一個(gè)包含Euler函數(shù)及k階Smarandache ceil函數(shù)的方程及其正整數(shù)解

朱敏慧

(西安工程大學(xué)理學(xué)院,陜西西安 710048)

一個(gè)包含Euler函數(shù)及k階Smarandache ceil函數(shù)的方程及其正整數(shù)解

朱敏慧

(西安工程大學(xué)理學(xué)院,陜西西安 710048)

設(shè)k≥2為給定的整數(shù).對(duì)任意正整數(shù)n,k階Smarandache ceil函數(shù)Sk(n)定義為Sk(n)=min{x:x∈N,n|xk}.本文的主要目的是利用初等方法研究函數(shù)方程Sk(n)=φ(n)的可解性,并給出該方程的所有正整數(shù)解,其中φ(n)為Euler函數(shù).

k階Smarandache ceil函數(shù)；Euler函數(shù)；方程；正整數(shù)解

1 引言

其中d(n)為Dirichlet除數(shù)函數(shù),ζ(s)為Riemann zeta-函數(shù).

本文的主要目的是研究函數(shù)方程Sk(n)=φ(n)的可解性,并求出該方程的所有正整數(shù)解,其中φ(n)為Euler函數(shù).具體地說(shuō)也就是利用初等方法證明下面兩個(gè)結(jié)論:

定理1對(duì)任意正整數(shù)n,方程S2(n)=φ(n)成立當(dāng)且僅當(dāng)n=1,4,8,18,54.

定理2設(shè)k≥3為給定的整數(shù).則對(duì)任意正整數(shù)n,方程Sk(n)=φ(n)成立當(dāng)且僅當(dāng)n=1,4,18.

2 定理的證明

這節(jié)利用初等方法直接給出定理的證明.文中所用到Euler函數(shù)的性質(zhì)均可以在文[7-8]中找到,所以這里不必重復(fù)！

首先證明定理1.

顯然n=1是方程S2(n)=φ(n)的一個(gè)解.n=2不是該方程的解！如果該方程有其它解n≥3,那么n一定為偶數(shù),因?yàn)楫?dāng)n≥3時(shí),Euler函數(shù)φ(n)為偶數(shù)!

綜合以上分析,推出方程S2(n)=φ(n)成立當(dāng)且僅當(dāng)n=1,4,8,18,54.于是完成了定理1的證明.

現(xiàn)在證明定理2.

當(dāng)k≥3時(shí),容易驗(yàn)證n=1滿(mǎn)足方程Sk(n)=φ(n).于是假定n＞1.以下分兩種情況討論.

[1]Smarandache F.Only problems,Not Solutions[M].Chicago:X iquan Publishing House,1993.

[2]Sabin Tabirca,Tatiana Tairca.Some new results concerning the Smarandache ceil function[J].Smarandache Notions Journal,book series,2002,13:30-36.

[3]Ren Dongmei.On the Smarandache Ceil Function and the Dirichlet Divisor Function[C]//Research on Smarandache problems in number theory(Vol.II),Phoenix:Hexis,2005:51-54.

[4]Xu Zhefeng.On the Smarandache Ceil Function and the Number of Prim e Factors[C]//Research on Smarandache problems in number theory,Phoenix:Hexis,2004:71-76.

[5]Lu Yaming.On a Dual Function of the Smarandache Ceil Function[C]//Research on Smarandache problems in number theory(Vol.II).Phoenix:Hexis,2005:54-57.

[6]沈虹.一個(gè)新的數(shù)論函數(shù)及其它的值分布[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2007,23(2):235-238.

[7]張文鵬.初等數(shù)論[M].西安:陜西師范大學(xué)出版社,2007.

[8]Apostol T M.Introduction to Analytic Number Theory[M].New York:Springer-Verlag,1976.

An equation involving the Euler function and the Smarandache ceil function of korder and its positive in teger solu tions

ZHU min-hui

(School of Science,Xi’an Polytechnic University,Xi’an 710069,China)

Letk be afixed positive integer with k≥2.For any positive integer n,the Smarandache ceil function of k order is defined as Sk(n)=min{x:x∈N,n|xk}.The main purpose of this paper is using the elementary method to study the solvability of the equation Sk(n)=φ(n),and obtain its all positive integer solutions,whereφ(n)is the Euler function.

Smarandache ceil function of k order,Euler function,equation,positive integer solutions.

O156.4

A

1008-5513(2009)02-0414-03

2008-06-23.

國(guó)家自然科學(xué)基金(10671155),陜西省教育廳科研專(zhuān)項(xiàng)基金(07JK 267).

朱敏慧(1977-),講師,研究方向：數(shù)論.

2000M SC:11B83