# Associative function

## 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
Staff Emeritus
Gold Member
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 Helper

## 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)?

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