ON THE PERMUTATION SEQUENCE AND ITS SOME PROPERTIES

ON THE PERMUTATION SEQUENCE AND ITS SOME PROPERTIES*

ZHANG ,\VENPENG

Research Center for Basic Science, Xi'an Jiaotong University Xi'an, Shaanxi, PoR.China The main purpose of this paper is to prove that there is no any perfect power among the permutation sequence: 12, 1342, 135642, 13578642, 13579108642, ....... This answered the question 20 of F.Smarandach in [1].

ABSTRACT.

~tflf:'f ) pa-r~ I

1.

INTRODUCTION

For any positive integer n, we define the permutation sequence {pen)} as follows: P(I) = 12, P(2) = 1342, P(3) = 135642, P( 4) = 13578642, P(5) = 13579108642, .0 .... , Pen) = 135· .. (2n - 1)(2n)(2n - 2)· 0·42, .... o. ,. In problem 20 of [1], Professor F.Smaranda:ch asked us to answer such a question: Is there any perfect power among these numbers? Conjecture: no! This problem is interesting, because it can help us to find some new properties of permutation sequence. In this paper, we shall study the properties of the permutation sequence PC n), and proved that the F.Smarandach conjecture is true. This solved the problem 20 of [1], and more, we also obtained some new divisible properties of P( n). That is, we shall prove the following conclusion: Theorem. There is no any perfect power among permutation sequence, and n

P( n)

=

n

-."--,,..-."--, -811 ( 11· 102 n - 13·10n+) ,2. . = 11· . ·1 X 122· . . 2,

2.

. ;..J..

-re-'"

'" C, I,

nt.. ~

PROOF OF THE THEOREM

In this section, we complete the proof of the Theorem. First for any positive integer n, we have Pen) =

=

+ ... + (2n -1) x IOn + 2n X lOn-l + (2n - 2) X lOn-2 + .. ·4 X 10 + 2 [10 2n - 1 + 3 X 10 2n - 2 + 0.. + (2n -1) X IOn] + [271 X IO n- 1 + (2n - 2) X IO n - 2 + ···4 X 10 + 2]

102n-l

+3 X

10 2n -

2

Key words and phrases. Permutation sequence; Perfect power; A problem of F.Smarandach. work is supported by the N.S.F. and the P.S.F. of P.R.China.

* This

153

(1)

Now we compu te SI and S2 in (1) respec tively. Note that 951 = 10S1 - S1 = I0 2n + 3 X 102n-1 + ... (2n - 1) x lOn+1 _102n -l _ 3 X 102n - 2 - ••. - (2n - 1) x IOn = I0 2n = 10 2n

+2X +2X

10 2n -

1

+2 X

lOn+1

X

952 = 1052 - S2 = 2n

X

I0 2n -

IO n - 1 -

2

1

9

+ ... + 2 x IO n + 1 -

- (2n - 1)

X

(2n - 1) x IOn

IOn

and

- 2n

X

= 2n

X

= 2n

X

10 n -

1

IOn

+ (2n - 2)

X

10n -

l

+ .. ·4 X 102 + 2 X 10

(2n - 2) X lO n - 2 - .• ·4 X 10 - 2 lOn - 2 X 10 n- 1 - 2 X lOn-2 - .. ·2 X 10 - 2 lOn-1 -

IOn - 2

X ---

9

So that

(2) and

(3)

1

S2 = 81 [18n

X

+ 2] .

IOn - 2 X lOn

Thus combi ning (1), (2) and (3) we have

pen) = Sl +S2 1

+ 81

= 8\

X

[11

X

10 2n -18n

[18n X IOn - 2 X lOn

X

-11 X

lOn]

+ 2] n

(4)

lOn

n

1 ( 11·102n ) =11· ----.. . ~ =-13·1 0 n +2 ··1x1 22··· 2.

c...Cf

I

81 . l~fs From (4) we can easily find that 2 1 P(n), but 4 f pen), if n ~ 2, -'J So that pen) can not be a perfec t power , if n ~ 2. In fact, if we assum e P( n) be a perfec t power , then Pen) = mk, for some positiv e intege r m ~ 2 and k ~ 2.

Since 2 1 P(n), so that m must be an even numbe r. Thus we have 41 P(n). This contra dictio n with 4 t P( n), if n ~ 2: Note that P(1) is not a perfec t power, so that P( n) can be a perfec t power for all n ~ 1) ~s compl etes the proof of the Theor em.

~~q.)

REFER ENCES

1. F. Smaran dache, Only problems, not Solutio ns, Xiquan Pub!.

House, Chicago , 1993, pp. 2l.

2. Tom M. Aposto l, Introdu ction to Analyti c Numbe r Theory, Springe r-Verla g, New York, 1976.

3. R. K. Guy, Unsolved Problem s in Numbe r Theory, Springe r-Verla g, New York, Heidelb erg, Berlin, 198!.

4. "Smara ndache Sequen ces" at http://w ww.gal Iup.unm .eduj" smaran dache/s naqint. txt.

5. "Smara ndache Sequen ces" at http://w ww.gal Iup.unm .eduj"s marand ache/sn aqint2. txt.

6. "Smara ndache Sequen ces" at http://w ww.gal lup.unm .eduj"s marand ache/sn aqint3. txt.

154