Thread: Interesting variations on the Collatz Conjecture

1. Interesting variations on the Collatz Conjecture

After reading http://xkcd.com/710/ and straining to recall my number theory classes I had many many moons ago, I started doing a somewhat serious analysis of the collatz conjecture. after asking some of my questions about similar systems I came up with one of my own which appears to behave like collatz's, as well as a 2nd one that causes numbers to fall into one of two orbits, unlike collatz.

*excuse the pseudo-code. I'm a medic by profession and amateur coder by hobby.

Collatz Conjecture:
$f(n)=($ If n is odd $(3*n+1)$ else $(n/2))$
all integers >=1 will eventually equal 1

My variation on Collatz:
$f(n)=($ If n is odd $(3*n+3)$ else $(n/2))$
all integers >=1 will eventually equal 3

My 2 orbit variation on collatz:
$f(n)=($ If n is odd $(3*n+7)$ else $(n/2))$
all integers n such that n mod(7) !=0 will eventually equal 5
all integers n such that n mod(7) =0 will eventually equal 7

I've noticed that for pretty much any system who's "odd" half does not have 3 as it's co efficient will have multiple "stable orbits" as well as infinitely increasing numbers that never orbit, yet I have not seen a system that has both a single stable orbit and infinite orbits

I've also noticed that for the negative numbers, multiple stable orbits seemingly without infinite orbits occur more frequently[/edit]

When working with such prime coefficients for the odd half (ex: 5n+1 or 11n+1) I noticed very frequently that when an orbit would end in 6 (excluding 16), it would turn into an infinite orbit, due to the numbers' end fluctuating between 6 and 3. This makes me wonder if the collatz conjecture works somehow because of the "Rule of 9's" as the collatz conjecture itself doesn't seem to generate any multiple of 9 unless the originating number itself is a multiple of 9 (correct me if I'm wrong here, I'm working with spreadsheets of the first 1,000 numbers). This has made me wonder if the collatz conjecture couldn't be extrapolated to work on a different number base system, such as base 6 (odd=2n+1 ?). Any legitimate attempt at a proof of the conjecture, or even proof of properties of collatz-like systems is well beyond my abilities, but maybe someone on here would know how to begin approaching these ideas.

I don't have access to (or know which kind) of resources would show if this kind of work with orbits based off the Collatz Conjecture have been done before, but I would like to find out more either way. Also, I'd love to find a program that could map out conjectures of this nature since I find that a lot of the graphical representations of mathematical systems are beautiful from an artistic standpoint (watching a Collatz tree "grow" is somewhat mesmerising Dailymotion - Collatz conjecture - une vidéo Hi-Tech et Science)

Could someone possibly point me in the right direction for finding other published variations on the Collatz Conjecture, and a (preferably free) program that I could use to map out my own Trees of Collatz conjectures?

2. Originally Posted by justinwcurtiss
Could someone possibly point me in the right direction for finding other published variations on the Collatz Conjecture, and a (preferably free) program that I could use to map out my own Trees of Collatz conjectures?
I think you bring up some interesting questions, but I'll only be responding to the quoted part above.

Searching for "Collatz conjecture" on Google Scholar revealed the article The 3x+ 1 problem and its generalizations by JC Lagarias (1985), among others. (You will need JSTOR access to see that full article from the link.)

Also, a quick search on regular Google gave a site you might be interested in, Collatz Conjecture Calculation Center.

I'd like to use these questions as programming exercises, but I have limited time. I'll let you know if I come up with anything that could be useful.