AP has a newer proof of Collatz with that of only 3N+1, the old original statement--added on new chapter to AP's proof of Collatz published book 2016. A new chapter of clarity for it has been 6 years since the proof of Collatz.

Alright, well, I discovered a Mathematical Induction proof of Collatz for the singular statement, the original Collatz of 1937. Of course the AP proof of a generalized Collatz using 3N+-1 simultaneously 3N+-3 is the stronger proof for it exposes the actual mechanism of why division by 2 rather than 3N+1 of all odd numbers converges to 1.

But on reflection this morning, I realized I have a mathematical induction proof over just the singular statement of 3N+1, all facilitated by the fact the true numbers of mathematics are Decimal Grid Numbers. This allows a mathematical induction proof of 10 Grid then 100 Grid and because computers have already shown 3N+1 is true out to 10^20, we have a reverse Mathematical Induction (perhaps a very first time application of Reverse Mathematical Induction where instead of proving for 1, assume true for K, prove true for K+1, the reverse is prove true for 10^1, prove true for 10^2 prove true for 10^3, assume true for K, 10^19, prove true for K+1 which is 10^20.

The reason I come upon this proof of Collatz original is that I was looking for a Practical Application. All true math has a practical application in the sciences.

The instant recognizable application of Collatz is in biology of nerves and communication to organs of the body. Or in the case of what I was thinking earlier back in 2016, the mitochondria and ribosomes of cells. So I made the mental picture as a Application of Collatz as a Odd number is a nerve impulse that goes to a mitochondria as Even number and releases energy or other substance and the amount released is how factorable it is by 2. So the nerve impulse or communication is the odd number multiplied by 3N add 1, while once there on a even number the mitochondria or ribosome release or produce energy or proteins. I see Collatz math as being electrical signals as odd numbers signalling a mitochondria to release packets of energy as even numbers factorable by 2 multiple times. So I see odd numbers as electricity magnetic monopoles telling a mitochondria to release energy in even numbers..

All True Math has applications to Physics, biology, chemistry.

This application helps me now in proving the original Collatz, 3N+1.

And I start the proof by recognizing what hindered me, or thwarted me in seeing that 3N+1 alone can prove Collatz, without the need to cover all 4 Consecutive Even numbers which one of them has a even factorable at least by 8..

What impeded my understanding that 3N+1 alone has a proof-- a proof by mathematical induction is this mix-up in thinking that there are 3 phenomenon involved-- the convergence to 1, the convergence to a number not 1 and internal spinning around, and then the divergence to infinity. That is what thwarted me in seeing that 3N+1 alone has a proof. For now I understand there are but two categories of outcome. Either 3N+1 all converge to 1, or, if there is a odd number that spins around inside is a divergence. You see, just this morning I realized divergence is the same as spinning around and not landing on 1. For example 5N+5 start with 3, goes to 20, goes to 5, means 5N+5 diverges. It does not go to infinity which is 10^604, but in Collatz, we need no polynomial go to infinity. In Collatz we define divergence as not ending up being 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, approved

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

5 goes to 16, goes to 1, approved

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, approved

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

That encompasses the entire 10 Grid.

Computers have already verified all the odd numbers in 100 Grid.

We assume true for K, and must prove true for K+1.

Since all the Decimal Grid Numbers are created out of Mathematical Induction we easily see it is true for 10^19 Grid and the computers have proven true for 10^20 allows us to say, that Collatz 3N+1 is true for the entirety of Decimal Grid Systems of Counting Numbers from 1 to 1*10^604 and the ability to go to 1*10^1208 of Algebraic Completeness.

Now I have not the time to list all the 50 odd numbers from 1 to 100 in the 100 Grid and show that 3N+1 converges to 1.

It is worth saying again. What hindered and delayed my seeing the proof of 3N+1 alone is all convergent upon 1, is my misunderstanding of what Divergence means in Collatz. It does not mean going to infinity without ever spinning around inside. No, divergence in Collatz means it never converges to 1. Another example 3N+3 starting with 3 goes to 12, goes to 3, hence diverges..

AP, King of Science, especially Physics-Chemistry

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..

AP has a newer proof of Collatz with that of only 3N+1, the old original statement--added on new chapter to AP's proof of Collatz published book 2016. A new chapter of clarity for it has been 6 years since the proof of Collatz.

