MathDB

1

Part of 2003 Pan African

Problems(2)

Function N-->N

Source: Pan African 2003

10/4/2005
Let N0={0,1,2⋯ }N_0=\{0, 1, 2 \cdots \}. Find all functions: N0→N0N_0 \to N_0 such that: (1) f(n)<f(n+1)f(n) < f(n+1), all n∈N0n \in N_0; (2) f(2)=2f(2)=2; (3) f(mn)=f(m)f(n)f(mn)=f(m)f(n), all m,n∈N0m, n \in N_0.
functionlogarithmsinduction
functional equation

Source: Pan African 2003

10/4/2005
Let N0={0,1,2⋯ }\mathbb{N}_0=\{0,1,2 \cdots \}. Does there exist a function f: \mathbb{N}__0 \to \mathbb{N}_0 such that: f2003(n)=5n,∀n∈N0 f^{2003}(n)=5n, \forall n \in \mathbb{N}_0 where we define: f1(n)=f(n)f^1(n)=f(n) and fk+1(n)=f(fk(n))f^{k+1}(n)=f(f^k(n)), ∀k∈N0\forall k \in \mathbb{N}_0?
functionfunctional equation