AP proved the math Collatz conjecture in 2016 but has another proof in 2022..

Archimedes Plutonium<plutonium.archimedes@gmail.com>
3:49 PM
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This is 6 years later, from 2016 when I proved Collatz by 3N+-1 simultaneously 3N+-3 pointing to the mechanism of Collatz as that every 4 Consecutive Evens has at least one of those Evens divisible by 8, so that a 3 times upward climb by a Odd number is met with a 8 times downward slide by a Even number that one is forced to pick. But now is 2022, and having a total review of my 2016 Collatz, and now I realize I can indeed prove the original 1937 Collatz of just 3N+1 alone. I can prove it by mathematical induction. And the huge stumbling block before-- the 2016 proof, is that I thought there were 3 outcomes of converge to 1, spin around and converge but not to 1, and diverge to infinity. Turns out that is not true. There is but 2 outcomes of converge to 1, or converge to a number not 1 which we must call divergence of 3N+1. So the proof of the 1937 Collatz is a question of whether 3N+1 can converge to a number not 1, and thus spin around inside all the numbers from 1 to 1*10^604. This spinning around is Divergence. And this recognition that divergence is the spinning around hindered my understanding of Collatz. Hindered it so much that I thought no proof was possible for 3N+1 alone..
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
4:59 PM (3 hours ago)
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In hindsight, now that I have proven Collatz, the generalized Collatz of 3N+-1 simultaneously with 3N+-3 and the stand alone Collatz of 3N+1. Now that I have proven both Collatz, in hindsight, let me reinforce my misconception of divergence with asking whether 99N+97 diverges without spinning around on a convergence point of 1 or any odd number from 1 to 97 or higher.

In other words take 99N+97 a polynomial for a test ride of convergence. Back in 2016, I would have expected this polynomial to never converge on 1, and sometimes converge on 97 and higher odd numbers and expected it to go to infinity of 1*10^604 without any convergence at all.

Here in 2022, with a second proof of Collatz, the original Collatz of 3N+1, I am expecting 99N+97 to not go to infinity of 1*10^604 but to converge on 97 or higher odd numbers and thus divergent, because never a converging on 1.

Starting with 1 in 99N+97 goes to 196, goes to 49, goes to 4948, goes to 1237, goes to 122560, goes to 1915, goes to 189682, goes to 94841, no, this is too big for me to tackle.

Let me start with 1 on 19N+7 goes to 26, goes to 13, goes to 254, goes to 127, goes to 2420, goes to 605, goes to 11502, goes to 5751, no, this one is too big for my hand held calculator.

Let me try 9N +5 starting with 1 goes to 14, goes to 7, goes to 68, goes to 17, goes to 158, goes to 79, goes to 716, goes to 179, goes to 1616, goes to 101, goes to 914, goes to 457, goes to 4118, goes to 2059, goes to 18536, goes to 2317, goes to 20858, goes to 10429, goes to 93866, goes to 46933, goes to 422402, goes to 211201, goes to 1900814, goes to 950407, goes to 8553668, goes to 2138417, goes to 19245758, goes to 9622879, goes to 86605916, goes to 21651479, goes to 194863316, goes to 48715829, goes to 438442466, goes to --- ran out of hand calculator. But what I am driving at is that in one of the iterations it lands on a highly factorable even number divisible by 2 numerous times that it falls back and then eventually spins round and round inside of all the numbers from 1 to 10^604.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
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I described a physical interpretation of Collatz previously as the odd numbers being nerves in biology impacting organs to produce as being even numbers.

But perhaps the most relevant application of Collatz is in physics where we describe the Evens as opposed to Odds as the curvature of a closed loop that ends at 1. So that the polynomial phrase of 3N+1 is like a closed loop circuit of physics all ending at a point of 1 where the loop is going upwards in odd numbers and swing downward in even numbers.

As I said, all true math has application in sciences and that means physics for physics gives rise to mathematics and math is a junior player where physics is all of science.

AP
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
7:53 PM (7 minutes ago)
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In New Math, there are boundaries, such as infinity starts at 1*10^604, and its Algebraic Completeness at 1*10^1208. So if we ask whether the Perfect Squares are an infinite set, we have to have at least 1^10^604 perfect squares from 1 to 1*10^1208. And indeed that playground is so big that there are an infinity of perfect squares 1,4,9, 16, 25, ....

