Discussion on the positive integer solution of equation \(k\varphi (m) = S(m^t)\). (Chinese. English summary) Zbl 1488.11012

Zhang, Sibao; Jiang, Lianxia

J. Henan Univ., Nat. Sci. 51, No. 3, 367-372 (2021).

Summary: Euler function \(\varphi (n)\) and Smarandache function \(S(n)\) are two important arithmetic functions in number theory. The solvability of equations involving Euler function \(\varphi (n)\) and Smarandache function \(S(n)\) has attracted the attention of many number theory enthusiasts, and has obtained rich research results. The solvability of the equation \(k\varphi (m) = S(m^{31})\) was discussed in this note. Based on the properties of Euler function \(\varphi (n)\) and Smarandache function \(S(n)\) and the elementary method, the equation has positive integer solutions only when \(k = 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 24, 32, 33\), and all positive integer solutions of it were given.

MSC:

11A25	Arithmetic functions; related numbers; inversion formulas
11D41	Higher degree equations; Fermat’s equation

Keywords:

Euler function; Smarandache function; solvability; positive integer solution