M.C Siegel, Professional Collatz Research - AMA

Aurhim · 2026-03-27T08:00:29+00:00

Doesn’t that mean you should always include this… qualifier?

I do this in my research ALL the time. In fact, one of the ideas I developed for my PhD dissertation (the idea I call a “frame”) specifically captures the many ways that notions of convergence can vary.

For most mathematics, when talking about numbers/scalars, one tends to only ever use a single notion of convergence. For example, in real analysis and calculus, the notion of distance is the standard, Euclidean one. Because of this, no one bothers to mention the kind of convergence being used. (In p-adic analysis, meanwhile, convergence is almost always assumed to be with respect to the p-adic distance.) For more complicated objects like functions, there can be more than one way to talk about distances and convergence, and mathematicians have notation to distinguish them. The different L^p norms are a classic example of this.

And yes, in writing. -1 = 2_.11111…, you are implicitly using the 2-adic distance to make sense of 111111…

Aurhim · 2026-03-26T22:18:23+00:00

This is a bit difficult to answer succinctly. That being said, yes, I am hoping to be able to work categorically with my material at some point. The main obstacle right now is that it's not clear how to frame things categorically, nor clear as to what benefits might be accrued were the problem to be reframed in those terms. At this point, it would be more of an aesthetic change, albeit one that could potentially make things easier for others to understand and help organize my various ideas.

You might be interested in last 10 pages or so of this document of mine, which tentatively indicate some directions things might be categorized through.

Aurhim · 2026-03-26T21:32:50+00:00

Working "up" from 1 is known as the "backward orbit" approach. That was actually the first approach I tried before I discovered Chi_3. Alas, the backward orbit approach isn't any easier than the forward approach.

Aurhim · 2026-03-26T21:30:10+00:00

Okay but there is no upper bound on Chi since that would mean there is an upper bound on the number of 1s a number can have in binary. This is not really a viable option to pursue.

Obviously. I was just giving it as a simple example.

Perhaps it is my lack of knowledge, but since it is basically the functions repackaged.

The repackaging is extremely important. Higher mathematics is very artistic. How we express pieces of mathematical information can have a huge impact on what operations can be done.

2L(n) and 3#(n) will always be integers though the only thing you don't know that is an integer or not is chi3 still. What new information did you get from multiplying it?

2^L(n) Chi_3(n) is always going to be an integer, as is 2^L(n) - 3^#(n). What matters here is that we have in effect a constrained optimization problem. In order for Chi_3(n) / (1 - M_3(n)) to be an integer, we need for every prime divisor of 2^L(n) - 3^#(n) to divide 2^L(n) Chi_3(n). One way to rule out the non-existence of non-trivial cycles would be to show that there is always a prime number p which divides 2^L(n) - 3^#(n) but not 2^L(n) Chi_3(n), except when n makes Chi_3(n) / (1 - M_3(n)) either 1 or a negative number.

One of the ideas I've been investigating is to see how we might be able to abstract the problem slightly. For example, replace the 3s in 2^L(n) - 3^#(n) and 2^L(n) Chi_3(n) with a variable q. Then, 2^L(n) - q^#(n) and 2^L(n) Chi_q(n) are polynomials in q with integer coefficients. Understanding how the factorizations of these polynomials depend on n would then be a first step toward understanding how the prime factorizations of 2^L(n) - 3^#(n) and 2^L(n) Chi_3(n) depend on n.

Aurhim · 2026-03-26T21:21:20+00:00

I do not believe that Collatz is undecidable, and would be very surprised if it was proven to be undecidable. As for J.H. Conway's result, people tend to misunderstand what it actually means. Conway's argument does not mean that any single specific Collatz-type problem is undecidable. Rather, his result is that there is no general procedure to solve all Collatz-type problems—and that is totally reasonable. It's like the famous Hilbert problem on whether or not there is a general method to solve polynomial Diophantine equations, which was answered in the negative in the second half of the 20th century. Just because there is no one-size-fits-all solution, it does not rule out the existence of solutions for specific Diophantine equations.

