ON THE DECIDABILITY OF THE COLLATZ CONJECTURE: A
SORTING ALGORITHM FOR (2,4) MOD 6. THE COLLATZ
CONJECTURE SOLUTION
ANDREW S. BOUNDS
Part 1. INTRODUCTION
Part 2. DOMAINS
The Domain of the seed values, x, are the integers, and the range are the natural
numbers.
Given: 3n + 1 = y1 where: if n = 2×1, y
2 , if n = 2×2 + 1, then 3n+1
2 = y2
This domain definition yields three trivial loops, two loops at x1 = 0, and another
at x1 = 1 for y1 = 2x as it then feeds into the odd trivial loop of x2 = 0, which
gives x1 = 2 is equivalent to x2 = 0. It also yields a single domain for all x.
Every power of two (2a) will collapse to the trivial odd loop if we divide by 2,
so we need to ensure that our algorithm can cycle through all possible evens that
can have powers of two, divide by two if it is a power of two, and kick out of our
”slot machine” if it is not.
We need to account for every even and odd input, and every possible output.
Our function cannot have solutions that repeat, but the more values it covers, the
more likely we are to find a power of two.
Ideally, we would also be able to extend the approach to the domain of negative
integers. If there is a linear solution, or set of solutions, then we should be able
to extend those to the negative integer domain while maintaining the same range,
just negative.
Part 3. COPRIME/MOD 6/BASE 6 NOTATION
With the realization that only powers of two will collapse to the trivial loop of
x=0 for all n = 2x and is the only point in which they crossover, we can use the
expanded equation of the case where n = 2x + 1:
(1) 3(2x + 1) + 1 = 6x + 4
Received by the editors 2024.
2020 Mathematics Subject Classification. Number Theory.
©XXXX American Mathematical Society
1
2 HAILSTONE PROBLEM
0 mod 6 1 mod 6 2 mod 6 3 mod 6 4 mod 6 5 mod 6
2(3x) 2(3x+1) 2(3x+2)
6x+0 6x+1 6x+2 6x+3 6x+4 6x+5
3(2x) 3(2x)+1 3(2x)+2 3(2x+1) 3(2x+1)+1 3(2x+1)+2
Table 1. Positive coprime expansion
x 0 1 2 3 4 5 6 7
3(2x+1)+1
21 ) mod 6 2 5 2 5 2 5 2 5
3(2x+1)+1
22 mod 6 1 4 1 4
Table 2. Oscillating values
to begin understanding the possible combinations. This would lead to the use of
a 6-coprime expansion, also a C mod 6 notation. This also leads to the expansion
of all 6-coprimes listed in the diagram (Table 1). This coprime expansion will also
be the basis for expanding to the negative integers. As this expansion is inclusive
of all the coprime expansions of 2 and 3, all possibilities for the Collatz expansion
are represented.
It is a nominally trivial notion that this can also be expanded to work with any
linear expressions with odd coefficients and constants less than the coefficient simply
by doubling the lead coefficient to determine the different factors when dividing by

