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!

Abstract Algebra Proof needed

  1. Oct 3, 2005 #1

    I was reading a journal and an interesting problem came up. I believe the journal was in the American Mathematics Society publications. Well, here's the statement.

    "For all integers, n greater than or equal to 3, the number of compositions of n into relatively prime parts is a multiple of 3."

    Example : For 4: the compositions of relatively prime parts are:

    (1,3), (3,1), (2,1,1), (1,2,1), (1,1,2), (1,1,1,1).

    This is what I have so far for a "proof":

    Let n be an integer greater than or equal to 3.
    Then the first composition will be given by (n-1, 1), (1, n-1); since for all k, an integer, (k, 1) and (1, k) are always relatively prime.
    Also, (1, 1, ..., 1) where the composition adds to n is also an obtainable composition.

    (So basically, I've gotten the end points of the compositions to be a multiple of 3, then I need to prove that the "in-between" compositions will also be a multiple of 3.)

    Well, obviously I'm stuck there. I've tried to split it into two cases afterwards where the cases involve n - odd and n - even but it has come to no avail. Also I've tried to find a formula where the compositions of relatively prime parts is a multiple of 3 but it fails at "6". Here was the formula I came up with that failed, if it could be potentially be improved upon.

    Formula: Starting at n=1, where i=3, i being the starting point.


    For 3, 3! = 6 divided by 2^1 = 2 will equal 3 compositions- a multiple of 3
    For 4, 4! = 24 divided by 2^2 = 4 will equal 6 compositions - a multiple of 3
    For 5, 5! = 120 divided by 2^3 = 8 will equal 15 compositions - multiple of 3

    Well, hopefully people will post their ideas...
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted

Similar Discussions: Abstract Algebra Proof needed
  1. Geometry Proof (Replies: 0)

  2. Boolean Algebra (Replies: 0)

  3. Delta-Epsilon Proof (Replies: 0)