# Interesting variations on the Collatz Conjecture

#### justinwcurtiss

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

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

My 2 orbit variation on collatz:
$$\displaystyle f(n)=($$ If n is odd $$\displaystyle (3*n+7)$$ else $$\displaystyle (n/2))$$
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?

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#### undefined

MHF Hall of Honor
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.

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