For Collatz, is a different picture, for we can only go out to 1*10^604 as the last counting number, but its algebraic completeness 1*10^1208 allows us to still play with the numbers from 10^604 all the way to 10^1208.

Most would be more comfortable in Old Math where none of these boundaries existed, and in that view of numbers, it is safe to say that any odd number in 3N+1 cycles down to 1, because it is just a matter of time for any odd number to hit and land upon a even number of the sequence 2,4,8,16,.... and that is all she wrote. So the Collatz without boundaries and simply 3N+1, is just a spinning through numbers until it lands on the 2,4,8,16, .... highway and down she slides to 1. There are some patterns in 3N+1 that if the odd number has a run, it slowly gets onto the offramp highway onto the 2,4,8,16,.... expressway.

So Old Math is easier to comprehend Collatz as a spinning expressway just waiting for the odd number to be a on ramp to 2,4,8, 16, .... expressway.

In New Math there is some analysis as to whether the boundaries interfer.

AP

AP proved the math Collatz conjecture in 2016 but has another proof in 2022.. And this is probably important for physics for Collatz math is describing closed loop circuits with Even and Odd numbers.

The mathematics of Collatz can be seen as a closed loop circuit in 3Dimensional space, where it lands on 1 can converges there on 1. As the Odd numbers are a ascent in a closed loop, the even numbers a downslide in the closed loop which eventually converges on 1.

I do not know if we can describe all closed loops as a Collatz math such as a cylinder winding of copper wire as a Collatz math. But perhaps we can, all forms of closed loops, symmetrical or asymmetrical, loops with long smooth runs and loops with no smoothness. Perhaps the circle and ellipse and oval all have Collatz like math structure. Totally reasonable when we consider the Collatz 3N+1 converges to 1, meaning it closes the loop.

AP

On Saturday, June 4, 2022 at 8:13:14 PM UTC-5, Archimedes Plutonium wrote:
> AP proved the math Collatz conjecture in 2016 but has another proof in 2022. And this is probably important for physics for Collatz math is describing closed loop circuits with Even and Odd numbers.
>
> The mathematics of Collatz can be seen as a closed loop circuit in 3Dimensional space, where it lands on 1 can converges there on 1. As the Odd numbers are a ascent in a closed loop, the even numbers a downslide in the closed loop which eventually converges on 1.
>
> I do not know if we can describe all closed loops as a Collatz math such as a cylinder winding of copper wire as a Collatz math. But perhaps we can, all forms of closed loops, symmetrical or asymmetrical, loops with long smooth runs and loops with no smoothness. Perhaps the circle and ellipse and oval all have Collatz like math structure. Totally reasonable when we consider the Collatz 3N+1 converges to 1, meaning it closes the loop.
>

Alright, before this day is over, let me get started on what could be Collatz math greatest practical use to Physics and then of course all of science.. For I speculate that the upward climb in odd landing versus downward fall in even number landing is a math form of electrical physics closed loop coils and other closed loop figures of geometry.

Here is the odd numbers in 10 Decimal Grid for the Polynomial 3N+1.

So, now, my chore is simply to list all the 10 Grid possibilities of 3N+1 starting with the odd number 1

1 goes to 4, goes to 1

3 goes to 10, goes to 5, goes to 16, goes to 1

5 goes to 16, goes to 1

7 goes to 22, goes to 11, goes to 34, goes to 17, goes to 52, goes to 13, goes to 40, goes to 5 which goes to 1

9 goes to 28, goes to 7 which goes to 1

That encompasses the entire 10 Grid.

So now that 10 Decimal Grid in Evens, starting in Evens.

2 goes to 1

4 goes to 1

6 goes to 3, goes to 10, goes to 5, goes to 16, goes to 1

8 goes to 1

10 goes to 5, goes to 16, goes to 1

Now how do I craft that above 10 starting points as geometry? Do I say the odds are 1/2 of a closed loop circuit in 3Dimension, where the odds are upward straightline curves and evens are downward straightline curves?

