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]
方程φ3(n)=n/d 的可解性
- Title:
- On the Solvability of the Equation φ3(n)=n/d
- 文章编号:
- 2096-1618(2017)01-0095-07
- Keywords:
- applied mathematics; applied number theory; generalized euler function; equation; positive integer solution; Mö; bius function; congruence
- 分类号:
- O156.4
- 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β1且素数p≡2(mod 3),αi1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 β1 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)