AP's 19th published book -- Proof of the Collatz Conjecture by AP in 2016. Add on a new chapter.
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Archimedes Plutonium<plutonium.archimedes@gmail.com>
Jun 1, 2022, 2:34:54 PM (12 hours ago)
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AP's 19th published book -- Proof of the Collatz Conjecture by AP in 2016. Add on a new chapter.

I saw a post in sci.math recently concerning the Collatz conjecture, only the author refused to admit AP had proven it in 2016.

But since 2016, I much forgot the reasoning and rationale of why my proof worked so well. Maybe my age of 72 is wearing on me. Maybe I have so so much science work that I can no longer keep track of it all. And so from the fact that I could not instantly remember how the proof of mine went. I have decided to add this additional clarity to the published proof.

The clarity that 4 Consecutive Even numbers must have at least one of the four divisible by 8. When we have one of the four consecutive evens divisible by 8 and are Mandatory to select that even number, we slide down close to 1 by a factor of 8, while the odd number raises us upward by no more than a factor of 3 in 3n+1. I had forgotten that it is Mandatory to chose and pick the Even number out of 3n-3, 3n-1,3n+1, and 3n+3.

Old Math Collatz is unprovable unless and until it sharpens its Statement.

Whenever a number is selected, whether odd or even. If Even, then you divide by 2 until you come to a odd. If we land on a Odd number you must review 3n-3, 3n-1,3n+1, and 3n+3. Once you have those 4 consecutive Evens, you must pick the one which is at least minimum divisible by 8. And hence you slide down at least by 8 times whereas you rise by no more than 3 times. Hence the proof for you always converge to 1.

Now the world is full of assinine people who will try to prevent AP from getting credit for proving Collatz, they attack you anyway possible to deny you credit. And the usual attack of AP's proof is that I altered the Statement so much that it is no longer Collatz. Not true. For the original Collatz is "indeterminate in proof". No-one can prove the original Collatz for it is a statement of "fog". I can dream up millions of math statements that looks like a "well formed math statement" but is actually fog. For instance, the No Odd Perfect Number Statement, when you place the definition of factors in a fog. Is 9 in No Odd Perfect Number, is 9 to be 1+3 or is it to be 1+3+3 for that 6 in Even Perfects is 1+2+3. And so why are idiot mathematicians with no logical brains forcing 9 to be 1+3 when it should be 1+3+3 to follow the same logical rules as 6 is 1+2+3. Same for Collatz, you have mathematicians with no logical brains saying that if you land on a Odd Number, you must only consider 3n+1, with no link or connection to the surrounding even numbers of 3n+1. A fog conjecture that can never be proven one way or another unless the idiot mathematician goes through ever entire integer from 1 to that of 1*10^604, which no computer can ever do for that is longer than the age of the Universe.

Here is another fog statement which looks like math only it is foggy and indeterminate. Prime numbers can be placed in a string of any length you want and separated by a Even number. Here is a string of 3 primes separated by distance 2-- 3, 5, 7. That appears to be a valid well defined statement of mathematics. But it is not. It is fog and indeterminate. It is fog because primes never existed in the first place for Counting Numbers are not well defined on division. To be well defined, means that every counting number A, B in A/B returns another counting number. Counting Numbers are well defined on addition and multiplication, but division no. Same as Subtraction is not defined well on Counting Numbers.

The reason that Mathematics has no pattern for Primes with a single formula that encompasses all Primes, is the reason that Primes are not defined for division.

The true numbers of mathematics are Decimal Grid Numbers and in them-- there does not exist a concept of "prime". Primes in mathematics are as goofball fakery as to think that we have Santa Claus in physics classroom.

Here is another example of what appears like being math statement but is in fact fog. The sine function of Old Math forms a sinusoid curve, so says every Old Math textbook taught in High School. Yet if we had a logical brain which is few and rare in modern times. We understand the sine function as being the opposite / hypotenuse of a right triangle that goes around a unit circle. That means the unit circle makes the hypotenuse be 1 in length. That means the right triangle for sine function follows the exact same track as the path of a unit circle. That means the graph of the sine function is never a sinusoid but a semicircle graph.

Almost every conjecture in Old Math that was unproven, needed first, a doctoring up of its initial Statement. The statement was deeply flawed and never allowed a proof.

I stated all of that in my book and even on Usenet sci.math by 2016. But all of us in science are required to constantly update and make more and more clear as to what we discovered. So in that light I will add a new chapter of a 2022 review of my 2016 proof of Collatz.

19th published book

World's First Proof of Collatz Conjecture// Math proof series, book 6 Kindle Edition
by Archimedes Plutonium (Author)

Last revision was 20Aug2021. This is AP's 19th published book.

Preface: Old Math's Collatz conjecture, 1937, was this: If you land on an even number, you divide by 2 until you come to an odd number. If you come to or land on an odd number, you do a 3N+1 then proceed further. The conjecture then says that no matter what number you start with, it ends up being 1..

What the Collatz proof of math tells us, is that so very often mathematicians pose a conjecture in which their initial formulation of the conjecture is murky, obfuscation and poorly designed statement. Such poorly designed statements can never be proven true or false. An example that comes to mind of another poorly designed conjecture is the No Odd Perfect Conjecture, in which the statement is obfuscation of factors. So for the odd number 9, is it 1+3, or is it 1+ 3 + 3. So when a mathematics conjecture is full of obfuscation and error in the statement, then these type of conjectures never have a proof. And takes a person with a logical mind to fix and straighten out the conjecture statement and then provide a proof, thereof.

