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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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