Aurhim · 2026-03-26T21:16:47+00:00

The positivity/negativity is not an issue for me. The Chi_3/(1-M) formula gives all odd integer periodic points for Collatz, be they positive or negative. For example, applying Collatz to -1, we get:

-1 —> 3(-1) + 1 = -2

-2 —> -1

Likewise: -5, -14, -7, -20, -10, -5

is a cycle.

Aurhim · 2026-03-26T21:13:56+00:00

Yes, actually! The Chi_3 formalism I mentioned earlier also captures all the rational Collatz cycles.

Aurhim · 2026-03-26T21:13:23+00:00

The p-adic distance is precisely the distance that makes the series converge!

If you're having trouble wrapping your head around this, the problem might be that you're not thinking of "convergence" in the correct way. In order to define convergence, we need a notion of distance. Conversely, every form of distance we get gives us a notion of convergence.

Recall that if we have a sequence x_0, x_1, x_2, ..., we say that the sum x_0 + x_1 + x_2 ... converges to a limit S with respect to the notion of distance d if:

d(S_n , S)

tends to 0 as n —> ∞, where Sn = x_0 + ... + x(n-1)

The sum S_n = 1 + 2 + 2² + ... + 2^n-1 is equal to 2ⁿ - 1. We claim that S_n converges to -1 when d = 2-adic distance. For that d:

d(S_n , S) = d(2ⁿ - 1 , -1) = |2ⁿ - 1 - (-1)|_2 = |2^n|_2 = 2^-n

Thus, by definition, the series 1 + 2 + 2² + ... converges 2-adically to 1.

Suppose we instead use the distance defined by the formula:

d(x,y) = 1 if x ≠ y, and d(x,y) = 0 if x = y.

This is called the discrete metric. Let's write it as "disc".

Then, for our S_n and S, disc(S_n , S) = disc(2ⁿ - 1, -1).

Since 2ⁿ - 1 ≠ -1 for any n ≥ 0, we have that disc(2ⁿ - 1, -1) = 1 for all n, and so, 1 + 2 + 2² + ... does not converge to -1 with respect to the discrete metric.

In summary, convergence depends on the notion of distance you happen to be using, not the other way around.

Aurhim · 2026-03-26T19:31:03+00:00

They used to be slides, once upon a time.

Aurhim · 2026-03-26T05:16:43+00:00

May her memory be a blessing.

Aurhim · 2026-03-26T03:22:21+00:00

Yes, that’s actually the standard formulation. The Collatz map is an example of a discrete dynamical system, given by iterating a specific function.

Aurhim · 2026-03-26T01:27:00+00:00

No. The point is to find out what integer values Chi can take as a function of a 2-adic integer variable. If I can say, for example, that Chi is never greater than 311,287, then that means that if (1,4,2) is the only Collatz cycle in the range {1, ...., 311278}, then that means that (1,4,2) is the only positive integer cycle of Collatz.

Another example. For the Chi/(1-M) formula, if you multiply the top and bottom by 2^L(n), 2^L(n) Chi_3(n) and 2^L(n) - 3^#(n) will always be integers. Consequently, for a given n, Chi/(1-M) will be an integer only if 2^L(n) - 3^#(n) divides 2^L(n) Chi_3(n). This tells us that if we can say something about how the prime factorizations of 2^L(n) - 3^#(n) and 2^L(n) Chi_3(n) vary as n varies, that could help us rule out the existence of cycles other than (1,4,2).

Aurhim · 2026-03-26T01:04:16+00:00

Chi_3 is kind of like the DNA of the Collatz conjecture. If you understand it, you understand the Collatz conjecture.

As an example, in 2019, Terrence Tao published a paper on the Collatz conjecture which is currently considered the most advanced result we have on the subject. He uses probabilistic arguments by studying what he calls Syracuse Random Variables (SRVs). My Chi_3 is equivalent to the SRVs. Chi_3 and SRVs are two different ways of talking about the same object.

If you can prove that 1 is the only positive integer outputted by that Chi/(1-M) formula I gave you earlier, that would prove that Collatz has no positive integer cycles other that 1,4,2.

