Functions Questions #2

I am currently thinking of how to solve this following question, and will encourage any undergraduates to discuss this problem together. I’ll share a bit of what I’ve thought.

If f: \mathbb{N} \rightarrow \mathbb{N} is a bijective function that satisfies f(xy) = f(x)f(y) and f(2015) = 42, what is the minimum value of f(2000)

So here the definition of f means that f is an automorphism of a free commutative monoid on an infinite generating set (of prime numbers). By the fundamental theorem of arithmetic, we know the set of \mathbb{N} are a free commutative monoid over the set of prime numbers. It is a direct consequence of the general theory of monoids, that f send primes to primes there.

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