I am thinking of a cylinder type of coil in Faraday law with copper wire windings, not a symmetrical coil and of course closed loop at the 1 convergence.

AP

So here I am going to try to make this a closed loop geometry figure. And what I am going to do is collect all the upward movement numbers in Odd numbers 1, 3, 5, 7, 9 and then all the downward movement numbers.
> 1 goes to 4, goes to 1
>
> 3 goes to 10, goes to 5, goes to 16, goes to 1
>
> 5 goes to 16, goes to 1
>
> 7 goes to 22, goes to 11, goes to 34, goes to 17, goes to 52, goes to 13, goes to 40, goes to 5 which goes to 1
>
> 9 goes to 28, goes to 7 which goes to 1

Upward motion 4, 10, 16, 16, 22, 34, 52, 40, 28. Now I eliminate repeats and then place them in order of size.

4, 10, 16, 22, 28, 34, 40, 52. Now I am going to do the same for the Evens on downward motion.

> So now that 10 Decimal Grid in Evens, starting in Evens.
>
> 2 goes to 1
>
> 4 goes to 1
>
> 6 goes to 3, goes to 10, goes to 5, goes to 16, goes to 1
>
> 8 goes to 1
>
> 10 goes to 5, goes to 16, goes to 1

2, 4, 6, 3, 5, 8, 10, 16

And arranging in order of size 2,3,4,5,6,8,10,16

Surprisingly I have 8 in the upward group and 8 in the downward group.

But I need to add a 1 to both groups.

And so I have a figure that is probably a closed loop asymmetric-oval in 2nd dimension.

I graph this in 2D from point 1 and on the one side are distance lengths of 4, 10, 16, 22, 28, 34, 40, 52 and on the other side are points of distance length 2, 4, 6, 3, 5, 8, 10, 16. There are 17 points of the graph of this closed loop figure.

Now this is not what I really had in mind, for I want something more immediate than a analysis and plotting from that analysis. I want some way of directly translating Collatz up and down motion directly into a geometry figure. And I suspect that have to be in 3rd dimension, not 2nd dimension.

AP

On Saturday, June 4, 2022 at 10:58:43 PM UTC-5, Archimedes Plutonium wrote:
> So here I am going to try to make this a closed loop geometry figure. And what I am going to do is collect all the upward movement numbers in Odd numbers 1, 3, 5, 7, 9 and then all the downward movement numbers.
> > 1 goes to 4, goes to 1
> >
> > 3 goes to 10, goes to 5, goes to 16, goes to 1
> >
> > 5 goes to 16, goes to 1
> >
> > 7 goes to 22, goes to 11, goes to 34, goes to 17, goes to 52, goes to 13, goes to 40, goes to 5 which goes to 1
> >
> > 9 goes to 28, goes to 7 which goes to 1
> Upward motion 4, 10, 16, 16, 22, 34, 52, 40, 28. Now I eliminate repeats and then place them in order of size.
>
> 4, 10, 16, 22, 28, 34, 40, 52. Now I am going to do the same for the Evens on downward motion.
> > So now that 10 Decimal Grid in Evens, starting in Evens.
> >
> > 2 goes to 1
> >
> > 4 goes to 1
> >
> > 6 goes to 3, goes to 10, goes to 5, goes to 16, goes to 1
> >
> > 8 goes to 1
> >
> > 10 goes to 5, goes to 16, goes to 1
> 2, 4, 6, 3, 5, 8, 10, 16
>
> And arranging in order of size 2,3,4,5,6,8,10,16
>
> Surprisingly I have 8 in the upward group and 8 in the downward group.
>
> But I need to add a 1 to both groups.
>
> And so I have a figure that is probably a closed loop asymmetric-oval in 2nd dimension.
>
> I graph this in 2D from point 1 and on the one side are distance lengths of 4, 10, 16, 22, 28, 34, 40, 52 and on the other side are points of distance length 2, 4, 6, 3, 5, 8, 10, 16. There are 17 points of the graph of this closed loop figure.
>
> Now this is not what I really had in mind, for I want something more immediate than a analysis and plotting from that analysis. I want some way of directly translating Collatz up and down motion directly into a geometry figure. And I suspect that have to be in 3rd dimension, not 2nd dimension.
>

Alright, I am not satisfied with my first attempt of a geometry for translating Collatz runs. Let me try a different approach.

