## The solutions of arithmetic function equation $$k\varphi(Y) = {\varphi_2}(Y) + S(Y^8)$$.(Chinese. English summary)Zbl 07448418

Summary: The solvability of the equation $$k\varphi(Y) = {\varphi_2}(Y) + S(Y^8)$$ involving $$\varphi(n)$$, $${\varphi_e}(n)$$ and $$S(n)$$ three arithmetic functions is discussed. By using the properties of these three arithmetic functions, it is obtained that the equation has positive integer solutions only when $$k = 1, 2, 4, 5, 9, 11$$, and its specific positive integer solutions are given, where the arithmetic function $$\varphi(n)$$ is Euler function, the arithmetic function $${\varphi_2}(Y)$$ is generalized Euler function and the arithmetic function $$S(n)$$ is Smarandache function.

### MSC:

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