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Find all N-to-N functions-Iran 3rd round-Number Theory 2007

Source:

July 28, 2010
functioninductionnumber theoryprime numbersnumber theory unsolved

Problem Statement

Find all functions f:N→Nf : \mathbb N \to \mathbb N such that:
i) f2000(m)=f(m)f^{2000}(m)=f(m) for all m∈Nm \in \mathbb N,
ii) f(mn)=f(m)f(n)f(gcd⁡(m,n))f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}, for all m,n∈Nm,n\in \mathbb N, and
iii) f(m)=1f(m)=1 if and only if m=1m=1.
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