Cover picture: when I think of Collatz, I think of a slide, a slide down and so my French curve is the best slide I can think of, other than a slide-ruler, but a slide ruler is slide across.

Product details
• ASIN ‏ : ‎ B07PS98K5H
• Publication date ‏ : ‎ March 16, 2019
• Language ‏ : ‎ English
• File size ‏ : ‎ 1955 KB
• Text-to-Speech ‏ : ‎ Enabled
• Screen Reader ‏ : ‎ Supported
• Enhanced typesetting ‏ : ‎ Enabled
• X-Ray ‏ : ‎ Not Enabled
• Word Wise ‏ : ‎ Not Enabled
• Print length ‏ : ‎ 61 pages
• Lending ‏ : ‎ Enabled
• Best Sellers Rank: #212,131 in Kindle Store (See Top 100 in Kindle Store)
◦ #4 in 45-Minute Science & Math Short Reads
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Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium.archimedes@gmail.com>
Jun 1, 2022, 2:42:27 PM (12 hours ago)
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Yes, I need to write a new chapter as a 2022, clarification on my 2016 proof of Collatz. When I cannot remember the mechanism of the proof, some 6 years later, means I need to add more clarity.

Dan Joyce being opaque about the Collatz conjecture. Probably because Dan does not want to give credit to AP for proving Collatz in 2016.

Dan Joyce being opaque about the Collatz conjecture. Probably because Dan does not want to give credit to AP for proving Collatz in 2016.
On Monday, May 30, 2022 at 4:02:28 PM UTC-5, Dan joyce wrote:
> Simplifying the 3n+1 problem where any starting odd number will always

Here is AP's proof of Collatz--
World's First Proof of Collatz Conjecture// Math proof series, book 6 Kindle Edition
by Archimedes Plutonium (Author)

Last revision was 20Aug2021. This is AP's 19th published book.

Preface: Old Math's Collatz conjecture, 1937, was this: If you land on an even number, you divide by 2 until you come to an odd number. If you come to or land on an odd number, you do a 3N+1 then proceed further. The conjecture then says that no matter what number you start with, it ends up being 1.

What the Collatz proof of math tells us, is that so very often mathematicians pose a conjecture in which their initial formulation of the conjecture is murky, obfuscation and poorly designed statement. Such poorly designed statements can never be proven true or false. An example that comes to mind of another poorly designed conjecture is the No Odd Perfect Conjecture, in which the statement is obfuscation of factors. So for the odd number 9, is it 1+3, or is it 1+ 3 + 3. So when a mathematics conjecture is full of obfuscation and error in the statement, then these type of conjectures never have a proof. And takes a person with a logical mind to fix and straighten out the conjecture statement and then provide a proof, thereof.

Cover picture: when I think of Collatz, I think of a slide, a slide down and so my French curve is the best slide I can think of, other than a slide-ruler, but a slide ruler is slide across.

Alright, in adding on a new chapter of my existing book of proof of Collatz, I should add on some new research, not just making the existing book packed with more clarity. So the new research I am adding on concerns how much are needed the 3N+3, the 3N-1 and the 3N-3. Do any of those for a odd number landing either diverge or spin around endlessly. I see that computers have verified Collatz of 3N+1 out to 10^20. But 10^20 is a small fry compared to what needs to be verified out to 10^604 in which we can declare true and proven. The AP Collatz which includes 3N+-1 simultaneously 3N+-3 and forces one to pick the even number with at minimum factorable by 8, is a instanteous proof and needs no computer checking.

So in this new research I have this.

3N+3
start with 7 goes to 24 goes to 3 goes to 12 goes to 3 and so internal spinning

3N-1
start with 7 goes to 20 goes to 5 goes to 14 goes to 7 and internal spinning

3N-3
start with 7 goes to 18 goes to 9 goes to 24 goes to 3 goes to 3 and internal spinning

Of course the original Collatz goes to 1.

3N+1
start with 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 goes to 16 goes to 1.

So, we have to ask the logical question, and perhaps I covered this in my book dating back to 2016. The logical question that either 3N+1 also has internal spin or worse yet diverge to infinity, or, the second possibility that the form 3N+1 is special and always converges to 1.

I am guessing it is not special and that somewhere in the large numbers of perhaps 300 digits or 400 or 500 or 600 digits we see where 3N+1 has internal spinning or worse yet diverges to infinity which is 1*10^604 and beyond.

I have the feeling that 3N+1 is the closest to the sequence 2,4, 8, 16, 32, 64, etc combined with the sequence 3, 6, 12, 24, 48, etc. These two sequences are the two major sequences and somehow 3N+1 is linked up in some manner. But when we get to very extreme large numbers of 600 digits, 3N+1 will break down.