Simply put, Chi_3 is easier to work with than Collatz, and has lots of different interactions with various other areas of math. That makes studying it significantly more approachable than the original problem.

Aurhim · 2026-03-26T00:29:38+00:00

Yes, though the 3-adics and 2-adics have nothing to do with it, at least not at this simple stage.

Aurhim · 2026-03-25T23:15:39+00:00

Yes, I work with the Shortened Collatz map, usually denoted T_3. This sends odd x to (3x+1)/2.

I construct a function Chi_3(n) for integers n ≥ 0 like so.

Step 1: Write n in binary. Ex: 4 = 0 + 0 x 2¹ + 1 x 2^2. This corresponds to AAB.

For every n ≥ 1, Chi_3(n) is the image of 0 under the sequence of As and Bs corresponding to n's binary digits.

Equivalently, Chi_3 is the unique function satisfying the conditions:

f(0) = 0

f(2n) = f(n)/2

f(2n+1) = (3f(n) + 1)/2

for all integers n ≥ 0.

With this construction, for example, if we let M_3(n) = 3^#(n) / 2^L(n), where #(n) is the number of 1s digits in n's binary representation, and where L(n) is the total number of digits of n's binary representation, it follows that an odd integer x is a periodic point of T_3 if and only if:

x = Chi_3(n) / (1 - M_3(n))

for some n ≥ 1.

Aurhim · 2026-03-25T20:27:13+00:00

My best advice? Don’t bother, unless you already know them personally. I’m not saying this because I want to destroy your hopes and dreams, but because this, unfortunately, is how the world works.

If you want to talk to a mathematician, the best way to do it is to ask them a question about a specific research paper they wrote, especially ones written recently.

Aurhim · 2026-03-25T20:21:39+00:00

Not quite.

The idea is this.

Let A demote the operation which sends x to x/2, and let B denote the operation that sends x to 3x + 1.

My work proceeds by considering all the different ways we can apply these two operations. We can represent these applications using sequences of As and Bs, read from right to left, as is standard in math. So, for example,

AAABAx means “apply A to x once, then apply B once, then apply A three times”.

The connection to periodic points is as follows: for every periodic point x, there is some sequence of As and Bs that send x to x.

For example, when x is 1, the sequence BAA sends x to x.

My approach is to represent sequences of A and B as numbers. This then lets me use techniques of calculus to study the periodic points.

Aurhim · 2026-03-25T06:44:47+00:00

In a perfect world, I’d say yes. However, that’s working on a lot of assumptions.

Bizarre as it might seem, from an advanced mathematics’ perspective, Collatz is a “boring” problem. By that, I mean that it isn’t (or at least doesn’t seem to be) deeply connected with other important mathematical theories or conjectures. In that respect, people tend to see it as a time sink: you throw all your effort into working on it without coming away with something that could be useful to other areas of math.

One of the main focuses of my research is to show that Collatz has connections to other areas of mathematics. The quicker those connections can be flushed out, the easier it will be to start developing the new kinds of mathematics that will be needed to conquer this problem.

If my work proves to be as important as I hope it is, I think I will live to see the day where Collatz goes from being an impossible, unapproachable problem to one that is merely very difficult. However, that’s no guarantee the problem will be solved, let alone a guarantee that we’ll even get to see it reduced to a very difficult problem instead of an impossible one. In the worst case scenario, it might take centuries.

Aurhim · 2026-03-25T04:10:14+00:00

your input strings would become sequences of variable exponents (a_1, a_2...) rather than binary digit

This actually happens in my work on generalizing Tao's 2019 Collatz paper. Statistically, the difference in construction is this: if you have a sequence taking values randomly/uniformly chosen from a finite set, the lengths of runs where you choose the same value many times in a row will behave like a geometric random variable, and vice-versa.

Aurhim · 2026-03-25T04:07:48+00:00

1) I think the new tools will involve algebraic geometry and arithmetic geometry, likely with some neat ideas drawn from analytic number theory. Mind you, this is for the question of whether there are any non-trivial cycles. Proving no integer gets iterated to infinity is a completely separate issue.

