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An equation involving the F.Smarandache multiplicative function (1).

Abstract For any positive integer n, we call an arithmetical function f(n) as the F.Smarandache multiplicative function if f(1) = 1, and if n > 1, n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] the fractorization of n into prime powers, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. The main purpose of this paper is using the elementary methods to study the solutions of an equation involving the F.Smarandache multiplicative function, and give its all positive integer solutions.

Keywords F.Smarandache multiplicative function, function equation, positive integer solution, elementary methods.

[section] 1. Introduction and result

For any positive integer n, we call an arithmetical function f(n) as the F.Smarandache multiplicative function if f(1) = 1, and if n > 1, n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the fractorization of n into prime powers, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. For example, the function S(n) = min{m : m [member of] N, n|m!} is a F.Smarandache multiplicative function. From the definition of S(n), it is easy to see that if n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the fractorization of n into prime powers, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

So we can say that S(n) is a F.Smarandache multiplicative function. In fact, this function be the famous F.Smarandache function, the first few values of it are S(1) = 1, S(2) = 2, S(3) = 3, S(4) = 4, S(5) = 5, S(6) = 3, S(7) = 7, S(8) = 4, S(9) = 6, S(10) = 5,.... About the arithmetical properties of S(n), some authors had studied it, and obtained some valuable results. For example, Farris Mark and Mitchell Patrick [2] studied the upper and lower bound of S([p.sup.[alpha]]), and proved that

(p - 1)[alpha] + 1 [less than or equal to] S([p.sup.[alpha]]) [less than or equal to] (p - 1)[[alpha] + 1 + [log.sub.p] [alpha]] + 1.

Professor Wang Yongxing [3] studied the mean value properties of S(n), and obtained a sharper asymptotic formula, that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Lu Yaming [4] studied the solutions of an equation involving the F.Smarandache function S(n), and proved that for any positive integer k [greater than or equal to] 2, the equation

S([m.sub.1] + [m.sub.2] + ... + [m.sub.k]) = S([m.sub.1]) + S([m.sub.2]) + ... + S([m.sub.k])

has infinite groups positive integer solutions ([m.sub.1], [m.sub.2],..., [m.sub.k]).

Jozsef Sandor [5] proved for any positive integer k [greater than or equal to] 2, there exist infinite groups of positive integer solutions ([m.sub.1], [m.sub.2],..., [m.sub.k]) satisfied the following inequality:

S([m.sub.1] + [m.sub.2] + ... + [m.sub.k]) > S([m.sub.1]) + S([m.sub.2]) + ... + S([m.sub.k]).

Also, there exist infinite groups of positive integer solutions ([m.sub.1], [m.sub.2],..., [m.sub.k]) such that

S([m.sub.1] + [m.sub.2] + ... + [m.sub.k]) < S([m.sub.1]) + S([m.sub.2]) + ... + S([m.sub.k]).

In [6], Fu Jing proved more general conclusion. That is, if the positive integer k and m satisfied the one of the following conditions:

(a) k > 2 and m [greater than or equal to] 1 are all odd numbers.

(b) k [greater than or equal to] 5 is odd, m [greater than or equal to] 2 is even.

(c) Any even numbers k [greater than or equal to] 4 and any positive integer m, then the equation

m x S([m.sub.1] + [m.sub.2] + ... + [m.sub.k]) = S([m.sub.1]) + S([m.sub.2]) + ... + S([m.sub.k])

has infinite groups of positive integer solutions ([m.sub.1], [m.sub.2],..., [m.sub.k]).

In [7], Xu Zhefeng studied the value distribution of S(n), and obtained a deeply result. That is, he proved the following Theorem:

Let P(n) be the largest prime factor of n, then for any real numbers x > 1, we have the asymptotic formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (1)

where [zeta](s) is the Riemann zeta-function.

On the other hand, if n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] k be the fractorization of n into prime powers, we define

SL(n) = max{[p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2],..., [p.sup.[[alpha].sub.k].sub.k]}.

Obviously, this function is also a Smarandache multiplicativa function, which is called F.Smarandache LCM function. About the properties of this function, there are many scholars have studied it, see references [8] and [9].

Now, we define another arithmetical function [bar.S](n) as follows: [bar.S](1) = 1, when n > 1 and if n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] be the fractorization of n into prime powers, then we define

[bar.S](n) = max{[[alpha].sub.1][p.sub.1], [[alpha].sub.2][p.sub.2], [[alpha].sub.3][p.sub.3],..., [[alpha].sub.k][p.sub.k]}.

It is easy to prove that this function is also a F.Smarandache multiplicative function. About its elementary properties, we know very little, there are only some simple properties mentioned in [7]. That is, if we replace S(n) with [bar.S](n) in (1), it is also true.

