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

Zhang, Sibao; Jiang, Lianxia

J. Jiangxi Norm. Univ., Nat. Sci. Ed. 45, No. 2, 194-197 (2021).

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

Keywords:

Euler function; generalized Euler function; Smarandache function; positive integer solution