AP is proving the old original Collatz of 1937, with no change in statement..

Alright, well, I discovered a Mathematical Induction proof of Collatz for the singular statement, the original Collatz of 1937. Of course the AP proof of a generalized Collatz using 3N+-1 simultaneously 3N+-3 is the stronger proof for it exposes the actual mechanism of why division by 2 rather than 3N+1 of all odd numbers converges to 1.

But on reflection this morning, I realized I have a mathematical induction proof over just the singular statement of 3N+1, all facilitated by the fact the true numbers of mathematics are Decimal Grid Numbers. This allows a mathematical induction proof of 10 Grid then 100 Grid and because computers have already shown 3N+1 is true out to 10^20, we have a reverse Mathematical Induction (perhaps a very first time application of Reverse Mathematical Induction where instead of proving for 1, assume true for K, prove true for K+1, the reverse is prove true for 10^1, prove true for 10^2 prove true for 10^3, assume true for K, 10^19, prove true for K+1 which is 10^20.

The reason I come upon this proof of Collatz original is that I was looking for a Practical Application. All true math has a practical application in the sciences.

The instant recognizable application of Collatz is in biology of nerves and communication to organs of the body. Or in the case of what I was thinking earlier back in 2016, the mitochondria and ribosomes of cells. So I made the mental picture as a Application of Collatz as a Odd number is a nerve impulse that goes to a mitochondria as Even number and releases energy or other substance and the amount released is how factorable it is by 2. So the nerve impulse or communication is the odd number multiplied by 3N add 1, while once there on a even number the mitochondria or ribosome release or produce energy or proteins. I see Collatz math as being electrical signals as odd numbers signalling a mitochondria to release packets of energy as even numbers factorable by 2 multiple times. So I see odd numbers as electricity magnetic monopoles telling a mitochondria to release energy in even numbers..

All True Math has applications to Physics, biology, chemistry.

This application helps me now in proving the original Collatz, 3N+1.

And I start the proof by recognizing what hindered me, or thwarted me in seeing that 3N+1 alone can prove Collatz, without the need to cover all 4 Consecutive Even numbers which one of them has a even factorable at least by 8..

What impeded my understanding that 3N+1 alone has a proof-- a proof by mathematical induction is this mix-up in thinking that there are 3 phenomenon involved-- the convergence to 1, the convergence to a number not 1 and internal spinning around, and then the divergence to infinity. That is what thwarted me in seeing that 3N+1 alone has a proof. For now I understand there are but two categories of outcome. Either 3N+1 all converge to 1, or, if there is a odd number that spins around inside is a divergence. You see, just this morning I realized divergence is the same as spinning around and not landing on 1. For example 5N+5 start with 3, goes to 20, goes to 5, means 5N+5 diverges. It does not go to infinity which is 10^604, but in Collatz, we need no polynomial go to infinity. In Collatz we define divergence as not ending up being 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, approved

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

5 goes to 16, goes to 1, approved

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, approved

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

That encompasses the entire 10 Grid.

Computers have already verified all the odd numbers in 100 Grid.

We assume true for K, and must prove true for K+1.

Since all the Decimal Grid Numbers are created out of Mathematical Induction we easily see it is true for 10^19 Grid and the computers have proven true for 10^20 allows us to say, that Collatz 3N+1 is true for the entirety of Decimal Grid Systems of Counting Numbers from 1 to 1*10^604 and the ability to go to 1*10^1208 of Algebraic Completeness.

Now I have not the time to list all the 50 odd numbers from 1 to 100 in the 100 Grid and show that 3N+1 converges to 1.

It is worth saying again. What hindered and delayed my seeing the proof of 3N+1 alone is all convergent upon 1, is my misunderstanding of what Divergence means in Collatz. It does not mean going to infinity without ever spinning around inside. No, divergence in Collatz means it never converges to 1. Another example 3N+3 starting with 3 goes to 12, goes to 3, hence diverges..

AP, King of Science, especially Physics-Chemistry

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

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

🕷 of Math and 🕸️ of Physics Archimedes "Court Jester of Math"
Plutonium <plutonium.archimedes@gmail.com> fails at math and science:
> In hindsight, now that I have proven Collatz,

But you haven't proven the Collatz Conjecture. You proved something
else, the Arky Conjecture, where the next term for odd n is
{3n-3,3n-1,3n+1,3n+3}, whichever is divisible by 8, while the real
Collatz Conjecture only uses 3n+1 as the next term.

