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The positive integer solutions to multivariate Euler function equation \(\varphi({x_1}{x_2} \cdots {x_n}) = {k_1}\varphi({x_1}) + {k_2}\varphi({x_2}) + \cdots + {k_n}\varphi({x_n}) \pm l\). (Chinese. English summary) Zbl 07448803

Summary: Let \(\varphi(n)\) be the Euler’s function, we discussed the solvability of Euler function equation \(\varphi({x_1}{x_2} \cdots {x_n}) = {k_1}\varphi({x_1}) + {k_2}\varphi({x_2}) + \cdots + {k_n}\varphi({x_n}) \pm l\), and gave the more precise upper bound of all positive integer solutions to this equation with elementary method. As an application, we obtained all positive integer solutions to this equation when some positive integer sums as \({k_1}, \cdots, {k_n}\) and \(l\) were given.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11D72 Diophantine equations in many variables
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