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Some properties of the permutation sequence.

Abstract The main purpose of this paper is using the elementary method to study the properties of the permutation sequence, and prove some interesting conclusions.

Keywords Permutation sequence, perfect power, divisibility.

[section] 1. Introduction and result

For any positive integer n, the F.Smarandache permutation sequence {P(n)} is defined as P(n) = 135 ... (2n - 1)(2n) ... 42. For example, the first few value of the sequence {P(n)} are: P(1) = 12, P(2) = 1342, P(3) = 135642, P(4) = 13578642, ... This sequence was introduced by professor F.Smarandache in reference [1], where he asked us to study its elementary properties. About this problem, many people had studied it, and obtained a series valuable results, see references [2], [3] and [4]. In reference [5], F.Smarandache proposed the following problem: Is there any perfect power among the permutation sequences? That is, whether there exist positive integers n, m and k with k [greater than or equal to] 2 such that p(n) = [m.sup.k].

The main purpose of the paper is using the elementary method to study this problem, and solved it completely. At the same time, we also obtained some other properties of the permutation sequence JP(n)j. That is, we shall prove the following:

Theorem 1. There is no any perfect power among the permutation sequence.

Theorem 2. Among the permutation sequence JP(n)j, there do not exist the number which have the factor [2.sup.k], where k [greater than or equal to] 2; There exist infinite positive integers a [member of] {P(n)} such that [3.sup.2] divide a.

[section] 2. Proof of the theorems

In this section, we shall use the elementary method to prove our Theorems directly. First we prove Theorem 1. For any positive integer n [greater than or equal to] 2, note that P(n) be an even number and


So from (1) we know that


That is to say, 2 | P(n) and 4 [dagger] P(n). So P(n) is not a perfect power. Otherwise, we can write P(n) - [m.sup.k], where k and m [greater than or equal to] 2. Since P(n) be an even number, so m must be an even number. Therefore, P(n) = [m.sup.k] [equivalent to] 0 mod 4. Contradiction with 4 [dagger] P(n). This proves Theorem 1.

Now we prove Theorem 2. For any positive integer n [greater than or equal to] 2, from (2) we may immediately deduce that [4.sup.r] [dagger] P(n), where r [greater than or equal to] 2 is an integer.

Now we write P(n) as:


Note that 10 [equivalent to] 1 mod 9, from the properties of the congruence we may get [10.sup.t] [equivalent to] 1 mod 9 for all integer t [greater than or equal to] 1. Therefore,


From (4) we know that P(n) [equivalent to] 0 mod 9, if n [equivalent to] 0 or 4 mod 9. That is to say, for all integers n [greater than or equal to] 2, P(9n) and P(9n+ 4) can be divided by 9 = [3.sup.2]. This completes the proof of Theorems.


[1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 21, 1993.

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

[3] Zhang Wenpeng, The elementary number theory (in Chinese), Shaanxi Normal University Press, Xi'an, 2007.

[4] P.K.Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York, 1981.

[5] "Smarandache Sequences" at

[6] "Smarandache Sequences" at

[7] "Smarandache Sequences" at

Min Fang

Department of Mathematics, Northwest University, Xi'an, Shaanxi, P.R.China
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Author:Fang, Min
Publication:Scientia Magna
Article Type:Report
Geographic Code:1USA
Date:Sep 1, 2008
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