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 is a bijective function that satisfies and , what is the minimum value of
So here the definition of means that 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 are a free commutative monoid over the set of prime numbers. It is a direct consequence of the general theory of monoids, that send primes to primes there.