For this approach I need a graph paper for it is in 2Dimension.

I will tranlate 3N+1 for that of 1, 2, 3 runs.

That means

1 goes to 4, goes to 1
2 goes to 1
3 goes to 10, goes to 5, goes to 16, goes to 1

So I have a 1st quadrant only and Counting numbers on x and y axis.

So my first plotted coordinate point is (1,4), my next plotted coordinate point is (4,1), next is (2,1), next is (3,10), next (10,5), next (5,16) and finally (16,1)

Now I connect all those plotted points and yes it delivers a oblong closed loop

Now I look for a symmetrical closed loop. No luck with a perfectly symmetrical closed loop but as I plot 6 goes to 3, goes to 10, goes to 5, goes to 16, goes to 1 as that of coordinate points (6,3) (3,10) (10,5) (5,16) (16,1) I get almost a pencil ellipse.

So I believe Collatz translated into geometry are closed loop figures, most oblong and not symmetrical but a few that are symmetrical. This geometry plotting of Collatz may be able to tell us why a run converges and something about when-- if symmetrical enough, a run converges to 1.

AP

Alright, I am not satisfied with my first attempt of a geometry for translating Collatz runs. Let me try a different approach.

For this approach I need a graph paper for it is in 2Dimension.

I will tranlate 3N+1 for that of 1, 2, 3 runs.

That means

1 goes to 4, goes to 1
2 goes to 1
3 goes to 10, goes to 5, goes to 16, goes to 1

So I have a 1st quadrant only and Counting numbers on x and y axis.

So my first plotted coordinate point is (1,4), my next plotted coordinate point is (4,1), next is (2,1), next is (3,10), next (10,5), next (5,16) and finally (16,1)

Now I connect all those plotted points and yes it delivers a oblong closed loop

Now I look for a symmetrical closed loop. No luck with a perfectly symmetrical closed loop but as I plot 6 goes to 3, goes to 10, goes to 5, goes to 16, goes to 1 as that of coordinate points (6,3) (3,10) (10,5) (5,16) (16,1) I get almost a pencil ellipse.

So I believe Collatz translated into geometry are closed loop figures, most oblong and not symmetrical but a few that are symmetrical. This geometry plotting of Collatz may be able to tell us why a run converges and something about when-- if symmetrical enough, a run converges to 1.

Come to think about it, if I enumerate fully some of these runs then I suspect I can get far more symmetry. So in the 5 run of 5 goes to 16, goes to 8 goes to 4 goes to 2 goes to 1. Whereas before, I was just doing it for convenience and not listing each descent algebraically.

So then the 5 run is plotted as (5,16) (16, 8) (8,4) (4,2) (2,1)

Now plotting the 5 run delivers what appears to be either a triangle or a oval of straightline curves.

So this geometry Collatz has very much potential, and the best use as far as I can see, is to develop all forms of closed loops, probably just 2D, unless there is some means of 3 axes for 3rd D.

AP

To wrap and finish this up, I need a Geometrical Interpretation of Collatz rises in odd number landings in 3N+1 and falls in division by 2 in even number landings. Sort of reminds me of Ancient Greeks needing to define triangular numbers 1, 3, 6, 10 , square numbers 1, 4, 9, 16, pentagonal numbers 1, 5, 12, 22, ..

So what the Ancient Greeks did with point dots arrangement, I am attempting to do with straightline curves of closed loops.

I need to see a circle or ellipse to be confident these Collatz rises and falls delivers a geometry translation.

3N+1
________

1 goes to 4 to 2 to 1 translates to (1,4) (4,2) (2,1)
2 goes to 1 translates to (2,1)
3 goes to 10, to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1)

3N-1
_______

1 goes to 2 to 1 translates to (1,2) (2,1)
2 goes to 1 translates to (2,1)
3 goes to 8 to 4 to 2 to 1 translates to (3,8)(8,4)(4,2)(2,1)

So let me try for some symmetry by graphing in 2D, first quadrant only.