AP

On Thursday, June 2, 2022 at 5:27:00 AM UTC-5, Archimedes Plutonium wrote:
> Alright, in adding on a new chapter of my existing book of proof of Collatz, I should add on some new research, not just making the existing book packed with more clarity. So the new research I am adding on concerns how much are needed the 3N+3, the 3N-1 and the 3N-3. Do any of those for a odd number landing either diverge or spin around endlessly. I see that computers have verified Collatz of 3N+1 out to 10^20. But 10^20 is a small fry compared to what needs to be verified out to 10^604 in which we can declare true and proven. The AP Collatz which includes 3N+-1 simultaneously 3N+-3 and forces one to pick the even number with at minimum factorable by 8, is a instanteous proof and needs no computer checking.
>
> So in this new research I have this.
>
> 3N+3
> start with 7 goes to 24 goes to 3 goes to 12 goes to 3 and so internal spinning
>
> 3N-1
> start with 7 goes to 20 goes to 5 goes to 14 goes to 7 and internal spinning
>
> 3N-3
> start with 7 goes to 18 goes to 9 goes to 24 goes to 3 goes to 3 and internal spinning
>
> Of course the original Collatz goes to 1.
>
> 3N+1
> start with 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 goes to 16 goes to 1.
>
> So, we have to ask the logical question, and perhaps I covered this in my book dating back to 2016. The logical question that either 3N+1 also has internal spin or worse yet diverge to infinity, or, the second possibility that the form 3N+1 is special and always converges to 1.
>
> I am guessing it is not special and that somewhere in the large numbers of perhaps 300 digits or 400 or 500 or 600 digits we see where 3N+1 has internal spinning or worse yet diverges to infinity which is 1*10^604 and beyond.
>
> I have the feeling that 3N+1 is the closest to the sequence 2,4, 8, 16, 32, 64, etc combined with the sequence 3, 6, 12, 24, 48, etc. These two sequences are the two major sequences and somehow 3N+1 is linked up in some manner. But when we get to very extreme large numbers of 600 digits, 3N+1 will break down.
>

Now I do recall in some other aspects of Number theory where a subtraction of 1 was far different than a addition of 1 and of course the addition of 3 and subtraction of 3.

For example the perfect squares.

4, 9, 16, 25, 36, 49, etc

Now taking addition of 1 leads to 5, 10, 17, 26, 37, 50, etc
Taking subtraction of 1 leads to 3, 8, 15, 24, 35, 48, etc
Taking addition of 3 leads to 7, 12, 19, 28, 39, 52, etc
Taking subtraction of 3 leads to 1, 6, 13, 22, 33, 46, etc

No, this is not the special pattern I was looking for, it must have been something else in Number theory. I recall where the addition of 1 yielded abundant types of a kind while the subtraction of 1 yielded very few examples and had to go way out before it yielded a first of that kind. Something in the structure of Counting Numbers that the addition of 1 favored that type over and above addition of 3 or subtraction of 1 or 3.

Maybe if I combine 2,4,8,16,32, .... with 3,6,12,24,48 as a interwoven 2,3,4,6,8,12,16,24,32,48,.....

AP

Alright, my recollections and memories of proving Collatz in 2016 are coming back to me. I recall the funny feature of 5N+5 converging on 5 rather than on 1. And now I see the 3N+3 and 3N-3 converging on 3 rather than 1. Also I see the 3N-1 converging on 7 rather than 1.

So, here a immediate question comes to mind and a massive investigation. Can the 3N+3 or 3N-3 ever converge to 1 rather than to 3 when and if it converges? Can the 5N+5 ever converge to 1 and when does it diverge to infinity? Does the 3N+3 and 3N-3 ever diverge to infinity in a number less than 10^20??

And perhaps more importantly, why does the tack on of a constant k in 3N +-k, why is that constant a determinant of what the convergence point is? Why is the constant 3 in 3N+3 determining the convergence point???

So, a lot of new investigations.

None of these investigations however impact the validity of AP's valid proof of a well-defined Collatz conjecture proven in 2016. The key to Collatz is that every 4 consecutive even numbers has a factor of 8 in one of those 4 evens, for which every Collatz number started has only one direction-- converge to 1.

AP, King of Science, especially Physics -Chemistry

Alright, my recollections and memories of proving Collatz in 2016 are coming back to me. I recall the funny feature of 5N+5 converging on 5 rather than on 1. And now I see the 3N+3 and 3N-3 converging on 3 rather than 1. Also I see the 3N-1 converging on 7 rather than 1.

Take for example 5N+5 and starting with 7 so we have 40, goes to 5 goes to 30 goes to 15 goes to 80 goes to 5. So in this viewpoint one can say it is spinning internally to a convergence point of 5. Does it ever converge to 1?

Let me try 9 on 5N+5 and goes to 50, goes to 25, goes to 130 goes to 65, goes to 330, goes to 165, goes to 830, goes to 415, and here is perhaps an example of 5N+5 completely diverging.

Let me see if 9 can make 3N+3 diverge. Starting at 9 goes to 30, goes to 15, goes to 48, goes to 3, goes to 12, goes to 3 and spins internally.

So here I am looking for where I can make 3N+3 diverge completely.

So, here a immediate question comes to mind and a massive investigation. Can the 3N+3 or 3N-3 ever converge to 1 rather than to 3 when and if it converges? Can the 5N+5 ever converge to 1 and when does it diverge to infinity? Does the 3N+3 and 3N-3 ever diverge to infinity in a number less than 10^20??

And perhaps more importantly, why does the tack on of a constant k in 3N +-k, why is that constant a determinant of what the convergence point is? Why is the constant 3 in 3N+3 determining the convergence point???

So, a lot of new investigations.

None of these investigations however impact the validity of AP's valid proof of a well-defined Collatz conjecture proven in 2016. The key to Collatz is that every 4 consecutive even numbers has a factor of 8 in one of those 4 evens, for which every Collatz number started has only one direction-- converge to 1.

AP, King of Science, especially Physics -Chemistry

Alright, so in 2016, I was just happy to prove Collatz and move on, leaving many many questions unanswered.

