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An equation involving the Smarandache function.

Abstract For any positive integer n, let S(n) denotes the Smarandache function, and [empty set](n) is the Euler function. The main purpose of this paper is using the elementary method to study the solutions of the equation S(n) = [empty set](n), and give all solutions for it.

Keywords Smrandache function; Equation; Solutions.

[section] 1. Introduction

For any positive integer n, the Smarandache function S(n) is defined as the smallest integer m such that n|m!. From the definition and the properties of S(n), one can easily deduce that if n = [p.sup.[[alpha].sub.1.]sub.1][p.sup.[[alpha].sub.2].sub.2] x x x [p.sup.[[alpha].sub.k].sub.k] is the prime powers factorization of n, then


About the arithmetical properities of S(n), many people had studied it before, see references [3], [4] and [5].

If n [greater than or equal to] 1, the Euler function [empty set](n) is defined to be the number of all positive integers not exceeding n, which are relatively prime to n. It is clear that [empty set](n) is a multiplicative function.

In this paper, we shall use the elementary method to study the solutions of the equation S(n) = [empty set] (n), and give all solutions for it. That is, we shall prove the following:

Theorem. The equation S(n) = [empty set] (n) have only 4 solutions, namely,

n = 1, 8, 9, 12.

[section] 2. Proof of the theorem

In this section, we shall complete the proof of the theorem. Let n = [p.sup.[[alpha].sub.1].sub.1] [p.sup.[[alpha].sub.2].sub.2] x x x [p.sup.[[alpha].sub.k].sub.k] denotes the factorization of n into prime powers, and let


Then from the definitions of S(n) and [empty set](n) we have

[empty set](n) = [p.sup.[[alpha].sub.1]-1.sub.1]([p.sub.1] - 1)[p.sup.[[alpha].sub.2]-1.sub.2] ([p.sub.2] - 2) x x x [p.sup.[[alpha].sub.k]-1.sub.k]([p.sub.k] - 1) =[empty set]([p.sup.[alpha]-1])(p - 1])[empty set]([n.sub.1]) = S ([p.sup.[alpha])].

It is clear that n = 1 is a solution of the equation S(n) = [empty set](n). If n > 1, then we will discuss the problem in three cases:

(I) If [alpha] = 1 and n = p, then S(n) = p 6= p [not equal to] 1 = [empty set](n). That is, there is no any prime satisfied the equation. If [alpha] = 1 and n = [n.sub.1]p, then S(n) = p [not equal to] (p - 1) [empty set]([n.sub.1]) = [empty set] ([n.sub.1]p). So the equation has also no solution.

(II) If [alpha] = 2, then S([p.sup.2]) = 2p and [empty set]([p.sup.2][n.sub.1]) = p(p - 1)[alpha]([n.sub.1]). So in this case S(n) = [empty set](n) if and only if

(p - 1)[empty set]([n.sub.1]) = 2.

This time, there are two cases: p - 1 = 1, [empty set]([n.sub.1]) = 2; p - 1 = 2, [empty set]([n.sub.1]) = 1. That is, p = 2, [n.sub.1] = 3; p = 3, [n.sub.1] = 1. So in this case, the equation has two solutions: n = 12, 9.

(III) If [alpha] = 3, it is clear that S([2.sup.3]) = [empty set]([2.sup.3]) = 4, so n = 8 satisfied the equation. If [alpha] [greater than or equal to], 3 and p > 2, noting that

[p.sup.[alpha]-2] > [2.sup.[alpha]-2] = [(1 + 1).sup.[alpha]-2] = 1 + [alpha] - 2 + x x x + 1 > [alpha].

That is,

[p.sup.[alpha]-1] > [alpha]p [right arrow][p.sup.[alpha]-1](p - 1)[empty set]([n.sub.1]) > [alpha]p,


S([p.sup.[alpha]]) [less than or equal to] [alpha]p:

So this time, the equation has no solution.

Now combining the above three cases, we may immediately get all 4 solutions of equation S(n) = [empty set](n), namely

n = 1, 8, 9, 12.

This completes the proof of Theorem.


[1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 1993.

[2] Tom M. Apostol, Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976.

[3] Wang Yongxing, On the Smarandache function, Research on Smarandache problems in number theory, Hexis, 2005, pp. 103-106.

[4] Ma Jinping, The Smaranache Multiplicative Function, Scientia Magna, 1(2005), 125-128.

[5] Li Hailong and Zhao Xiaopeng, On the Smarandache function and the K-th roots of a positive integer, Research on Smarandache problems in number theory, Hexis, 2004, pp. 119-122.

Ma Jinping

Department of Mathematics, Northwest University

Xi'an, Shaanxi, P.R.China, 710069
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Author:Ma, Jinping
Publication:Scientia Magna
Date:Jun 1, 2005
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