Graphing both sets of coordinate points.

I graphed them separately and do not yet know what to make of them. Are both irregular ovals with (2,1) being the apex point?

3N+1
_______

9 goes to 28 to 14 to 7 to 22 to 11 to 34 to 17 to 52 to 26 to 13 to 40 to 20 to 10 to 5 to 16 to 8 to 4 to 2 to 1

Translates to (9,28) (28,14) (14,7) (7,22) (22,11) (11,34) (34,17) (17,52) (52,26) (26,13) (13,40) (40,20) (20,10) (10,5) (5,16) (16,8) (8,4) (4,2) (2,1)

3N-1

9 goes to 26 to 13 to 38 to 19 to 56 to 28 to 14 to 7 to 20 to 10 to 5 to 14
Translates to (9,26) (26,13) (13,38) (38,19) (19, 56) (56, 28) (28, 14) (14,7) (7, 20) (20,10) (10,5) (5,14)

I want to say they look somewhat parabolic if I leave them open with apex (2,1) and (5,14).

AP

On Sunday, June 5, 2022 at 9:01:15 AM UTC-7, Archimedes Plutonium wrote:
> To wrap and finish this up, I need a Geometrical Interpretation of Collatz

I suggest that to wrap and finish this up you need three things.

First you need to have actually settled the Collatz conjecture for every n out to infinity.
Since you seem to have settled on using induction you need to demonstrate an
argument that convinces everyone that since Collatz has been checked out to 10^n
then your contribution proves without doubt that every case out to 10^(n+1) has
been settled.

Go and find the most precise detailed definition in a logic textbook for "induction"
and present that and then show how you have exactly fulfilled the requirements
of that definition and thus settled Collatz for every n out to infinity, whether it is
your definition of infinity or everyone else's in the world definition of infinity.

If your argument is "well just have a computer check out to 10^(n+1)" isn't it.
You yourself have said that for large enough n this is impossible, that is why you
need to present a proof that settles this for all n+1 given a single specific n.

Your "reverse induction" doesn't seem to make any sense. Computers have checked
every number out to 10^20 and that includes all numbers out to 10^19. Your "reverse
induction" seems to say "since they have checked out to 10^20 that it must be true out
to 10^19." But 10^19 was already checked. You don't need induction, forward or
reverse to settle 10^19. What everyone is waiting for is your proof that convinces
everyone that it is true out to infinity, not that it is true for smaller numbers because
those and bigger have already been checked.

Second, for your urge to have a geometrical interpretation, consider a polar plot.
Let the radius be each of the numbers in the sequence of Collatz and let the angle
increase by a fixed amount with each number. If the sequence repeats with three
numbers 1,2,4 repeating then increase the angle by 120 degrees for each point.
If the sequence repeats with 10 numbers repeating then increase the angle by
36 degrees each time. Or maybe choose a better angle increment to really really
demonstrate what is hiding behind all those points.

The resulting graph won't be perfectly smooth, but it will (irregularly) decrease in
radius down to the repeating numbers and then loop there forever. Or it will
(irregularly) increase in radius as it spirals off to infinity. If two different starting
numbers happen to fall onto the same number at some point then the two curves
will merge and both will proceed on to the rest.

I don't think I've ever seen anyone present Collatz that way, but you should check and
give credit if you do find someone who has already done that.

It might be even more interesting geometrically if you could present all the plots for
every starting number in one graph and see what that cloud of curves looks like and
how many of those curves merge. What do the curves for the first 1000 integers look
like as they spiral around the origin and some merge sooner and some later and all
of them end up spiraling down to 1,2,4,1,2,4,1,2,4.........? Perhaps there will be some
information that can be seen when you carefully correctly present that graph.

Finally, third.

The goal of all of this is that you want to absolutely confirm for everyone else exactly
what their belief in what you are is. When you and everyone else in the world can agree
on what everyone else in the world believes about you, when that is completely settled,
then you can save vast amounts of time you are spending now writing insults about
everyone else, you and everyone else will have agreed what THEY think of you and all
your claims. Then you can get on with something else more important in your life.