But here in 2022, with a fresh new glimpse at Collatz, I have the super opportunity to uncover more of the mysteries of Collatz. Mysteries that I started to penetrate in 2016, but felt no drive or need to dig deeper.

One mystery is this convergence to the point I have called the constant k. The constant k in 5N+k where k = 5 and that this polynomial 5N+5 converges to 5 most often, when it does converge. When it diverges then a pattern is involved.

So the question never answered in 2016 is whether 5N+5 ever converges to 1, while 3N+1 seems to converge to 1 for all counting numbers checked by computer up to 10^20.

So in 2016, I puzzled over 5N+5 and 3N+3 converging to 5 and 3 respectively, but did not complete my research to find out if ever possible that 5N+5 or 3N+3 could sometimes converge to 1, which 3N+1 always converged to 1 for every counting number of 1 to 10^20.

So, let me explore that question for if I do indeed find that 5N+5 if it converges, can only converge to k constant of 5 and whether 3N+3, whenever it converges is allowed to only converge on k = 3 of the polynomial 3N+3.

If I do indeed find that to be the case, heavily implies that the "k value" determines the convergency point. That further implies there is a deep connection between odd numbers and what bridges or links those odd numbers can have on Even Number sequences starting with 2,4,8, 16, etc and next 3,6,12,24,etc and next 5,10,20,40, etc and on and on with Even number sequences.

In other words, if I find out that 5N+5 can only converge to 5 and no other number and 3N+3 can only converge to 3 and no other number, implies to me, my logical intelligence, that the Collatz conjecture is about bridging Odd numbers with only a certain specific Even number sequence. A 5N+5 landing on a odd number, can only end up if it converges with the Even sequence of 5,10,20,40, etc and the 3N+3 polynomial on a odd number if it converges can only end up at 3 from the 3,6,12,24,etc sequence.

If true, that is a very very remarkable hidden structure inside the Counting Numbers. And if true, let us not get cared away for it is 50 to 50 likely true, but if true, it means there is a hidden structure of Counting Numbers that they can only enter into a Even Number sequence determined by constant k in the polynomial.

I should research 3N-1.

Now the example problem I used in my proof was of the AP-Collatz proof of 3N+-3 simultaneous 3N+-1 and I started it with 75 odd number as the example proof. Where I obtain 222, 224, 226 and 228 for my 4 Consecutive Evens. And my proof forced me to accept 224 as the only acceptable of the 4 Evens in a Row, for 224 = 2*2*2*2*2*7. Now the only reason I bring that up is to ask what form of number is 224? Is it a 3N+3 or a 3N+1, or a 3N-1 or a 3N-3, and it happens to be a 3N-1 number. Telling me that perhaps in every convergence of 3N+1 involves either a 3N+1 or a 3N-1 Algebraic form of polynomial.

What I am alluding towards here, is the idea that we can give every even number of the Counting Numbers a form of one of the Four expressions-- 3N+3 or a 3N+1, or a 3N-1 or a 3N-3. And depending on what odd number we land land on, it forms a bridge or link to a specific even number due to those 4 expressions.

I am attempting to see if I can prove the Old Original Collatz, without using all 4 expressions of even numbers as in the AP proof.

If I can determine that only 3N+1 and 3N-1 link up in a bridge to the sequence 2,4,8,16,... then I will have proven the Old Math Collatz Conjecture only with 3N+1, 3N-1. A step closer but still not the original Collatz for it demands each Odd number landing include both 3N+1 in addition to 3N-1.

So, yes, I am very happy of reviewing my 2016 Collatz for I provide a huge amount of new research above and beyond Clarity.

AP, King of Science, especially Physics-Chemistry

Alright, let me get started on a new table of research data to see if there is any truth to the idea that a polynomial 5N+5 when it converges cannot converge to 1 but the lowest convergence is 5. Calling the constant "k" in 5N+k as the convergence constant. Then a table of 3N+3 where the convergence constant k is 3 and that no convergence goes to 1.

5N+5
_________

N= 1 goes to 10, goes to 5, goes to 30, goes to 15, goes to 80, goes to 5 and spins around endlessly
N= 3 goes to 20, goes to 5, and again spins to 5
N= 5 converges to 5
N=7 goes to 40, goes to 5, converges to 5
N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity.
N=11 goes to 60, goes to 15, goes to 75, goes to 380, goes to 95, goes to 480, goes to 15 and converges on 15 not 5.

Let me try 3N+3, for I am looking for a convergence to 1
________________

N=1, converges to 3
N=3 goes to 12, goes to 3
N=5 goes to 18, goes to 9, goes to 30, goes to 15, goes to 48, goes to 3, converges to 3
N=7 goes to 24, goes to 3, converges to 3
N=9 converges to 3
N= 11 goes to 36, goes to 9 converges to 3
N= 13 goes to 42, goes to 21, goes to 66, goes to 33, goes to 102, goes to 51, goes to 156, goes to 39, goes to 120, goes to 15, converges to 3
N=15 goes to 48, converges to 3
N=17 goes to 54, goes to 27, goes to 84, goes to 21, converges to 3
N= 19 goes to 60, goes to 15, converges to 3
N=21 goes to 66, goes to 33, converges to 3
N=23 goes to 72, goes to 9 converges to 3
N=25 goes to 78, goes to 39, converges to 3
N=27 goes to 84, goes to 21, converges to 3

Now in these two tables one can easily see that a computer on Collatz is easily making its way through counting numbers from 1 to 10^20 because we can program the computer to spot where the new number to be checked is already covered under previous try. And this pattern can perhaps tell us the bridge or link between Odd number and Even numbers that they will converge.

