Vol. 4 (2008), No. 2, 21-24
On the pseudo Smarandache square-free function Bin Cheng Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R.China Abstract For any positive integer n, the famous Pseudo Smarandache Square-free function Zw (n) is defined as the smallest positive integer m such that mn is divisible by n. That is, Zw (n) = min{m : n|mn , m ∈ N }, where N denotes the set of all positive integers. The main purpose of this paper is using the elementary method to study the properties of Zw (n), and give an inequality for it. At the same time, we also study the solvability of an equation involving the Pseudo Smarandache Square-free function, and prove that it has infinity positive integer solutions. Keywords The Pseudo Smarandache Square-free function, Vinogradov’s three-primes theorem, inequality, equation, positive integer solution.
§1. Introduction and results For any positive integer n, the famous Pseudo Smarandache Square-free function Zw (n) is defined as the smallest positive integer m such that mn is divisible by n. That is, Zw (n) = min{m : n|mn , m ∈ N }, where N denotes the set of all positive integers. This function was proposed by Professor F. Smarandache in reference [1], where he asked us to study the properties of Zw (n). From the αr 1 α2 definition of Zw (n) we can easily get the following conclusions: If n = pα 1 p2 · · · pr denotes the factorization of n into prime powers, then Zw (n) = p1 p2 · · · pr . From this we can get the first few values of Zw (n) are: 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, · · · . About the elementary properties of Zw (n), some authors had studied it, and obtained some interesting results, see references [2], [3] and [4]. For example, Maohua Le [3] proved that ∞ X
1 , α²R, α > 0 (Zw (n))α n=1 is divergence. Huaning Liu [4] proved that for any real numbers α > 0 and x ≥ 1, we have the asymptotic formula · ¸ ´ ³ X 1 ζ(α + 1)xα+1 Y 1 + O xα+ 2 +² , (Zw (n))α = 1− α ζ(2)(α + 1) p p (p + 1) n≤x
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Bin Cheng
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where ζ(s) is the Riemann zeta-function.
Ã
Now, for any positive integer k > 1, we consider the relationship between Zw
k Y
! mi
i=1
and
k X
Zw (mi ). In reference [2], Felice Russo suggested us to study the relationship between
i=1
them. For this problem, it seems that none had studied it yet, at least we have not seen such a paper before. The main purpose of this paper is using the elementary method to study this problem, and obtained some progress on it. That is, we shall prove the following: Theorem 1. Let k > 1 be an integer, then for any positive integers m1 , m2 , · · · , mk , we have the inequality k X v à ! à k ! Zw (mi ) u k u Y Y i=1 k t Zw mi < ≤ Zw mi , k i=1 i=1
and the equality holds if and only if all m1 , m2 , · · · , mk have the same prime divisors. Theorem 2. For any positive integer k ≥ 1, the equation à k ! k X X Zw (mi ) = Zw mi i=1
i=1
has infinity positive integer solutions (m1 , m2 , · · · , mk ).
§2. Proof of the theorems In this section, we shall prove our Theorems directly. First we prove Theorem 1. For any positive integer k > 1, we consider the problem in two cases: (a). If (mi , mj ) = 1, i, j = 1, 2, · · · , k, and i 6= j, then from the multiplicative properties of Zw (n), we have à Zw
k Y
! mi
i=1
=
k Y
Zw (mi ).
i=1
Therefore, we have v à ! u k u Y k t Zw mi = i=1
k X v à k ! Zw (mi ) u k k uY Y Y i=1 k t Zw (mi ) < < Zw (mi ) = Zw mi . k i=1 i=1 i=1
αir i1 αi2 (b). If (mi , mj ) > 1, i, j = 1, 2, · · · , k, and i 6= j, then let mi = pα 1 p2 · · · pr , βi1 βi2 βir αis ≥ 0, i = 1, 2, · · · , k; s = 1, 2, · · · , r. we have Zw (mi ) = p1 p2 · · · pr , where
( βis =
0, if αis = 0 ; 1, if αis ≥ 1.