On Sunday, June 5, 2022 at 11:01:15 AM UTC-5, Archimedes Plutonium wrote:
> To wrap and finish this up, I need a Geometrical Interpretation of Collatz rises in odd number landings in 3N+1 and falls in division by 2 in even number landings. Sort of reminds me of Ancient Greeks needing to define triangular numbers 1, 3, 6, 10 , square numbers 1, 4, 9, 16, pentagonal numbers 1, 5, 12, 22, ..
>
> So what the Ancient Greeks did with point dots arrangement, I am attempting to do with straightline curves of closed loops.
>
> I need to see a circle or ellipse to be confident these Collatz rises and falls delivers a geometry translation.
>
> 3N+1
> ________
>
> 1 goes to 4 to 2 to 1 translates to (1,4) (4,2) (2,1)
> 2 goes to 1 translates to (2,1)
> 3 goes to 10, to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1)
>
> 3N-1
> _______
>
> 1 goes to 2 to 1 translates to (1,2) (2,1)
> 2 goes to 1 translates to (2,1)
> 3 goes to 8 to 4 to 2 to 1 translates to (3,8)(8,4)(4,2)(2,1)
>
> So let me try for some symmetry by graphing in 2D, first quadrant only.
>
> Graphing both sets of coordinate points.
>
> I graphed them separately and do not yet know what to make of them. Are both irregular ovals with (2,1) being the apex point?
>
> 3N+1
> _______
>
> 9 goes to 28 to 14 to 7 to 22 to 11 to 34 to 17 to 52 to 26 to 13 to 40 to 20 to 10 to 5 to 16 to 8 to 4 to 2 to 1
>
> Translates to (9,28) (28,14) (14,7) (7,22) (22,11) (11,34) (34,17) (17,52) (52,26) (26,13) (13,40) (40,20) (20,10) (10,5) (5,16) (16,8) (8,4) (4,2) (2,1)
>
> 3N-1
>
> 9 goes to 26 to 13 to 38 to 19 to 56 to 28 to 14 to 7 to 20 to 10 to 5 to 14
> Translates to (9,26) (26,13) (13,38) (38,19) (19, 56) (56, 28) (28, 14) (14,7) (7, 20) (20,10) (10,5) (5,14)
>
> I want to say they look somewhat parabolic if I leave them open with apex (2,1) and (5,14).
>

Alright, I am guessing these Collatz polynomials which are straightlines in geometry of mx+b. That they form parabolas with straight line segments and the covergence point as apex of the parabola.

To check on that assertion I am going to try the "start with 3" in 3N-3, 3N-1, 3N+1, 3N+3

Then I am graphing these and also 5N+5 starting at 3 to see if they all follow the same graphing-- parabola shape, irregular parabola.

3N+1, start with 3 goes to 10 to 5 to 16 to 8 to 4 to 2 to 1

3N+3, start with 3 goes to 12 to 6 to 3

3N-1 , start with 3 goes to 8 to 4 to 2 to 1

3N-3, start with 3 goes to 6 to 3

5N+5, start with 3 goes to 20 to 10 to 5 to 30 to 15 to 80 to 40 to 20 to 10 to 5

let me also try 5N-5

5N-5, start with 3 goes to 10 to 5 to 20 to 10 to 5

Now I am trying to think of something in Quantum Electrodynamics in which we think of a closed loop coil of circle ellipse oval but is instead a parabola. Much like that of astronomy, planets and satellites are circle ellipse ovals but occasionally we see comets come through the neighborhood in parabolic orbits.

Of course it is a comet parabola orbit, but from the perspective of the comet, it is a elliptical orbit for the comet of a different star system where it originated from.

AP, King of Science, especially physics-chemistry

Graphing these
>
> 3N+1, start with 3 goes to 10 to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1)
>

Now I am fully aware that my structuring the coordinate points, is causing a parabola shape geometry. I need not go from 10 to 5 to 16 as that of (3,10)(10,5)(5,16). So I am aware that I create a parabola shape. But the thing is, there is no other way to geometrically place those numbers as coordinate points.

There is no other way of making Collatz numbers be coordinate points. So then the parabola shape is intrinsic to Collatz numbers.