I am trying hard to see why 3N+3 will not fall upon a Even Sequence of 2,4, 8,16, ....etc that will force it to converge to 1. Somehow, all the Odd numbers in 3N+3 cause them to avoid the 2,4,8,16, etc sequence that delivers a convergence to 1. Instead the odds in counting numbers forces 3N+3 onto the sequence of 3,6, 12, 24, etc track, avoiding the 2,4,8,16, etc track. That is a marvellous insight, for it is very difficult to see why 3N+3 avoids the 2,4,8,16,... track.

But perhaps it is not that difficult to see for we can see that 4 consecutive Evens MUST possess a even divisible by 8. In that same sense, a Odd number placed for N in 3N+3 must be inevitably on the 3,6,12,24 track and never the 2,4,8,16 track by simply the way 3N+3 is constructed to produce only Evens on the 3,6,12,24 track.

So well, unless I can not find a 3N+3 that diverges and all convergence is 3, I may have proven Old Math's Old Collatz also!!! And that the proof would simply be that 3N+1 always converges to k=1, and if 3N+3 never diverges then 3N+1 never diverges.

But I doubt I can show that 3N+3 never diverges. If I find a divergence in 3N+3, will automatically tell me that 3N+1 has a divergence starting with a huge number.

Now 5N+5 diverges and we see that in N=9. Here I am going to try to see if that divergence tells me where the first divergence of 3N+3 occurs if it in fact occurs. If it diverges, it may well predict where 3N+1 diverges. I vaguely remember some evidence exists that 3N+1 diverges at 500 digit counting numbers.

AP, King of Science, especially Physics-Chemistry

On Friday, June 3, 2022 at 2:27:31 AM UTC-5, Archimedes Plutonium wrote:
> Alright, let me get started on a new table of research data to see if there is any truth to the idea that a polynomial 5N+5 when it converges cannot converge to 1 but the lowest convergence is 5. Calling the constant "k" in 5N+k as the convergence constant. Then a table of 3N+3 where the convergence constant k is 3 and that no convergence goes to 1.
>
> 5N+5
> _________
>
> N= 1 goes to 10, goes to 5, goes to 30, goes to 15, goes to 80, goes to 5 and spins around endlessly
> N= 3 goes to 20, goes to 5, and again spins to 5
> N= 5 converges to 5
> N=7 goes to 40, goes to 5, converges to 5
> N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity.

Now I spotted something in this divergence that is immensely helpful, unless long runs occur in 5N+5 as they do in 3N+1, and long runs are where the convergence takes a long time to obtain. But in the 5N+5 notice that each odd number obtained is increasing larger, for N=9 starts with 9 then goes to 25, then goes to 65, then goes to 115 then goes to 145 (not shown) so it is safe to say that for N=9 in 5N+5 finds a upward divergent track instead of a downward convergent track to 5. This upward track is 9, 25, 65, 115, 145, ....etc.

Now this maybe a total new concept in mathematics-- a Upward divergent sequence of Counting Numbers. And the homework exercise would be to have the students find the Polynomial that fits the sequence.

Now the 3N+3 polynomial is what I am concerned about at this moment and whether it has a Upward divergent sequence. If so, it looks as though I have not yet found it in my table tonight, but tomorrow will look in my published book of 2019, my 19th published book

World's First Proof of Collatz Conjecture// Math proof series, book 6 Kindle Edition
by Archimedes Plutonium (Author)

Last revision was 20Aug2021. This is AP's 19th published book.

Preface: Old Math's Collatz conjecture, 1937, was this: If you land on an even number, you divide by 2 until you come to an odd number. If you come to or land on an odd number, you do a 3N+1 then proceed further. The conjecture then says that no matter what number you start with, it ends up being 1..

What the Collatz proof of math tells us, is that so very often mathematicians pose a conjecture in which their initial formulation of the conjecture is murky, obfuscation and poorly designed statement. Such poorly designed statements can never be proven true or false. An example that comes to mind of another poorly designed conjecture is the No Odd Perfect Conjecture, in which the statement is obfuscation of factors. So for the odd number 9, is it 1+3, or is it 1+ 3 + 3. So when a mathematics conjecture is full of obfuscation and error in the statement, then these type of conjectures never have a proof. And takes a person with a logical mind to fix and straighten out the conjecture statement and then provide a proof, thereof.

Cover picture: when I think of Collatz, I think of a slide, a slide down and so my French curve is the best slide I can think of, other than a slide-ruler, but a slide ruler is slide across.

Product details
• ASIN ‏ : ‎ B07PS98K5H
• Publication date ‏ : ‎ March 16, 2019
• Language ‏ : ‎ English
• File size ‏ : ‎ 1955 KB
• Text-to-Speech ‏ : ‎ Enabled
• Screen Reader ‏ : ‎ Supported
• Enhanced typesetting ‏ : ‎ Enabled
• X-Ray ‏ : ‎ Not Enabled
• Word Wise ‏ : ‎ Not Enabled
• Print length ‏ : ‎ 61 pages
• Lending ‏ : ‎ Enabled
• Best Sellers Rank: #212,131 in Kindle Store (See Top 100 in Kindle Store)
◦ #4 in 45-Minute Science & Math Short Reads
◦ #9 in Number Theory (Kindle Store)
◦ #32 in Number Theory (Books)

I will look in that book of 2016 if I perhaps had stumbled upon if 3N+3 diverges or not.