Vol.4
23
On the pseudo Smarandache square-free function
Thus k X
Zw (mi )
pβ1 11 pβ2 12 · · · prβ1r + pβ1 21 pβ2 22 · · · prβ2r + · · · + pβ1 k1 pβ2 k2 · · · pβr kr k k ! Ã k Y p 1 p2 · · · pr + p 1 p 2 · · · pr + · · · + p1 p2 · · · p r = p1 p2 · · · pr = Zw mi , k i=1 i=1
≤
=
and equality holds if and only if αis ≥ 1, i = 1, 2, · · · , k, s = 1, 2, · · · , r. v à ! u k q u Y √ k αr k tZ 1 α2 k m = p p · · · p ≤ pα w i 1 2 r 1 p2 · · · pr i=1 k X
≤
where αs =
pβ1 11 pβ2 12 · · · pβr 1r + pβ1 21 pβ2 22 · · · pβr 2r + · · · + pβ1 k1 pβ2 k2 · · · pβr kr = k k X
Zw (mi )
i=1
k
,
βis , s = 1, 2, · · · , r, but in this case, two equal sign in the above can’t be hold
i=1
in the same time. So, we obtain k X v à ! Zw (mi ) u k u Y i=1 k tZ mi < . w k i=1
From (a) and (b) we have k X v à à k ! ! Zw (mi ) u k u Y Y i=1 k tZ ≤ Zw mi < mi , w k i=1 i=1
and the equality holds if and only if all m1 , m2 , · · · , mk have the same prime divisors. This proves Theorem 1. To complete the proof of Theorem 2, we need the famous Vinogradov’s three-primes theorem, which was stated as follows: Lemma 1. Every odd integer bigger than c can be expressed as a sum of three odd primes, where c is a constant large enough. Proof. (See reference [5]). Lemma 2. Let k ≥ 3 be an odd integer, then any sufficiently large odd integer n can be expressed as a sum of k odd primes n = p1 + p2 + · · · + pk . Proof. (See reference [6]).
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Now we use these two Lemmas to prove Theorem 2. From Lemma 2 we know that for any odd integer k ≥ 3, every sufficient large prime p can be expressed as p = p 1 + p2 + · · · + p k . By the definition of Zw (n) we know that Zw (p) = p. Thus, Zw (p1 ) + Zw (p2 ) + · · · + Zw (pk ) = p1 + p2 + · · · + pk = p = Zw (p) = Zw (p1 + p2 + · · · + pk ). This means that Theorem 2 is true for odd integer k ≥ 3. If k ≥ 4 is an even number, then for every sufficient large prime p, p − 2 is an odd number, and by Lemma 2 we have p − 2 = p1 + p2 + · · · + pk−1 or p = 2 + p1 + p2 + · · · + pk−1 . Therefore, Zw (2) + Zw (p1 ) + Zw (p2 ) + · · · + Zw (pk−1 ) = 2 + p1 + p2 + · · · + pk−1 = p = Zw (p) = Zw (2 + p1 + p2 + · · · + pk−1 ). This means that Theorem 2 is true for even integer k ≥ 4. At last, for any prime p ≥ 3, we have Zw (p) + Zw (p) = p + p = 2p = Zw (2p), so Theorem 2 is also true for k = 2. This completes the proof of Theorem 2.
References [1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 1993. [2] F. Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press, USA, 2000. [3] Maohua Le, On the peseudo-Smarandache squarefree function, Smarandache Notions Journal, 13(2002), 229-236. [4] Huaning Liu and Jing Gao, Research on Smarandache problems in number theory, Chicago, Xiquan Publishing House, 2004, 9-11. [5] Chengdong Pan and Chengbiao Pan, Foundation of analytic number theory, Beijing: Science Press, 1997. [6] Yaming Lu, On the solutions of an equation involving the Smarandache function, Scientia Magna, 2(2006), No.1, 76-79. [7] Tom M. Apostol, Introduction to analytical number theory, Spring-Verlag, New York, 1976. [8] Wenpeng Zhang, The elementary number theory, Shaanxi Normal University Press, Xi’an, 2007.