1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Associative function

  1. Oct 22, 2013 #1
    1. The problem statement, all variables and given/known data
    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##.



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



    3. 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?
     
  2. jcsd
  3. Oct 22, 2013 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    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...
     
  4. Oct 22, 2013 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    What's wrong with f(p,q)=2. Or f(p,q)=min(p,q)?
     
  5. Oct 23, 2013 #4
    I intend to find an algebraic function of p and q.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Associative function
  1. Gcd associativity (Replies: 1)

  2. LCM is associative (Replies: 4)

Loading...