AP

Let me check 77 for 3N+3, taking a cue from 5N+5 that 9 diverges. Where 4 +5 = 9 and where 4+3 = 7.

Starting with 77 goes to 234, goes to 117, goes to 354, goes to 177, good signs so far, goes to 534, goes to 267, goes to 804, goes to 201, goes to 606, goes to 303, goes to 912, goes to 57, goes to 174, goes to 87, goes to 264 goes to 33, goes to 102, goes to 51, goes to 156 goes to 39, goes to 120, goes to 15, goes to 48, goes to 3, converges to 3.

No luck in a divergence but was able to spot the fact that in these long runs of Collatz, we see a cycling around where on one pass we have 177 and then on a different pass we have 174, and again on one pass we have 51 and another we have 48.

This is noteworthy of the mechanism of why Collatz works, if in a set of 4 consecutive Evens, a first swipe pass of that set does not fetch the multiple divisions by 2, that a second swipe of that same 4 set of consecutive Evens captures the Even with a multiple division by 2.

So slowly but surely we are viewing the mechanism of why Collatz works-- seeking the multiple factor of 2 within a set of 4 consecutive Evens. Until the odd number finds a downward slide of the 3,6,12, 24, 48, etc sequence, for once it finds that sequence 3N+3 converges to 3.

AP

So, well, I need a divergence in 3N+3 but none is apparent so far. But before I go onwards with 3N+3, perhaps 3N+5. I must be assured that starting with 9 for 5N+5 is a actual divergence and not just a "long run before convergence to 5".

5N+5
---------------
N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity. 115 goes to 580, goes to 145, goes to 730, goes to 365, goes to 1830, goes to 915, goes to 4580, goes to 1145, goes to 5730, goes to 2865. Yes, I am thoroughly convinced it diverges.

Now I need to study why it diverges. Of course, the 5N makes it unable to ever seek a track of the convergent sequences of 2,4,8,16,32, .... etc nor the 3,6,12,24,48,....etc sequence. The only lowest track the 5N+5 can utilize is 5,10,20,40,80,....etc But the starting 9 does not even put 5N+5 into that track but rather into a track of 25, 50, 100, 200, ....etc.

So I need to thorough study this divergence of 5N+5 to see whether my 2016 proof and its conclusion-- that the original Old Math Collatz is forever indeterminate as to whether 3N+1 alone can all converge to 1, for there is not enough time in the Universe for a computer to analyze all numbers from 1 to 1*10^604 as a proof. If we said that the Old Math Collatz statement was for all Counting Numbers from 1 to 10^20 have a solution of converging to 1 for 3N+1, we have a computer proof of that statement. But no proof is possible for 1 to 10^604.

However, if we have the statement of AP-- 3N+-3 simultaneously 3N+-1 and pick the largest factorable by 2 of 4 Consecutive Evens, we have a universal proof of Collatz from 1 to 10^604.

What I am doing here, my post proof of a well defined Collatz statement, is looking to see if I can override that need of a well-defined Collatz. I am looking to see if there cannot be a divergence of 3N+1 for all numbers from 1 to 10^604.

It is my opinion that there is a divergence of 3N+1 before we get to 10^604.. And the very way to check that is to see if there is a number from 1 to 10^604 that has to go beyond the infinity borderline at 1*10^604, go beyond into beyond that of 1*10^1208 the Algebraic Completion of Infinity Borderline. The Algebraic completion 1*10^1208 is how we measure if a set is an infinite set, such as the smallest infinite set of Perfect Squares 1, 4, 9, 16, 25, .....etc.

Clearly, no computer on Earth will ever search through all numbers from 1 to 10^604, let alone trespass into 10^604 through 10^1208.

So I have to make the most and best use of 5N+5 starting with 9.
We see that this run goes from 9 to 25, to 65 to 115 to 145, to 365 to 915, to 1145, to 2865 with never any downward slide, just a steady upward climb..

It is my opinion that 3N+3 has a huge number that it also diverges and also 3N+1 and even a huger number diverges. And that is why the Collatz must be strapped and saddled with a pick out of 3N+-1 simultaneously 3N+-3, of 4 Consecutive Evens for its largest factorable by 2 Even number.

AP, King of Science, especially Physics-Chemistry

On Friday, June 3, 2022 at 12:37:00 PM UTC-7, Archimedes Plutonium wrote:
> So, well, I need a divergence in 3N+3 but none is apparent so far. But before I go onwards with 3N+3, perhaps 3N+5. I must be assured that starting with 9 for 5N+5 is a actual divergence and not just a "long run before convergence to 5".
>
> 5N+5
> ---------------
> N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity. 115 goes to 580, goes to 145, goes to 730, goes to 365, goes to 1830, goes to 915, goes to 4580, goes to 1145, goes to 5730, goes to 2865. Yes, I am thoroughly convinced it diverges.

