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Some interesting properties of the Smarandache function.

Abstract The main purpose of this paper is using the elementary method to study the property of the Smarandache function, and give an interesting result.

Keywords Smarandache function; Additive property; Greatest prime divisor.

[section] 1. Introduction and results

Let n be an positive integer, the famous Smarandache function S(n) is defined as following:

S(n) = min{m : m [member of] N; n|m!}:

About this function and many other Smarandache type function, many scholars have studied its properties, see [1], [2], [3] and [4]. Let p(n) denotes the greatest prime divisor of n, it is clear that S(n) [greater than or equal to] p(n). In fact, S(n) = p(n) for almost all n, as noted by Erdos [5]. This means that the number of n [less than or equal to] x for which S(n) [not equal to] p(n), denoted by N(x), is o(x). It is easily to show that S(p) = p and S(n) < n except for the case n = 4, n = p. So there have a closely relationship between S(n) and [pi](x):

[pi](x) = - 1 + [[x].summation over (n=2)] [S(n) / n],

where [pi] (x) denotes the number of primes up to x, and [x] denotes the greatest integer less than or equal to x. For two integer m and n, can you say S(mn) = S(m) + S(n) is true or false? It is difficult to say. For some m an n, it is true, but for some other numbers it is false.

About this problem, J.Sandor [7] proved an very important conclusion. That is, for any positive integer k and any positive integers [m.sub.1],[m.sub.2], ... [m.sub.k], we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This paper as a note of [7], we shall prove the following two conclusions:

Theorem 1. For any integer k [greater than or equal to] 2 and positive integers [m.sub.1], [m.sub.2], ... [m.sub.k], we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Theorem 2. For any integer k [greater than or equal to] 2, we can find infinite group numbers [m.sub.1], [m.sub.2], ... [m.sub.k] such that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[section] 2. Proof of the theorems

In this section, we will complete the proof of the Theorems. First we prove a special case of Theorem 1. That is, for any positive integers m and n, we have

S(m)S(n) [greater than or equal to] S(mn).

If m = 1 (or n = 1), then it is clear that S(m)S(n) [greater than or equal to] S(mn). Now we suppose m [greater than or equal to] 2 and n [greater than or equal to] 2, so that S(m) [greater than or equal to] 2, S(n) [greater than or equal to] 2, mn [greater than or equal to] m + n and S(m)S(n) [greater than or equal to] S(m) + S(n). Note that m|S(m)!, n|S(n)!, we have mn|S(m)!S(n)!|((S(m)+S(n))!. Because S(m)S(n) [greater than or equal to] S(m)+S(n), we have (S(m) + S(n))!|(S(m)S(n))!. That is, mn|S(m)!S(n)!|(S(m) + S(n))!|(S(m)S(n))!. From the definition of S(n) we may immediately deduce that

S(mn) [less than or equal to] S(m)S(n):

Now the theorem 1 follows from S(mn) [less than or equal to] S(m)S(n) and the mathematical induction.

Proof of Theorem 2. For any integer n and prime p, if [p.sup.[alpha]][parallel]n!, then we have

[alpha] = [[infinity].summation over (j=1)] [n / [p.sup.j]].

Let [n.sub.i] are positive integers such that [n.sub.i] [not equal to] [n.sub.j], if i [not equal to] j, where 1 [less than or equal to] i, j [less than or equal to] k, k [greater than or equal to] 2 is any positive integer. Since

[[infinity].summation over (r=1)] [[p.sup.[n.sub.i]] / [p.sup.r]] = [p.sup.[n.sub.i-1]] + [p.sup.[n.sub.i-2]] + ... + 1 = [p.sup.[n.sub.i]] - 1 / p - 1.

For convenient, we let [u.sub.i] = [p.sup.[n.sub.i]] - 1 / p-1. So we have

S([p.sup.[u.sub.i]]) = [p.sup.[n.sub.i]], i = 1, 2,..., k. (1)

In general, we also have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

So

S ([p.sup.[u.sub.1+[u.sub.2]+ ... +[u.sub.k])= [k.summation over (i=1)] [p.sup.[n.sub.i]] (2)

Combining (1) and (2) we may immediately obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [m.sub.i] = [p.sup.[u.sub.i]], noting that there are infinity primes p and [n.sub.i], we can easily get Theorem 2.

This completes the proof of the theorems.

References

[1] C.Ashbacher, Some Properties of the Smarandache-Kurepa and Smarandache-Wagstaff Functions. Mathematics and Informatics Quarterly, 7(1997), 114-116.

[2] A.Begay, Smarandache Ceil Functions Bulletin of Pure and Applied Sciences, 16(1997), 227-229.

[3] Mark Farris and Patrick Mitchell, Bounding the Smarandache function Smarandache Notions Journal, 13(2002), 37-42.

[4] Kevin Ford, The normal behavior of the Smarandache function, Smarandache Notions Journal, 10(1999), 81-86.

[5] P.Erdos, Problem 6674 Amer. Math. Monthly, 98(1991), 965.

[6] Pan Chengdong and Pan Chengbiao, Element of the analytic number theory, Science Press, Beijing, (1991).

[7] J.Sandor, On a inequality for the Smarandache function, Smarandache Notions Journal, 10(1999), 125-127.

Kang Xiaoyu

Editorial Board of Journal of Northwest University

Xi'an, Shaanxi, P.R.China
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Author:Kang, Xiaoyu
Publication:Scientia Magna
Date:Jun 1, 2005
Words:963
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