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[1]王 容,罗文力,廖群英.方程φ3(n)=n/d 的可解性[J].成都信息工程大学学报,2017,(01):95-101.[doi:10.16836/j.cnki.jcuit.2017.01.017]
　WANG Rong,LUO Wen-li,LIAO Qun-ying.On the Solvability of the Equation φ3(n)=n/d[J].Journal of Chengdu University of Information Technology,2017,(01):95-101.[doi:10.16836/j.cnki.jcuit.2017.01.017]
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方程φ3(n)=n/d 的可解性

成都信息工程大学学报[ISSN:1006-6977/CN:61-1281/TN] 卷: 期数: 2017年01期 页码: 95-101 栏目: 数理科学 出版日期: 2017-01-20
Title:
On the Solvability of the Equation φ3(n)=n/d
文章编号:
2096-1618(2017)01-0095-07
作者:
王 容;  罗文力;  廖群英
(四川师范大学数学与软件科学学院,四川 成都 610066)
Author(s):
WANG Rong;  LUO Wen-li;  LIAO Qun-ying
(College of Mathematics and Software Science, Sichuan Normal University,Chengdu 610066,China)
关键词:
应用数学;  应用数论;  广义欧拉函数;  方程;  正整数解;  麦比乌斯函数;  同余式
Keywords:
applied mathematics;  applied number theory;  generalized euler function;  equation;  positive integer solution;  Mö; bius function;  congruence
分类号:
O156.4
DOI:
10.16836/j.cnki.jcuit.2017.01.017
文献标志码:
A
摘要:
为将Lehmer同余式从模素数的平方推广到模任整数的平方,蔡天新等人在2007年定义了广义欧拉函数.本文利用已有的广义欧拉函数的准确计算公式φ3(n),研究了方程φ3(n)=n/d的正整数解,并利用初等的方法和技巧给出正整数n=pβ,3pβ,pα11pα22,3pα11pα22时,方程φ3(n)=n/d的全部正整数解,其中β1且素数p≡2(mod 3),αi1且素数pi≡2(mod 3)(i=1,2).给出正整数n=3α∏ki=1pαii>3,方程φ3(n)=n/d有解的必要条件,其中α=0且存在某个i=1,…,k,使得pi≡1(mod 3)以及正整数n=3α∏ki=1pαii>3,其中k>2,α∈{0,1},αi1且素数pi≡2(mod 3)(i=1,…,k),方程φ3(n)=n/d有解的必要条件.
Abstract:
In order to generaliz Lehmer's congruences from modulo prime squares to modulo integer squares, Cai, et al, defined the generalized Euler function.By using elementary methods, the equation φ3(n)=n/d related with the generalized Euler function φ3(n) is studied, where n is a positive integer and d is a positive factor of n. For the positive ingteger n=pβ,3pβ,pα11pα22,3pα11pα22,all solutions for the Diophantine equation φ3(n)=n/d are given, where β1 and the prime p≡2(mod 3),αi1 and the primes pi≡2(mod 3)(i=1,2). Several necessary conditions for the solvability of the equation φ3(n)=n/d are obtained, when n=3α∏ki=1pαii>3, with α=0 and some primes pi≡1(mod 3),or n=3α∏ki=1pαii>3 with k>2,α∈{0,1},αi1 andpi≡2(mod 3)(i=1,…,k)are primes.

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备注/Memo

收稿日期:2016-05-26 基金项目:国家自然科学基金重大资助项目(11401408); 四川省教育厅重点资助项目(14ZA0034); 四川省科技厅应用基础研究计划资助项目(2016JY0134)

更新日期/Last Update: 2017-01-20