It is proved that if F and G are multiplicative functions such that the equation $ G(n+k) = F(n) ... more It is proved that if F and G are multiplicative functions such that the equation $ G(n+k) = F(n) $ is satisfied for some positive integer k and for every positive integer n, furthermore the sets¶¶ $ {\Cal S}_F:=\{n\in{\Bbb N} \mid F(n)\neq 0\} \text{ and } {\Cal S}_G:=\{n\in{\Bbb N} \mid G(n)\neq 0\} $ ¶are finite, then either $ {\Cal S}_F:=\{n\in{\Bbb N} \mid F(n)\neq 0\} \text{ and } {\Cal S}_G:=\{n\in{\Bbb N} \mid G(n)\neq 0\} $ are finite, then either $ {\Cal S}_F = \{1\} $ , or $ {\Cal S}_F=\{1,p^{\delta}\} $ holds with some prime power $ p^{\delta} $ except for the following cases:¶¶ $ k = 3;~8;~20;~90;~\pi=\hbox{Mersenne prime} $ ,¶in which cases $ {\Cal S}_F $ has exactly three elements.
We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5... more We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = Uℓ, where Uℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = Uℓ.
It is proved that if F and G are multiplicative functions such that the equation $ G(n+k) = F(n) ... more It is proved that if F and G are multiplicative functions such that the equation $ G(n+k) = F(n) $ is satisfied for some positive integer k and for every positive integer n, furthermore the sets¶¶ $ {\Cal S}_F:=\{n\in{\Bbb N} \mid F(n)\neq 0\} \text{ and } {\Cal S}_G:=\{n\in{\Bbb N} \mid G(n)\neq 0\} $ ¶are finite, then either $ {\Cal S}_F:=\{n\in{\Bbb N} \mid F(n)\neq 0\} \text{ and } {\Cal S}_G:=\{n\in{\Bbb N} \mid G(n)\neq 0\} $ are finite, then either $ {\Cal S}_F = \{1\} $ , or $ {\Cal S}_F=\{1,p^{\delta}\} $ holds with some prime power $ p^{\delta} $ except for the following cases:¶¶ $ k = 3;~8;~20;~90;~\pi=\hbox{Mersenne prime} $ ,¶in which cases $ {\Cal S}_F $ has exactly three elements.
We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5... more We prove that if a complex valued completely multiplicative function F and a positive integer ℓ≦5 satisfy the condition F(N) = Uℓ, where Uℓ is the set of all ℓ-th roots of unity, then {F(n+1) F(n) ∣ nε N} = Uℓ.
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