2. Example: Ax + c where A is an odd integer and c is a number where 0 ≤ c < A
   would give a working value of (2A)x + (2c) for all divisions of 2 by also including
   the odd expressions of x.
   Part 4. Coprime Expansion
   As illustrated in the above diagram (Table 1), the odd and even coprime multiples
   of 3 and their related doublets as even numbers are completely mapped. This
   represents every possible permutation of co-prime linear equations for 3’s. This
   also gives a structure for separating out that which can be a power of two and that
   which cannot. Although this does not limit every possibility, it significantly limits
   the number of possibilities that can be an output. This also has an impact on the
   sequencing of every possibility when dividing, by leaving off the multiples of 3 and
   their doubles.
   With the above table (Table 1), it is trivial to see that the odd and even outputs
   of the algorithm using 3n + 1 with n = 2x + 1 or n = 2x have distinct values. What
   is not so trivial to see is that the function 3n + 2 is the other half of the solutions.
   The coprime or modular form also allows us to see a recurring pattern in the
   seed values for x. As seen in Table 2.
   This also gives a maximum slope of 3(2x + 1) + 2 or 6x + 5 for any iteration
   path that would climb toward infinity. This would also require that every seed
   value produce an odd value for the seed value x and there is only one division by
   2 every iteration. The distribution of evens and odds in the seed value x supports
   all previous work that took the probability as abstract. This applies to all previous
   AMS ARTICLE TEMPLATE 3
   2 coprime 2(3x) 2(3x-1) 2(3x-2)
   6 coprime 6x-0 6x-1 6x-2 6x-3 6x-4 6x-5
   3 coprime 3(2x) 3(2x)-1 3(2x)-2 3(2x-1) 3(2x-1)-1 3(2x-1)-2
   Table 3. The proper negation of the Collatz function
   work on 3n + 1 and 3n + 2 on the Collatz Conjecture as these are the only outputs,
   and therefore secondary seed values, for the Collatz Conjecture.
   Part 5. Negative Integer Domain
   If we change our domain, we need to continue the negation across all steps in
   the expression, so all terms are negated in turn after x:
   (2) 2x + 1 = 2x − 1 = n, 3n + 1 = 3n − 1 = 3(2x − 1) − 1 = 6x − 4
   This gives us an expansion of the expression similar to the positive integers:
   This form also allows us to see a particular pattern (Table 3).
   This allows us to see the direct expansion to negative integers and that every
   iteration following the form outlined above will give the same result for the domain
   of negative integers. All of the above shows that the previous findings of other
   researchers and in this paper apply to the negative integers when using this form.
   Part 6. Path Between Iterations
   (3) 3(2x + 1) + 1
   2p where p = 2a and also 3(2x + 1) + 1
   2q where q = 2a + 1
   which yields:
   (4) 3(2x + 1) + 1
   2p = 3(2x + 1) + 1
   2q as the only path between iterations.
   Part 7. Conclusion
   With the above information, we can determine, for any given x:
   (1) The only outputs possible from any given input are divisions by 2a and the
   parity of a tells you if it is 6x + 1 or 6x + 5 and anything else results in
   another division by 2.
   (2) The only way for the process not to stop and reset is to find a power of 2.
   (3) The even numbers corresponding to 6x + 2 and 6x + 4 can be sorted by this
   process into 3 distinct outputs, the odd roots, q = 2a + 1, not a power of
   two correspond to 6x+5, and the even roots, p = 2a, which are not a power
   of 2 correspond to 6x + 1. The powers of two would be the third output.
   The x = 0 being the only crossover point with an odd and an even power
   of two. All other values are not an output of the described algorithm.
   4 HAILSTONE PROBLEM
   (4) If we change the domain of x to negative integers, then all terms and cor-
   responding operations must be properly negated to show the same results
   for that domain, but the algorithm can be completely mapped to the set of
   negative integers.
   (5) The only path that produces any loop is the one that is x = 0, which yields
   an even and odd parity in the same co-prime set (2 and 4).
   With the above points, we can determine that the 6-coprime expansion encompasses
   all the behaviors and examples of the algorithm output possibilities. The expansion
   also shows differentiation by the parity of the input value of x and by the number
   of divisions by 2, or 2a. The parity of a is also determinant to which output is
   achieved, pairing even values of a with even values of x. The only other outputs
   available are the divisions by 2, which yield the trivial loop when there are two
   multiples of 2 in the same set of 6.
   Abstract. ABSTRACT: In this it is shown that there is an algorithm for
   sorting (2,4) mod 6 (a.k.a. Syracuse problem, Collatz conjecture, Hailstone
   problem, etc.) is true through the use of 6 coprime notation and seeing the
   problem as an algorithm for sorting roots of even numbers to find powers of
   two from integers-reduced-to-odd inputs, a robust algorithm in fact. Previous
   work focused on the 3n+1 structure, but this approach shows that it only
   encompassed half of the solution. 3n+2 being the other half.

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I am so happy you stayed interested
And of course, apology accepted
And my equal reply
In full, in kind
Now let's see if I can't make good the investment

I'm going to give this a helluva try
cause if I'm even mostly right
There's lots to do
more than me and you
and I'm a fucking Jedi Knight!

The gap after the song was indeed a dramatic pause
I wanted to be clear, plus it's some of the best advice I ever got
When he was singin'
Keep their heads ringin'
And made a phone call, like it or not

Now, A phone call to you, is a party line to me
They wouldn't have believed, and they needed to see
A tail wag to you
Hi, I'm Andrew
and it doesn't get any more real than me

And now, Experiment successful! let's say twice
the third time was to show you what it was like
Mrs. look-you-in-the-eyes-when-I-do-it
He thought He could do it
Was the fourth time because of the fight?

