方程φ_(5)(n)=2^(ω(n))的可解性

The Solvability for the Equationφ_(5)(n)=2^(ω(n))

为将Lehmer同余式的模从素数的平方推广到任意整数的平方,Cai等(CAI T X,FU X D,ZHOU X.Acta Aritmetica,2007,130(3):203-214.)定义了广义欧拉函数φ_(e)(n),给出了e=3,4,6时广义欧拉函数φ_(e)(n)的计算公式.最近Zhu等(ZHU C Z,LIAO Q Y.arXiv:2105.10870v1,2021.)确定了e=5时φ_(e)(n)的准确计算公式.利用初等的方法和技巧,研究方程φ_(5)(n)=2^(ω(n))的可解性,确定其全部正整数解.

For a fixed integer n,to generalize the modulo from the square of a prime number to the square of an arbitrary integer for the well-known Lehmer congruence formula,in 2007,Cai et al(CAI T X,FU X D,ZHOU X.Acta Aritmetica,2007,130(3):203-214.),defined the generalized Euler functionφ_(e)(n)and determined the explicit formula forφ_(e)(n)(e=3,4,6).Recently,Zhu et al(ZHU C Z,LIAO Q Y.arXiv:2105.10870 v1,2021.)determined the explicit formula forφ_(5)(n).Using elementary methods and techniques,we discuss the solvability for the equationφ_(5)(n)=2^(ω(n)),and then all the positive integer solutions are obtained.