3n-1: More than a mirror formula of 3n+1?

[StaceyStarwolf]

Oh no, this is so complicated for me as a non mathematician, but I will try and find out what the formulas mean so I can better understand what you are saying.

I wanted to show how the 3n+1 and 3n-1 formulas intertwine and work together and got inspired by Failix his suggestion. It is based off my other artwork that shows this, but on that one I had taken the links apart so it doesn’t show that well how the formulas interact. This one does, this is the 17 loop of the x3-1. Can you spot her?

You can fill in any number and even the even ones. Both formulas are linked to all numbers, so what we get are intertwined fragments of both formulas for each number. I decided not to care about the fragments and how they interact, instead I focused on their start (x2+1 or x2-1) and end points (x3+2 or x3-2) and see where the two will line up.

For those who wonder what the other loops are: -5 is the -17 loop, -3 the -7 loop, -1 the -1 loop and of course 1 is the 1 loop, 3 the 7 loop and 5 the 17 loop. Go lower or higher and the results will expand and never link up again due to the numbers being bound by their x2 and x3 grids.

Feel free to play with this or use it as art, my art is always for free for everyone.

Edit: Funny, I can now also show that when using the 3n+1 formula rules we are ignoring the 1n+1 formula rules and compensating her results with 3n-1. And of course also vice versa:

[StaceyStarwolf]

Hey, it is me again. I am chaotic and forget a lot or accidentally delete stuff. Because I sometimes don’t know how my own confused mind works I decided to find out why I created this art piece in the Collatz art topic:

I didn’t even know what the green numbers meant. But now I think I do. Maybe you guys already figured it out, but the green numbers that I have put on the linear scale is the difference between 1n+1 and 3n+1, which is also the difference between the 1n-1 and 3n-1.

Oh no, don’t you dare!

But you already posted this Stacey, what is the difference? Yes!

See, there is a lot going on here when we are dividing. We are not only compensating our 3n+1 mistake with just 3n-1, but also with the mistake made by using 3n+1 to compensate for 3n-1!

So to avoid using the 3n formulas I apparently substituted them with the difference between the 1n and 3n formulas and as you can see there are these 2 predictable ways to do this. This makes any number either a 3n+1/4 number and thus also a 3n-1/2 number or a 3n-1/4 and 3n+1/2 number. There are no other options.

Higher numbers will have more differences and stacking results on top of each other doesn’t matter. Ups will become other ups and eventually downs due to the other formula using the same numbers to do the other thing. This will make any number always fall back into their base difference.

So looking at the base differences by putting them on a linear scale was actually a smart move. A loop will occur when two opposite numbers accidentally share the same differences when using one or both ways to avoid the 3n formulas.

For example, this is the 17 loop. Notice how both numbers have a difference of 5:

In the art piece I posted in the Collatz Art topic and here you can find the other loops and how they work and that the results fan out making sure there are no other loops besides the one we now know. I think it is because the differences become to big to be able to avoid the x3 part of the 3n formulas.

I am sorry for being chaotic again. I just wanted to elaborate on something I created but don’t really understand, because of all the things that are going on in my mind. But hopefully my art explains it and I hope you enjoyed my silliness again!

[StaceyStarwolf]

It has been awhile since I revisited this topic and I think this is the best place to post some of my new shenanigans.

I was playing with the he T (n) = 3n+1/2 (n odd) and T (n) = 3n-1/2 (n odd) formulas and found out that they work in pairs. For example:

For every other number they are helping each other reach their outcomes and they also mirror previous formula pairs.

However, the interesting thing is that there are whole new Collatz Conjectures behind the ones we are using. Instead of blindly dividing n by 2 we could follow the paired formula.

That would mean instead of T (n) = 3n+1/2 (n odd) and n/2 (n even) this becomes T (n) = 3n+1/2 (n odd) and n*-1/2 (n even) and for T (n) = 5n+1/2 (n odd) and n/2 (n even) this becomes T (n) = 5n+1/2 (n odd) and n*-3/2 (n even) and so on.

They are pretty interesting and nicely balanced Conjectures that I think proper mathematicians could look into, especially if we can prove they are responsible for connecting all the numbers for the ones we are researching now.

[StaceyStarwolf]

Oh my dear, why didn’t I discover this earlier? It would have made things so much easier for me to understand what I created.

I decided to take a proper look at this system mainly to try and see if I can predict the irregularities of the /2 part of the Collatz Conjecture that comes after the upward comparison. I found out I didn’t have to and I will try to explain why.

