有关广义欧拉函数的两个方程的解
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摘要
令φ_e(n)为广义欧拉函数,其中n与e都是正整数.讨论了与广义欧拉函数有关的两个方程φ_3(n)=2~(ω(n))与φ_4(n)=2~(ω(n))的正整数解.基于广义欧拉函数及欧拉函数的性质,利用分类分段的讨论方式获得了这两个方程的全部解,其中函数ω(n)为正整数n不同的质因数个数函数.
Letφe(n)be generalized Euler function,where n and e are positive integers.The positive integer solutions of two equations φ_3(n)=2~(ω(n)) and φ_4(n)=2~(ω(n)) with generalized Euler function were studied.Based on the properties of generalized Euler function and Euler function,all solutions of the two equations were obtained by using the mode of discussion on classification and segmentation,whereω(n)was the number of the different prime factor of positive integer n.
Letφe(n)be generalized Euler function,where n and e are positive integers.The positive integer solutions of two equations φ_3(n)=2~(ω(n)) and φ_4(n)=2~(ω(n)) with generalized Euler function were studied.Based on the properties of generalized Euler function and Euler function,all solutions of the two equations were obtained by using the mode of discussion on classification and segmentation,whereω(n)was the number of the different prime factor of positive integer n.
引文
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[2]张四保,刘启宽.关于Euler函数一个方程的正整数解[J].东北师范大学学报(自然科学版),2015,47(3):49-54.
[3]范盼红.关于F.Smarandache函数和欧拉函数的三个方程[J].黑龙江大学学报(自然科学版),2012,29(5):626-628.
[4]张四保,官春梅,席小忠.方程φ(xyz)=kφ(x)φ(y)φ(z)的解[J].南昌大学学报(理学版),2016,40(2):111-116.
[5]许霞,徐小凡.关于欧拉方程φ(ab)=2k(φ(a)+φ(b))的正整数解[J].西南师范大学学报(自然科学版),2016,41(4):6-9.
[6] CAIT X,FU X D,ZHOU X.A congruence involving the quotients of Euler and its applications(II)[J].Acta Arith,2007,130(3):203-214.
[7] CAI T X,SHEN Z Y,HU M J.On the parity of the Generalized Euler Function[J].Advances In Mathernatics,2013,42(4):505-510.
[8] SHEN Z Y,CAIT X,HU M J.On the parity of the Generalized Euler Function(II)[J].Advances In Mathernatics,2016,45(4):509-519.
[9]许宏鑫,赵西卿,张利霞,等.关于数论函数方程φ2(n)=S(n7)的解[J].江西科学,2016,34(3):297-300.
[10]蒋舒囡,沈忠燕.数论函数方程φ(n)=S(n11)和φ2(n)=S(n11)的解[J].浙江外国语学院学报,2014,(5):31-37.
[11]王容,廖群英.方程φe(n)=nd(e=1,2,4)的可解性[J].纯粹数学与应用数学,2016,32(5),481-494.
[12]金明艳,沈忠燕.方程φ2(n)=2Ω(n)和φ2(φ2(n))=2Ω(n)的可解性[J].浙江外国语学院学报(自然科学版),2013,(4):47-52.
[13]俞洪玲,沈忠燕.与广义欧拉函数有关的方程[J].浙江外国语学院学报(自然科学版),2012,(3):91-97.