And you already know I am not a witch
You just can't stop scratching that itch
Stop being so bold
You know you've been told
Now watch those stumbles Mitch

And I'm not trying to take down the man
It's more like Denny's, a real Country Slam
and Brother, t'was more than little chuckle
When I got to call ya all Hat and no Buckle
It really was good to finally see your fam

So many out there that I could mention
The little Tumblers attention
And a blush
for my Comic Crush
And the White Queen up in her mansion

This isn't a joke, it's gravely important
I love to laugh, but y'alls perspective is distorted
If this is real
do you really feel
That your satisfaction is to what it should be devoted?

Please.

For the dude whose head looks like a cue
And Dave the 8-ball that was with you
'"Hi guys", you heard that too right?'
Yeah, you heard right that night
And, you doubting bitches, I did too

Wait, Hold up there buddy, It smells a little Musky
Now I'm pretty sure that bet was for 2 big hunks of money
dunno if it was a Mil,
or was it Bill?
But that Saturday Night Live shit was funny

'And to take out the cast'
they say the best for last 😉

Moses Supposes His Toeses Are Roses Song

'bows'
but always, always, always, make them laugh…

oh yeah,  I had to look back but it made me smile
You guys were the reason that she had to lie
So, thanks for the shower, Big Willie Style…

Let's talk about Truth, it's Manner and Kind
I am sure there are ways you can find
to speak the unspoken, from mind to mind

Every secret is a Promise
Not every promise is a secret.
If you learn nothing else,
Learn how to keep it.

Now, get that out of your mind.
It's Water you fool.
The wedge that cracks
The hammer that grinds

Ha! It's the foundation of business
That is an agreement
Two Hands that are clasped
That's Trust, a firm contract.

It's Marriage my friend,
We must disagree.
That's me and my love,
Two as one, that's a "We"!

Nay, it's Parenting you say,
To lift them up and raise them higher, 
Not hold them down with selfish desires.

It is Philosophy, the cornerstone of the Greeks.
It's Therefore, logic leads to knowledge
And Because it is shared, strong are the meek.

Yes, it's the paradox of All, no mind can ever be alone.
If there is One, there must be Many.
And without the Many, there can be no One.

Defense of Gwyneth Paltrow’s Goop offers case study on how to sell snake oil – Ars Technica

Killer robots are a mainstay of science fiction. But unlike teleportation and flying cars, they are something that we are likely to see within our lifetime. The only thing that’s stopping countries like the USA, South Korea, the UK, or France from deploying autonomous killing machine in the very near term is that they’re likely…

via Taking Killer Robots Seriously — Hackaday

Another great post, thanks again Dr. Jen

As an account based (meaning it is true) on real-life events. I only will change the names, or omit them, in order to just tell the story, not disparage anyone in particular. It needs to be told, if only to have my version out there, as no one has ever asked me for mine. A lot of assumptions have been made, social, legal, and otherwise. This is the only way to set things straight.

I raised a child. Not one I wanted. That may be cold to say, but it is true. You see, I only wanted a child with the love of my life, if at all. I never dreamed or imagined just having a child. Not to boost my ego, not to feel like a king, and not to make someone in my own image. And most importantly, not to have a child with someone I didn’t love. There’s a story about how this child that wasn’t mine became the child I was forced to raise, but that is another post, this is the story of how his mother and I came to meet, and how the engine got started on the railroad that stole years from me, and still continues to this day.

I used to visit a lot of people, for a variety of reasons, in my hometown. I once stopped by a friend’s house (shared rental, a couple of single moms lived there) and took a young girl for a ride n the back of my motorcycle. I would later regret being nice to that girl, but I am getting ahead of myself. You see, later, because of this encounter, we would be on a “friendly” level of familiarity, to the point of another friend of mine and I would utilize her and her friends as a “messaging service” when they were just standing on the street downtown. Realize, she was about  15. I truly thought I was doing a kind thing. I took a young girl on her first motorcycle ride, and, before texting, pagers, and much else, gave a small purpose that would keep some young girls in a small town from digressing into too much trouble. If there’s anyone who doubts those motives, I have learned that there’s nothing I can do to change that mindset. There are people who believe that all interactions between girls and boys are sexual in nature, and that will never change, at least not with me arguing. As a regular thing, we would leave messages for one another with those girls. He would tell them something for me, and when I was driving through town, I would stop, and say hi, and ask if there were any messages. My friend and I were 18 or 19. after a couple months of this, we somewhat stopped, as the usefulness of it ran out, and our lives kept us out of the downtown area more. Until late November-early December, when I did something nice that would become something of a regret.