> 3N+3, start with 3 goes to 12 to 6 to 3 translates to (3,12)(12,6)(6,3)
>
> 3N-1 , start with 3 goes to 8 to 4 to 2 to 1 translates to (3,8)(8,4)(4,2)(2,1)
>
> 3N-3, start with 3 goes to 6 to 3 translates to (3,6)(6,3)
>
> 5N+5, start with 3 goes to 20 to 10 to 5 to 30 to 15 to 80 to 40 to 20 to 10 to 5 translates to (3,20)(20,10)(10,5)(5,30)
(30,15)(15,80)(80,40)(40,20)(20,10)(10,5)
>
> let me also try 5N-5
>
> 5N-5, start with 3 goes to 10 to 5 to 20 to 10 to 5 translates to (3,10)(10,5)(5,20)(20,10)(10,5)
>
> Now I am trying to think of something in Quantum Electrodynamics in which we think of a closed loop coil of circle ellipse oval but is instead a parabola. Much like that of astronomy, planets and satellites are circle ellipse ovals but occasionally we see comets come through the neighborhood in parabolic orbits.
>
> Of course it is a comet parabola orbit, but from the perspective of the comet, it is a elliptical orbit for the comet of a different star system where it originated from.
>

So I wish there was some competing way of making Collatz numbers be coordinate points in geometry graphing, but there is just one way and that causes me to think that these figures formed from Collatz numbers are actually and truly parabolas. This would then imply in a theorem of Collatz, that a apex (not a focal point but a apex) always exists, and that no Collatz polynomial is without a apex. Addressing my concerns in 2016 proof that a divergence went to infinity without ever spinning around inside of the polynomial. In 2016, I thought divergence could mean spinning around or pure divergence no apex. But now in 2022, with a mathematical induction proof on pure 3N+1 alone, I see that pure divergence is a error of reasoning. In Old Math where those silly cads had no boundaries on infinity, then pure divergence was even more of a mistake, a colossal mistake in Old Math, for as the geometry of parabola plays out-- on any polynomial of Collatz, they all have a apex even if you had the polynomial 10^604N +10^604 in the dumb stupid Old Math, that polynomial is still a parabola with apex.

In the next few hours I will graph the above to see that they indeed form parabolas.

So, yes, well, I invite any reader to see if they can overturn my conclusion-- the conclusion that making Collatz a geometry involves the way that I made the coordinate points above. And then this coordinization mechanism of Collatz as the singular only way to graph Collatz needs a proof of its uniqueness. To be clear.

Statement: the only way to graph Collatz 3N+1 starting at 3 is this coordinization: 3N+1, start with 3 goes to 10 to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1).

So that mathematics needs a proof of that uniqueness.

AP

Now we can go far far beyond the idea that Collatz numbers are a parabola of geometry to the idea that the Logarithm in Calculus is a parabola, is the formation of a parabola geometry of polynomials of Y = mx+b forming parabolas.

So we need not stop with parabolas of Geometry for Collatz numbers but can utilize Collatz numbers in Calculus for the logarithmic functions of Old Math. Keeping in mind, in New Math the logarithm function is not a function at all until transformed into polynomials. And this is exactly where the Collatz numbers comes into prime use. The Collatz numbers are logarithmic functions translated into polynomials.

So in New Math, say someone has a logarithmic or exponential function and wants to do calculus with it. So what they end up doing is fitting the function with a Collatz sequence of numbers. In this manner, we can appreciate the ups and downs in Collatz runs because the logarithmic function is a up and down graph.

But I, AP, much rather prefer to just translate a logarithmic function into a specific interval and use Lagrange transform into a polynomial. Collatz translation is for advanced mathematics.

