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On multiplicative functions with \(f(p+q+n_{0})=f(p)+f(q)+f(n_{0})\). (English) Zbl 1360.11012

Summary: Let \(f(n)\) be a multiplicative function such that \(f\) does not vanish at some prime \(p_0\). In this paper, it is proved that, for any given integer \(n_0\) with \(1 \leq n_0 \leq 10^6\), if \(f(p + q + n_0) = f(p) + f(q) + f(n_0)\) for all primes \(p\) and \(q\), then \(f\) must be the identity function: \(f(n) = n\) for all integers \(n \geq 1\). If a variation of Goldbach’s conjecture is true, then the result is true for any fixed integer \(n_0\).
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MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11A41 Primes
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