You are far far far too easily convinced. You made a mistake Arky, 330/2=165 not 115 so

9 goes to 50 goes to 25 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
130 goes to 65 goes to 330 goes to 165 goes to 830 goes to 415 goes to 2080 goes to 1040 goes to 520 goes to 260 goes to
....
so you can be assured that starting with 9 for 5N+5 is a not an actual divergence

On 6/3/2022 5:37 PM, bwr fml wrote:
> On Friday, June 3, 2022 at 12:37:00 PM UTC-7, Archimedes Plutonium wrote:
>> So, well, I need a divergence in 3N+3 but none is apparent so far. But before I go onwards with 3N+3, perhaps 3N+5. I must be assured that starting with 9 for 5N+5 is a actual divergence and not just a "long run before convergence to 5".
>>
>> 5N+5
>> ---------------
>> N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity. 115 goes to 580, goes to 145, goes to 730, goes to 365, goes to 1830, goes to 915, goes to 4580, goes to 1145, goes to 5730, goes to 2865. Yes, I am thoroughly convinced it diverges.
>
> You are far far far too easily convinced. You made a mistake Arky, 330/2=165 not 115 so
>

Also 5n+5 is 5(n+1) so all numbers except the initial will be multiples
of 5.

So I made a mistake in arithmetic here.
> On Friday, June 3, 2022 at 12:37:00 PM UTC-7, Archimedes Plutonium wrote:
> > So, well, I need a divergence in 3N+3 but none is apparent so far. But before I go onwards with 3N+3, perhaps 3N+5. I must be assured that starting with 9 for 5N+5 is a actual divergence and not just a "long run before convergence to 5".
> >
> > 5N+5
> > ---------------
> > N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 115, and this one diverges to infinity. 115 goes to 830, goes to 415, goes to 2080, goes to 65, so here we have the case of 5N+5 has its first internal spin around where it converges to 65 instead of converging to 5 and instead of a infinite divergence.

N=9 goes to 50, goes to 25, goes to 130, goes to 65, goes to 330, goes to 165 goes to 830, goes to 415, goes to 2080, goes to 65, and so here the 5N+5 does not diverge but rather elevates its point of convergence from 5 to that of 65. This implies that further on down the line of 5N+5 keeps raising the convergence point until somewhere in large numbers such as 400 digits or 500 digits we find a divergence to infinity 1*10^604.

AP was hoping 9 was the first divergence but looks like we need a larger number to find the first divergence in 5N+5. And we can perceive the mechanism as being one in which 4 Consecutive Evens seldom lands on a often factorable by 2 Even number in all sets of 4 Consecutive Evens. In the case of 5N+5, it will never land on the Even Sequence of 2,4,8,16, etc or 3,6,12,24,etc. It will land on Even sequence of 5,10,20,40, etc or higher Even Sequences, until it reaches a large enough number that it goes past 10^604 without a convergence or past 10^1208 Algebra Completeness of Counting Numbers.

AP

Alright, AP is going to give it one more try in this review of my 2016 proof of Collatz, one more try to see if I can find a divergence in a starting Odd Number for 5N+5. If I do not find such, then I am left with the resolve that a divergence number for 5N+5 needs to be found by Computers. And I am content with that resolve.

For it requires Computers to find where 3N+3 starts having convergence points to higher numbers than 3. Say a convergence to 15 and internally spin around on 15 rather than drop down to 3. In the same manner, as 5N+5, that 3N+3 keeps getting higher convergence points, higher than 3, higher than 15, and then we have to ask where does 3N+3 diverge to infinity. Taking infinity to be 1*10^604 and its Algebraic Completeness at 1*10^1208.

Mind you, all of this newer research has no impact on AP's proof of Collatz of 2016, where it is stated: 3N+-1 simultaneously 3N+-3 over all Counting Numbers converges to 1 when landing on odd number and dividing by 2 when landing on even number. That is a solid proof and based on the fact that 4 Consecutive Evens has at least one number which is divisible by 8. This even number is forced to be selected of the 4 numbers and forces a convergence to 1 because all evens downslide by 8 times while all odds have only a 3 times uptick.

The original Collatz with its singular 3N+1 is indeterminate in proof, for there is never the possibility of going through all numbers from 1 to 10^604 or at most 1 to 10^1208. If a computer finds a 400 digit number that spins around internally and never falls to 1 convergence is a counterexample to the original Collatz. But I suspect no-one with computers has even dared plug in a 400 digit odd number for inspection. It maybe the case that such a single example takes 100 years to check.

AP, King of Science, especially Physics-Chemistry

On Friday, June 3, 2022 at 6:58:31 PM UTC-7, Archimedes Plutonium wrote:
>> The original Collatz with its singular 3N+1... But I suspect no-one with computers has even dared plug in a 400 digit odd number for inspection. It maybe the case that such a single example takes 100 years to check.

You have no idea what you are talking about Arky, as always.

Choosing a random 400 digit number and plugging that into 3N+1 seems to often take about 8000...10,000 iterations to reduce that to 1 and that takes a few seconds to complete on a very ordinary computer.

For example:
46185865952139850149301069254872966409486544263251646041017730214126145748130130~
45948812131675818835497319103249839546315055538248896666930423462001308917800907~
80240086110183936562298807624361106978536231034013043835184909292949243168188123~
57226067044231097057181993039947889132784321737415886855700057348948423672010111~
630337403509940157825782824761576575067117078772532881979607972155560392492300413

reduces to 1 under 3N+1 in 9050 iterations.

Let me try one more example before I leave it up to computers.

5N+5
________

481 goes to 2410, goes to 1205, goes to 6030, goes to 3015, goes to 15080, goes to 1885, goes to 9430, goes to 4715, goes to 23580, goes to 5895, goes to 29480, goes to 3685, goes to 18430, goes to 9215, goes to 46080, goes to 45, goes to 230, goes to 115, goes to 580, goes to 145, goes to 730, goes to 365, goes to 1830, goes to 915, goes to 4580, goes to 1145, goes to 5730, goes to 2865, goes to 14330, goes to 7165, goes to 35830, goes to 17915, goes to 89580, goes to 22395, goes to 111980, goes to 27995, goes to 139980, goes to 34995, goes to 174980, goes to 43745, goes to 218730, goes to 109365, goes to 546830, goes to 273415, goes to 1367080, goes to 170885, goes to 854430, goes to 427215, I conclude that this will go beyond 10^604 and not converge even at 10^1208. If true then this is the first 5N+5 to diverge to infinity.