AP, King of Science, especially Physics-Chemistry

On Sunday, June 5, 2022 at 5:49:26 PM UTC-5, Archimedes Plutonium wrote:
> Graphing these
> >
> > 3N+1, start with 3 goes to 10 to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1)
> >
>
> Now I am fully aware that my structuring the coordinate points, is causing a parabola shape geometry. I need not go from 10 to 5 to 16 as that of (3,10)(10,5)(5,16). So I am aware that I create a parabola shape. But the thing is, there is no other way to geometrically place those numbers as coordinate points.
>
> There is no other way of making Collatz numbers be coordinate points. So then the parabola shape is intrinsic to Collatz numbers.
>
>
> > 3N+3, start with 3 goes to 12 to 6 to 3 translates to (3,12)(12,6)(6,3)
> >
> > 3N-1 , start with 3 goes to 8 to 4 to 2 to 1 translates to (3,8)(8,4)(4,2)(2,1)
> >
> > 3N-3, start with 3 goes to 6 to 3 translates to (3,6)(6,3)
> >
> > 5N+5, start with 3 goes to 20 to 10 to 5 to 30 to 15 to 80 to 40 to 20 to 10 to 5 translates to (3,20)(20,10)(10,5)(5,30)
> (30,15)(15,80)(80,40)(40,20)(20,10)(10,5)
> >
> > let me also try 5N-5
> >
> > 5N-5, start with 3 goes to 10 to 5 to 20 to 10 to 5 translates to (3,10)(10,5)(5,20)(20,10)(10,5)
> >
> > Now I am trying to think of something in Quantum Electrodynamics in which we think of a closed loop coil of circle ellipse oval but is instead a parabola. Much like that of astronomy, planets and satellites are circle ellipse ovals but occasionally we see comets come through the neighborhood in parabolic orbits.
> >
> > Of course it is a comet parabola orbit, but from the perspective of the comet, it is a elliptical orbit for the comet of a different star system where it originated from.
> >
> So I wish there was some competing way of making Collatz numbers be coordinate points in geometry graphing, but there is just one way and that causes me to think that these figures formed from Collatz numbers are actually and truly parabolas. This would then imply in a theorem of Collatz, that a apex (not a focal point but a apex) always exists, and that no Collatz polynomial is without a apex. Addressing my concerns in 2016 proof that a divergence went to infinity without ever spinning around inside of the polynomial.. In 2016, I thought divergence could mean spinning around or pure divergence no apex. But now in 2022, with a mathematical induction proof on pure 3N+1 alone, I see that pure divergence is a error of reasoning. In Old Math where those silly cads had no boundaries on infinity, then pure divergence was even more of a mistake, a colossal mistake in Old Math, for as the geometry of parabola plays out-- on any polynomial of Collatz, they all have a apex even if you had the polynomial 10^604N +10^604 in the dumb stupid Old Math, that polynomial is still a parabola with apex.
>
> In the next few hours I will graph the above to see that they indeed form parabolas.
>
> So, yes, well, I invite any reader to see if they can overturn my conclusion-- the conclusion that making Collatz a geometry involves the way that I made the coordinate points above. And then this coordinization mechanism of Collatz as the singular only way to graph Collatz needs a proof of its uniqueness. To be clear.
>
> Statement: the only way to graph Collatz 3N+1 starting at 3 is this coordinization: 3N+1, start with 3 goes to 10 to 5 to 16 to 8 to 4 to 2 to 1 translates to (3,10)(10,5)(5,16)(16,8)(8,4)(4,2)(2,1).
>
> So that mathematics needs a proof of that uniqueness.
>

So yes, I got out a fresh new graph paper, one with tiny squares because some numbers are big like 80, and so I wanted a accurate graphing.

And sure enough the polynomials transform into a parabola shaped geometry with a apex for the parabola as point of convergence.

And as I noted earlier, there is only one method of graphing the Collatz, the above method. So these parabolas are not a result of making coordinate points of Collatz numbers, but a result of the fact Collatz can only have a unique coordinate system. A one and only coordinate system.

And the parabola is a form of Y = 1/x and so we have Yx = 1. Where the Y is a polynomial of mx+b and the x is a polynomial of mx+b.

The Collatz number run is a form of calculus, the calculus of the logarithm function.

So, well that pretty much wraps up my journey through Collatz for the time being. And in this stretch of research I have found a proof of the original 1937 Collatz conjecture, plus a geometry interpretation of Collatz.

I am very pleased of my research and 2 proofs of the Collatz conjecture. Should I need to return to Collatz in the future, I want to dig deeper into this calculus application of Collatz.

AP