As explained in one of my previous posts we have 2 types of numbers we can compare:

3n+1 /2 and 3n-1 /4 numbers, for example number 23:

Perfect! This fits inside the system and we can continue with numbers 35 and 17 for our formulas without any problems. You will find 23 on the blue 3n+1 /2 and 3n-1 /4 side or as I like to call it the 3n+1 up side:

3n-1 /2 and 3n+1 /4 numbers, for example number 21:

But wait a minute Stacey, this is incorrect. We are not done with dividing after 21x3+1 /4 = 16. A 21 should become a 3n+1 /64 number which will causes the /2 irregularities in the sequences we can’t predict.

Well, something extremely simple happens, something a lot of people here have already figured out. The 21 is connected to the 5 by -1/4 (remember this is the 3n-1 /2 and 3n+1 /4 side or the 3n+1 down side, hence the -1)

This very simple fact is a fail-safe system the Collatz Conjecture is using in the background to link up all of our uneven numbers. Any number that exceeds her 3n+1 /4 when dividing will get caught by this system and gets automatically linked to another uneven number which goes back into the system. This will repeat itself until the number fits again.

Now consider this:

• 50% of all uneven numbers will go up 3n+1 /2 and will hit a higher uneven number.
• 25% of all uneven numbers will go down 3n+1 /4 and will stay within the system to either go back up 3n+1 /2 or go down 3n+1 /4 again.
• The remaining 25% of all uneven numbers will get caught by the fail-safe system and gets assigned a new number to take over the sequence.

But that is not all. The fail-safe system is stacked by x3 for every uneven number x3-2 and because of this other sequences that are stacked on top of other numbers by x2-1 will drop into the same system!

As you can see the 21 is on the yellow side and it will stay within the system by turning into a 5. Now we can compare the 3n+1 and 3n-1 sequences without having to worry about a number not being connected or going rogue to create new loops. In this case our base number happens to be a multiple of 5, which increases or decreases by 2 for every next even difference.

Unfortunately Nully is the culprit and responsible for the loops. She is the extra slice tugged in within the system and she is pushing away all the numbers on both sides.

So I adjusted the artwork accordingly. It is hard to see because it is such a tiny part of the bigger picture, but the fail-safe numbers are now in there:

Personally I think it is such a beautiful way to show the 3n+1 and 3n-1 loops and all the other numbers stacked on top of them. It reminds me of an MRI device giving you the slices you want to study, except that this one goes on into infinity for both - and + sides.

I just had to make sure all of the numbers were included and now I know they are because of the fail-safe system. One that was staring me in my face for so long that I felt silly for a whole week for not figuring it out sooner. It is just something I wanted to share with you and I hope you enjoyed the read.

[StaceyStarwolf]

I decided to add an example of the 17 for the 3n-1 formula and the 19 for the 3n+1 formula:

The fail-safe system is working on both sides as intended and connecting every uneven number. I extended the 19 just to show the other fail-safe connection 3 /3 = 1. No runaway numbers anymore, no rogue numbers, everything is connected.

The second picture are all the positive differences starting with Nully up to 6 just to show that:

• For every +1 difference the x2-1 and x2+1 go up by +2 and a multiple of +2 for every next x2-1 and x2+1 .
• For every +1 difference the x3-2 and x3+2 go up by +3 and a multiple of +3 for every next x3-2 and x3+2.

As stated before in previous posts it means there will be no more loops because the gap increases in all directions for every +1 difference. (and of course by -1 when going negative) All because of Nully who is an even number tugged in between the other numbers and pushing them aside. Without her the x2-1 and x2+1 would be x2 and x3-2 and x3+2 would be x3.

But don’t worry Nully, we still love you!

I think this system is only perfect for the 3n+1 and 3n-1 formulas though because the differences are compensated with one another. I was working on the 5n+1 formula and not only are the differences not compensated properly, the fail-safe system is not working properly either and letting numbers trough. I can’t connect the numbers that go rogue and into infinity … or not and that is why I think a comparison between 5n+1 and 5n-1 is not possible like this.

I tried something silly by connecting the 5n+1 with 3n+1 trough 3n+1 /2,5 to see if that could help me. And although it created some crazy results, I think it does not. I am going to let it go for now, but who knows. I am just a silly pixel artist, so maybe you can find something I didn’t?

Anyways, I hope you enjoyed the read again and big hugs from Stacey.