive integer m such that y | n! for all integers 1 ≤ y ≤ m. The main purpose of this paperis to study the solvability of some equations involving the pseudo Smarandache function Z(n)and the Smarandache reciprocal function Sc(n), and propose some interesting conjectures.Keywords The pseudo Smarandache function, the Smarandache reciprocal function, the d-ual function, equation, positive integer solutions, conjecture.§1. Introduction and resultsFor any positive integer n, the famous pseudo Smarandache function Z(n) is defined asthe smallest positive integer m such that n |m(m+ 1)2. That is,Z(n) = min{m : n |m(m+ 1)2, n ∈ N}.Its dual function Z∗(n) is defined as Z∗(n) = max{m :m(m+ 1)2| n, m ∈ N}, where Ndenotes the set of all positive integers. From the definition of Z(n) we can find that the firstfew values of Z(n) are: Z(1) = 1, Z(2) = 3, Z(3) = 2, Z(4) = 7, Z(5) = 4, Z(6) = 3, Z(7) = 6,Z(8) = 15, Z(9) = 8, Z(10) = 4, Z(11) = 10, Z(12) = 8, Z(13) = 12, Z(14) = 7, Z(15) = 5,Z(16) = 31, · · · · · · . About the elementary properties of Z(n), many authors had studied it,and obtained some interesting results, see references [1], [2], [3], [4], [5] and [6]. For example,the first author [6] studied the solvability of the equations:Z(n) = S(n) and Z(n) + 1 = S(n),and obtained their all positive integer solutions, where S(n) is the Smarandache function.On the other hand, in reference [7], A.Murthy introduced another function Sc(n), whichcalled the Smarandache reciprocal function. It is defined as the largest positive integer m such2 Wenpeng Zhang and Ling Li No. 2that y | n! for all integers 1 ≤ y ≤ m. That is, Sc(n) = max{m : y | n! for all 1 ≤ y ≤m, and m+ 1 † n!}. For example, the first few values of Sc(n) are:Sc(1) = 1, Sc(2) = 2, Sc(3) = 3, Sc(4) = 4, S(5) = 6, Sc(6) = 6, Sc(7) = 10,Sc(8) = 10, Sc(9) = 10, Sc(10) = 10, Sc(11) = 12, Sc(12) = 12, Sc(13) = 16,Sc(14) = 16, S5(15) = 16, Sc(16) = 16, Sc(17) = 18, Sc(18) = 18, · · · · · · .A.Murthy [7], Ding Liping [8] and Ren Zhibin [9] also studied the elementary properties ofSc(n), and obtained some interesting conclusions, one of them is that if Sc(n) = x and n 6= 3,then x+ 1 is the smallest prime greater than n.The main purpose of this paper is to study the solvability of some equations related to theSmarandache function, and propose some interesting problems. That is, we have the following:Unsolved problem 1. Whether there exist infinite positive integers n such that theequationSc(n) + Z(n) = 2n. (1)Unsolved problem 2. Find all positive integer solutions of the equationSc(n) = Z∗(n) + n. (2)§2. Some results on these unsolved problemsIn this section, we shall give some new progress on these unsolved problems. First for theproblem 1, it is clear that n = 1 satisfy the equation (1). n = 3 does not satisfy the equation (1).If p ≥ 5 and pα+2 are two odd primes, then n = pα satisfy the equation (1). In fact this time,we have Z (pα) = pα−1, Sc (pα) = pα+1. Therefore, Sc (pα)+Z (pα) = pα+1+pα−1 = 2 ·pα.So n = pα satisfy the equation (1). For example, n = 1, 5, 11, 17, 29 and 41 are six solutions ofthe equation (1). We think that the equation (1) has infinite positive integer solutions. Evenmore, we have the following:Conjecture 1. For any positive integer n, the equationSc(n) + Z(n) = 2nholds if and only if n = 1, 3α and p2β+1, where α ≥ 2 be any integer such that 3α + 2 be aprime, p ≥ 5 be any prime, β ≥ 0 be any integer such that p2β+1 + 2 be a prime.For the problem 2, it is clear that n = 3 does not satisfy the equation (2). If p ≥ 5be a prime, n = p2α+1 such that n + 2 be a prime, then Sc(n) = n + 1, Z∗(n) = 1, soSc(n) = Z∗(n)+n. Therefore, n = p2α+1 satisfy the equation (2). Besides these, whether thereexist any other positive integer n satisfying the equation (2) is an open problem. We believethat the following conjecture is true.Conjecture 2. For any positive integer n, the equationSc(n) = Z∗(n) + nVol. 4 Two problems related to the Smarandache function 3In our conjectures, if prime p ≥ 5, then p2β + 2 can be divided by 3. So if pα + 2 be aprime, then α must be an odd number.From our conjectures we also know that there exists close relationship between the solutionsof the equations (1), (2) and the twin primes. So we think that the above unsolved problemsare very interesting and important.References[1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House,1993.[2] Kenichiro Kashihara, Comments and topics on Smarandache notions and problems,Erhus University Press, USA, 1996.[3] David Gorski, The pseudo Smarandache function, Smarandache Notions Journal, 13(2002),No.1-2-3, 140-149.[4] Lou Yuanbing, On the pseudo Smarandache function, Scientia Magna, 3(2007), No.4,48-50.[5] Zheng Yani, On the Pseudo Smarandache function and its two conjectures, ScientiaMagna, 3(2007), No.4, 74-76.[6] Zhang Wenpeng, On two problems of the Smarandache function, Journal of NorthwestUniversity, 38 (2008), No.2, 173-176.[7] A. Murthy, Smarandache reciprocal function and an elementary inequality, SmarandacheNotions Journal, 11(2000), No.1-2-3, 312-315.[8] Liping Ding, On the Smarandache reciprocal function and its mean value, ScientiaMagna. 4(2008), No.1, 120-123.[9] Zhibin Ren, On an equation involving the Smarandache reciprocal function and itspositive integer solutions, Scientia Magna, 4 (2008), No.1, 23-25.[10] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal Uni-versity Press, Xi’an, 2007.[11] TomM. Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag,1976.[12] I. Balacenoiu and V. Seleacu, History of the Smarandache function, SmarandacheNotions Journal, 10(1999), No.1-2-3, 192-201.

Comments and topics on Smarandache notions and problems,

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On an equation involving the Smarandache reciprocal function and its positive integer solutions,

Two problems related to the Smarandache function

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