## Discussion on the positive integer solution of equation $$k\varphi (m) = S(m^t)$$.(Chinese. English summary)Zbl 07448407

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:

 11D41 Higher degree equations; Fermat’s equation 11B68 Bernoulli and Euler numbers and polynomials 11B83 Special sequences and polynomials
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