The main purpose of this paper is using the elementary methods to study the solutions of an equation involving [bar.S](n). That is, we shall study all positive integer solutions of the equation

[summation over (d|n)] [bar.S](d) = n, (2)

where [summation over (d|n)] denotes the summation over all positive factors of n.

Obviously, there exist infinite positive integer n, such that [summation over (d|n)] [bar.S](d) > n . For example, let n = p be a prime, then [summation over (d|n)] [bar.S](d) = 1 + p > p. At the same time, there are also infinite positive integer n, such that [summation over (d|n)] [bar.S](d) < n.

In fact, let n = pq, p and q are two different odd primes with p < q, then we have [summation over (d|n)] [bar.S](d) = 1 + p + 2q < pq. So a natural problem is whether there exist infinite positive integer n satisfying (2)? We have solved this problem completely in this paper, and proved the following conclusion:

Theorem. For any positive integer n, the equation (2) holds if and only if n = 1, 28.

[section] 2. Proof of the theorem

In this section, we shall complete the proof of the theorem. Firstly, we prove some special cases:

(i) If n = 1, [summation over (d|n)] [bar.S](d) = [bar.S](d) = S(1) = 1, then n = 1 is a solution of equation (2).

(ii) If n = [p.sup.[alpha]] is the prime powers, then (2) doesn't hold.

In fact, if (2) holds, then from the definition of [bar.S](n), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (3)

Obviously, the right side of (3) is a multiple of p, but the left side is not divided by p, a contradiction. So if n is a prime powers, (2) doesn't hold.

(iii) If n > 1 and the least prime factor powers of n is 1, then the equation (2) also doesn't hold. Now, if n = [p.sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k] = [p.sub.1][n.sub.1] satisfied (2), then from the conclusion (ii), we know that k [greater than or equal to] 2, so from the definition of [bar.S](n), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (4)

Obviously, two sides of (4) has the different parity, it is impossible.

We get immediately from the conclusion (iii), if n is a square-free number, then n can't satisfy (2).

Now we prove the general case. Provided integer n > 1 satisfied equation (2), from (ii) and (iii), we know that n has two different prime powers at least, and the least prime factor power of n is larger than 1. So we let n = [p.sup.[[alpha].sub.1].sub.1][p.sup.[[alpha].sub.2].sub.2] ... [p.sup.[[alpha].sub.k].sub.k], [[alpha].sub.1] > 1, k [greater than or equal to] 2. Let [bar.S](n) = [alpha]p, we discuss it in the following cases:

(A) [alpha] = 1. Then p must be the largest prime factors of n, let n = [n.sub.1]p, note that, if d|[n.sub.1], we have [bar.S](d) [less than or equal to] p - 1, so from [summation over (d|n)] [bar.S](d) = n we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (5)

or

[n.sub.1] + 1 < 2d([n.sub.1]), (6)

where d([n.sub.1]) is the Direchlet divisor function. Obviously, if [n.sub.1] [greater than or equal to] 7, then 2 [less than or equal to] [n.sub.1] [less than or equal to] 6. It is also because the least prime factors power of [n.sub.1] is bigger than 1, we have [n.sub.1] = 4, and n = [n.sub.1]p = 4p, p > 3. Now, from

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

we immediately obtain p = 7 and n = 28.

(B) [bar.S](n) = [alpha]p and [alpha] > 1, now let n = [n.sub.1][p.sup.[alpha]], ([n.sub.1], p) = 1, if n satisfied (2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

If 1 < [n.sub.1] < 8, we consider equation (2) as follows:

(a) If [n.sub.1] = 2. That is, n = 2[p.sup.[alpha]](p > 2), from the discussion of (iii), we know that n = 2[p.sup.[alpha]] isn't the solution of (2).

(b) If [n.sub.1] = 3. That is, n = 3[p.sup.[alpha]], since ([n.sub.1], p) = 1, we have p [not equal to] 3.

If p = 2, n = 3 x [2.sup.[alpha]] satisfied (2). That is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

S(d) + 3 is an odd number, but [summation over (d|[2.sup.[alpha]])] [bar.S](d) + 3 is an even number, so n = 3 x [2.sup.[alpha]] is not the solution of (2).

If p > 3. That is, n = 3 x [p.sup.[alpha]] satisfied (2), then the least prime factor powers of n is 1, from (iii), we know that n = 3 x [p.sup.[alpha]] is not the solution of (2).

