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.