Discovering Associative Functions: A Prime Puzzle for Homework

[eddybob123]

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?

 

[Office_Shredder]

As a quick example of such a function f, let f(p,q) = pq. Then d(f(p,q)) = min(p,q). And g(p,g(q,r)) = g(g(p,q),r) = min(p,q,r).

It's unlikely they intend for the image of f to be the primes (which your post seems to imply) as that would make composing it with d fairly boring...

 

[Dick]

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?

What's wrong with f(p,q)=2. Or f(p,q)=min(p,q)?

 

[eddybob123]

I intend to find an algebraic function of p and q.