On Sat Jun 4 14:56:52 2022 Archimedes Plutonium wrote:
> 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

*
What about numbers greater than 10^604 ?

earle
*

On Saturday, June 4, 2022 at 5:52:52 PM UTC-5, Earle Jones wrote:
> On Sat Jun 4 14:56:52 2022 Archimedes Plutonium wrote:
> > 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
> *
> What about numbers greater than 10^604 ?
>
> earle
> *

On Saturday, June 4, 2022 at 5:52:52 PM UTC-5, Earle Jones wrote:
> On Sat Jun 4 14:56:52 2022 Archimedes Plutonium wrote:
> > 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
> *
> What about numbers greater than 10^604 ?
>
> earle
> *

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

On Sat Jun 4 18:48:53 2022 Michael Moroney wrote:
> ?? of Math and ??? of Physics Archimedes "Court Jester of Math"
> Plutonium <plutonium.archimedes@gmail.com> fails at math and science:
> > In hindsight, now that I have proven Collatz,
>
> But you haven't proven the Collatz Conjecture. You proved something
> else, the Arky Conjecture, where the next term for odd n is
> {3n-3,3n-1,3n+1,3n+3}, whichever is divisible by 8, while the real
> Collatz Conjecture only uses 3n+1 as the next term.

*
Nice try, Michael!

I dont think that AP will understand your comments.

earle
*

On 6/5/2022 12:57 AM, Earle Jones wrote:
> On Sat Jun 4 18:48:53 2022 Michael Moroney wrote:
>> ?? of Math and ??? of Physics Archimedes "Court Jester of Math"
>> Plutonium <plutonium.archimedes@gmail.com> fails at math and science:
>>> In hindsight, now that I have proven Collatz,
>>
>> But you haven't proven the Collatz Conjecture. You proved something
>> else, the Arky Conjecture, where the next term for odd n is
>> {3n-3,3n-1,3n+1,3n+3}, whichever is divisible by 8, while the real
>> Collatz Conjecture only uses 3n+1 as the next term.
>
> *
> Nice try, Michael!
>
> I dont think that AP will understand your comments.
>
Given that he decided to spam your response, it's quite obvious he
didn't understand them. Not that I expected anything different.

On Sat Jun 4 15:15:27 2022 Archimedes Plutonium wrote:
> 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

*
Actually, what it really means that you, with your lack of understanding of mathematics, tries to grab some phyiscs to bail you out. The Collatz conjecture is a mathematical proposition. It has nothing to do with science, except in your distorted imagination.

earle
*

On 6/5/2022 3:55 PM, Michael Moroney wrote:
> On 6/5/2022 12:57 AM, Earle Jones wrote:
>> On Sat Jun 4 18:48:53 2022 Michael Moroney  wrote:
>>> ?? of Math and ??? of Physics Archimedes "Court Jester of Math"
>>> Plutonium <plutonium.archimedes@gmail.com> fails at math and science:
>>>> In hindsight, now that I have proven Collatz,
>>>
>>> But you haven't proven the Collatz Conjecture.  You proved something
>>> else, the Arky Conjecture, where the next term for odd n is
>>> {3n-3,3n-1,3n+1,3n+3}, whichever is divisible by 8, while the real
>>> Collatz Conjecture only uses 3n+1 as the next term.
>>
>> *
>> Nice try, Michael!
>>
>> I dont think that AP will understand your comments.
>>
> Given that he decided to spam your response, it's quite obvious he
> didn't understand them. Not that I expected anything different.

And now he started to spam THIS post.

AP is so illogical...

On Sat Jun 4 17:52:12 2022 Archimedes Plutonium wrote:
> On Saturday, June 4, 2022 at 5:52:52 PM UTC-5, Earle Jones wrote:
> > On Sat Jun 4 14:56:52 2022 Archimedes Plutonium wrote:
> > > 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
> > *
> > What about numbers greater than 10^604 ?
> >
> > earle
> > *
>
> 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: Thank you for your response. It appears that your "proof" of the Collatz conjecture is limited to numbers up to 10^604 or perhaps 10^1208. The Collatz conjecture states that ALL numbers will eventually end up in the simple loop.

earle
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