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## Homework Statement

Let ##d(n)## denote the least prime factor of a positive integer ##n##, and let ##p## and ##q## be prime numbers. Find all functions ##f## such that ##d(f(p,q))## is associative for all ##p## and ##q##.

## Homework Equations

##f:\Bbb{P}\times \Bbb{P}\to \Bbb{P}## is a binary mapping of prime numbers.

## The Attempt at a Solution

For clarity, we shall call the function composition ##(d\cdot f)(p,q)## simply ##g(p,q)##

To be honest, I'm not even sure such a function exists, let alone try and find it. My first instinct was to expand it out and try to "force" the solution:

$$g(p,g(q,r))))=g(g(p,q),r))$$

which gives us two cases: either ##g## is surjective or ##p=g(p,q)## and ##g(q,r)=r##.

What do you guys think?