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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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