I was driving through the small town I lived in, it was a one-street town and has since become a two-street town, when I saw one of the young girls walking on the sidewalk. It was about 9 or 10 pm, dark, and frosty out. Not a lot of snow in those parts, but still chilly temperatures. She was wearing only stretch pants, what looked like a thin hoodie and thin jacket, and was walking alone. Not knowing where she lived, and being the kind person I like to believe myself to be, I pulled over and asked if she needed a ride home, instead of walking home in the cold. Now, I had no intentions of anything else. To be bluntly honest, let’s just say she was, and still is, far from what I consider an attractive woman to be. I had been spending time with other people I was attracted to, but not been with. The reasons for that is another post, and story, entirely, one I may never tell in its entirety. When this young girl, we were both young, but I was definitely the elder, told me she didn’t live far. I offered to drive her around a little bit so she could get warm before going home, as she was shivering noticeably. I drove approximately 2-3 miles. A nice little loop just barely out of town. SHe and I talked while we drove, not about anything in particular, She even asked me if I wanted anything for doing this favor. I replied, with a laugh, “You mean something like, ‘this car runs on ass, not gas, so put out or get out’?”. I laughed a bit more before answering, “No, I’m not like that, you don’t owe me anything for this”. After a couple more miles, the entire loop was approximately 7 miles round trip, about 10 minutes, enough to get someone warm with the heater on high, she grabbed my hand, pulled it into her crotch while saying, “I have a hole here…” and pulled my hand into what was indeed a hole in her pants. Let it be said that at this time I was involved with Ecstasy and Meth. And a LOT of weed. I broke what had been a long battle of willpower. I never have been much of a promiscuous man. I gave in. Whatever alignment of the stars that happened, I attribute it to the fact that I was mostly thinking of someone else during this ride, someone I was VERY attracted to, and I pulled over. Now mind you, to this day, it remains the only time I have ever had sex without taking off a single article of clothing. I never finished. I lost my erection quite quickly, made some excuse to the effect of, “That was the quickest ever…” and redid my pants, found myself filling with regret, and got back to driving the girl home. It was all very short-lived, and I hoped I would be able to forget all about it, which I did, until some years later when I was served with a paternity suit from the state. I was the last person named on it that hadn’t been tested. Not even a one-night stand. I found out many years later that pre-ejaculate has only 0.4% chance of impregnation, and that is only if there was a previous ejaculation within 72 hours. And that is a sliding scale, as sperm do not die off all at once. I can’t provide any proof it had been much longer for me, but it was more like 6 weeks. In case the divergent “proof” isn’t clear right now, I eventually had 2 very strong kinds of scientific evidence, one was a DNA paternity test, the other was clinical evidence of impossible impregnation. Only one of those pieces of evidence was dependent on human error, and that was the DNA test. People don’t determine the longevity of sperm once they’ve left the testes, and without viable sperm, there can be no fertilization. That left only one possibility, one I didn’t realize until much later, because I had indeed, forgotten all about the incident, until it came back to rear its ugly head. By the time I remembered the details (I was sober & clean quite shortly after the incident, for other reasons) I had already been given a summary judgement, and had a parenting plan on file with the courts. And a young boy calling me dad. A child I never wanted, but had in front of me. So I did the best I could. I do have an apology to that young boy, I was never completely in that parenting 100%. I hope I didn’t do him a disservice. I did the best I was able to give him a “normal” parent, and be a good dad. But, I always looked to that light at the end of the tunnel that was his 18th birthday. For better or worse, good or indifferent, that is true. I am a dad, and, I am not. I have no children, but I raised one. I  want children with the love of my life, I hope I find her before too long. and I am scarred.

To the wonderful, honorable, and intelligent young man that came out of that mess, I am sorry I couldn’t be more of what you wanted. I hope I was what you needed. I am proud of who you are. There may never be a storybook ending, but you have the integrity to live a good life, and it is yours to choose. Work hard, don’t give up. I tried to teach you that first off, when you finally chopped that single round of pine. You are a good man. Live that truth.

I’m sorry I couldn’t be more for you, but I must find my happiness, and live my life. I hope you understand. When it is all said and done, for a little while in this big world, I was a dad, and you were a wonderful son.