This type of work is easily to make a arithmetic mistake, so I need a check on the above. If the above is correct, what the message is, is that there are some descents, but it keeps ratcheting upward higher than any descent.

AP

On Friday, June 3, 2022 at 10:20:39 PM UTC-5, Archimedes Plutonium wrote:
> Let me try one more example before I leave it up to computers.
>
> 5N+5
> ________
>
> 481 goes to 2410, goes to 1205, goes to 6030, goes to 3015, goes to 15080, goes to 1885, goes to 9430, goes to 4715, goes to 23580, goes to 5895, goes to 29480, goes to 3685, goes to 18430, goes to 9215, goes to 46080, goes to 45, goes to 230, goes to 115, goes to 580, goes to 145, goes to 730, goes to 365, goes to 1830, goes to 915, goes to 4580, goes to 1145, goes to 5730, goes to 2865, goes to 14330, goes to 7165, goes to 35830, goes to 17915, goes to 89580, goes to 22395, goes to 111980, goes to 27995, goes to 139980, goes to 34995, goes to 174980, goes to 43745, goes to 218730, goes to 109365, goes to 546830, goes to 273415, goes to 1367080, goes to 170885, goes to 854430, goes to 427215, I conclude that this will go beyond 10^604 and not converge even at 10^1208. If true then this is the first 5N+5 to diverge to infinity.
>
> This type of work is easily to make a arithmetic mistake, so I need a check on the above. If the above is correct, what the message is, is that there are some descents, but it keeps ratcheting upward higher than any descent..
>

Alright, I started the above at 481, and had no serious downslide until I came to 46080 and fell back all the way to 45. And 46080 comes from the sequence 45, 90, 180, 360, etc etc. Now the question hanging on my mind, will this example of starting at 481, encounter another member of the 45, 90, 180,..... sequence and thus converge to 45?

So here I am trying to fathom how a cycling through of 5N+5 encounters one member of the sequence 45, 90, etc etc and then later on, encounter another member of that same sequence and thus end up converging. So the cycling encountered 9215 forcing it to become 46080 which fell to 45, now will the newer numbers also encounter a member of 45, 90, 180, etc and why is that possible? Almost as if ordained or fated to encounter a member of the doubling sequence 45, 90, 180, etc, etc. So can we capture or catch why a number 481 in 5N+5 is going to land on the 45,90,180, etc sequence and why or why not it will repeat land on that sequence again in the future? Sort of like the anatomy of a doubling sequence compared to a 5N+5 generator.

AP

Still looking for a Collatz polynomial that is clearly divergent. So far tried only 5N+5, thinking it escapes more readily the 4 Consecutive Evens that proof the AP-Collatz. But maybe I be better served with 3N-1. I have hesitated on using 3N-1 for 5N+5 is more intuitive. But perhaps 3N-1 may give a faster solution of a divergent Collatz.

--- quoting from already published book of Collatz proved in 2016 ---
Using 3N+3 start with 41 goes to 126 goes to 63, goes to 192, goes to 96 to 48 to 24 to 12 to 6 to 3.

Let me try 3N-1 starting with 10 goes to 5 to 14, to 7 to 20 to 5, converging to 5.

Starting with 41 on 3N-1 goes to 122, to 61 to 182, to 91, goes to 272 to 136 to 68 to 34 to 17 to 50 to 25 to 74, to 37 to 110 to 55 to 164 to 82 to 41.

So yes, I found a hint of a counterexample of 1937 Collatz of his 3N+1 in that 3N-1 has a circular spinning around forever that proves the Collatz statement, hinting there is a counterexample/s.

--- end quoting ---

In the above, what I was implying is that I needed not a divergence of 3N+1 but simply a spinning around inside of 3N+1 to a convergency that was not 1. If 3N-1 can converge on something other than 1 but on 41, strongly suggests that 3N+1 can breakdown to a other than 1 convergency. So if 3N-1 can converge on 41, suggests that 3N+1 for some huge number encounters the same double sequence for a second time in which the convergency is not 1 but something different.

AP

Let me try one more for divergence, 3N-1 starting with 1001, goes to 3002, goes to 1501, goes to 4502, goes to 2251, goes to 6752, goes to 211, goes to 632, goes to 79, goes to 236, goes to 59, goes to 176, goes to 11, goes to 32, goes to 1.

Let me try 999, goes to 2996, goes to 749, goes to 2246, goes to 1123, goes to 3368, goes to 421, goes to 1262, goes to 631, goes to 1892, goes to 473, goes to 1418, goes to 709, goes to 2126, goes to 1063, goes to 3188, goes to 797, goes to 2390, goes to 1195, goes to 3584, goes to 7, goes to 20, goes to 5, goes to 14, goes to 7 converge to 7.

So there are two ways of proving Old Math Collatz is false. A spinning around inside of 3N+1 with a convergency that is not 1. And a divergence of 3N+1. But a divergence seems difficult in even 5N+5 polynomial, in 3N+3 polynomial and in 3N-1 polynomial. Not even trying 3N-3 polynomial.

AP