(c) If [n.sub.1] = 4, n = 4 x [p.sup.[alpha]] (p [greater than or equal to] 3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

If p = 3. That is, n = 4 x [3.sup.[alpha]] satisfied equation (2), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Since [3.sup.2] [summation over (d|[3.sup.[alpha]])] [bar.S](d)), and [3.sup.2] | 4 x [3.sup.[alpha]], then [3.sup.2] | 12, this is impossible.

If p > 3. That is, n = 4 x [p.sup.[alpha]], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

or 4 x [3.sup.[alpha]] - 3/2[alpha]([alpha]+1)p+11 = 0. Now we fix [alpha], and let f(x) = 4 x [x.sup.[alpha]] - 1/2[alpha]([alpha]+1)x+11, if x [greater than or equal to] 3, f(x) is a increased function. That is,

f(x) [greater than or equal to] f(3) = 4 x [3.sup.[alpha]] - 3/2[alpha]([alpha] + 1) + 11 = g([alpha]).

So when x [greater than or equal to] 3, f(x) = 0 has no solutions, from which we get if p > 3, then equation (2) has no solutions.

(d) If [n.sub.1] = 5, we have n = 5 x [p.sup.[alpha]] (p [not equal to] 5).

If p > 5, then from (iii), we know that n = 5 x [p.sup.[alpha]] is not a solution of equation (2). If p = 2, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [2.sup.2] | 2 [summation over (d|[2.sup.[alpha]])] [bar.S](d)), and [2.sup.2] |5 x [2.sup.[alpha]], so we have [2.sup.2] | 10, this is impossible. Hence n = 5 x [2.sup.[alpha]] unsatisfied the equation (2).

If p = 3, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where 2 [summation over (d|[3.sup.[alpha]])] [bar.S](d)) + 6 is even, and 5 x [3.sup.[alpha]] is odd, so n = 5 x [3.sup.[alpha]] unsatisfied the equation (2).

(e) When [n.sub.1] = 6, n = 2 x 3 x [p.sup.[alpha]], from the discussion of (3), n unsatisfied (2).

(f) When [n.sub.1] = 7, we have n = 7 x [p.sup.[alpha]] (p [not equal to] 7).

If p > 7, then from (iii), n = 7 x [p.sup.[alpha]] isn't the solution of (2).

If p = 2, we must have [alpha] [greater than or equal to] 4. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

where 2 [summation over (d|[2.sup.[alpha]])] [bar.S] (d) + 15 is odd, but n = 7 x [2.sup.[alpha]] is even. So n = 7 x [2.sup.[alpha]] unsatisfied (2).

If p = 3, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

in above equation, 3 | 2 [summation over (d|[3.sup.[alpha]])] [bar.S] (d), and 3 | 7 x [3.sup.[alpha]]. If it satisfied (2), we must obtain 3 [dagger] 13, a contradiction! So n = 7 x [3.sup.[alpha]] is not a solution of (2) either.

If p = 5, since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

in the above equation, 2 [summation over (d|[5.sup.[alpha]])] [bar.S] (d) + 8 is even, 7 x [5.sup.[alpha]] is odd. So n = 7 x [5.sup.[alpha]] is not a solution of (2) either.

(g) When [n.sub.1] [greater than or equal to] 8, we have n = [n.sub.1] x [p.sup.[alpha]] and [p.sup.[alpha]] > [alpha]([alpha]+1)/2 p, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

then if [n.sub.1] [greater than or equal to] 8, = [n.sub.1][p.sup.[alpha]] is not a solution of (2) either.

In a word, equation (2) only has two solutions n = 1 and n = 28.

This completes the proof of the theorem.

(1) This work is supported by the N.S.F.C.(10671155)

Received March 11, 2007

References

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

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

[3] Wang Yongxing, On the Smarandache function, Research on Smarandache Problem In Number Theory (Edited by Zhang Wenpeng, Li Junzhuang and Liu Duansen), Hexis, Vol.II(2005), 103-106.

[4] Liu Yaming, On the solutions of an equation involving the Smarandache function, Scientia Magna, 2(2006), No. 1, 76-79.

[5] Jozsef Sandor, On certain inequalities involving the Smarandache function, Scientia Magna, 2(2006), No.3, 78-80.

[6] Fu Jing, An equation involving the Smarandache function, Scientia Magna, 2(2006), No. 4, 83-86.

[7] Xu Zhefeng, The value distribution property of the Smarandache function, Acta Mathematica Sinica (Chinese Series), 49(2006), No.5, 1009-1012.

[8] Pan Chengdong and Pan Chengbiao, The elementary number theory, Beijing University, Beijing, 1988.

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

Jianbin Chen

School of Science, Xi'an Polytechnic University

Xi'an, 710048, P.R.China
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Date:Jun 1, 2007
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