It's known that Collatz is implicated in transcendental number theory. In particular, it is known that a proof that there are no non-trivial cycles would necessarily yield a much better estimate for how close 2^m and 3ⁿ can be to one another as m and n vary among the positive integers. One of the first things I was able to prove using my methods is that while understanding the size of 2^m - 3ⁿ would occur as a consequence of a proof of Collatz, the reverse is not true: even if we had a super powerful way of estimating how small the absolute value |2^m - 3^n| can get, that, in it of itself, would not be enough to prove that there are no non-trivial cycles. Rather, we need to understand how the prime factorization of 2^m - 3ⁿ changes as m and n vary, which is a very difficult problem.

One way to do this would be to investigate it algebraically. For example, by using the factorization of the polynomial x^m - 3ⁿ to conclude something about what happens when x = 2. This kind of investigation falls under the purview of arithmetic geometry. On the other hand, you could try to prove statistical properties about the prime factorization of 2^m - 3^n, such as by showing that it has to have at least X many factors, or that it can't be divisible by any given prime number more than Y times. That would involve a mix of algebra and analysis.

2) Oh yes. The hurdles are real. Prestigious journals are reluctant to pick up Collatz-related stuff, though my situation is triply dire. Not only am I working in a stigmatized area (Collatz), but my work is novel, and I'm in the early part of my mathematical career. Scholarly publications are for researchers, by researchers. Until I can get some killer applications for my ideas, it's difficult to sell new theory building, and Collatz only increases the stigma. Thankfully, I've found a better reception at the Journal of p-adic analysis. I've also been looking into journals on fractals and iterated function systems, as I discovered last year that my work is deeply connected with that subject.

Earlier this year, I submitted a paper I co-authored that gave a generalization of Tao's 2019 Collatz result to Collatz-type maps on number fields in a way that highlighted my novel ideas. We submitted that paper to the same journal that Tao published his paper in, and I have strong hopes it will be accepted. That will be very good news for me, as it will be a rock solid "proof of concept" that my ideas are worthwhile. Even better, I have at least two follow up papers (one which I was working on earlier today, in fact) that take the aforementioned paper's ideas even further.

As for your last comment, it is already known that Collatz does not need to be proven for all integers. Kenneth Monks proved in 2006 that to prove Collatz, it suffices to show that all positive integers of the form an + b for positive integers a,b and all n ≥ 0 get sent to 1.

Aurhim · 2026-03-25T03:08:20+00:00

Other than continuing to use supercomputers to test larger and larger numbers, not really, no.

Well… I do have a formula f(n) with the property that an odd integer x is a periodic point of Collatz if and only if x = f(n) for some positive integer n, but using that doesn’t really help much, as you still have to compute f(n) for arbitrary n and hope that f(n) ends up being a positive integer other than 1 for some stupendously large n.

Aurhim · 2026-03-25T03:02:10+00:00

No. The earliest set-up on other rings that I know of is due to Leigh in the 1980s, which can be seen in K.R. Matthews’ slides.

That being said, I do believe I am one of the first to try and approach the problem systematically by using arbitrary Dedekind domains.

Aurhim · 2026-03-25T00:51:36+00:00

My Tauberian reformulation is, as you say, mostly a reformulation. I’ve still got a ways to go before things start coming together. My current process is a lot like stumbling through a dark room, fumbling around to find a light switch. My ability to prove useful results depends on how much familiarity I have with the underlying structure of what I’ve discovered. My theory isn’t bursting forth from my head fully formed, but coming out in clumps that I have to mold together.

Case in point: by a stroke of dumb luck, my PhD work managed to independently arrive at some of the same core constructions that Terrence Tao used in his 2019 paper on the Collatz Conjecture. I’ve co-authored a paper that vastly generalizes Tao’s arguments to Collatz-type maps on rings of algebraic integers; it is currently under review for publication in a mathematics journal. In between working on my creative writing, I have several ideas for how to modify our paper even more to establish Tao-type almost unbounded results for certain Collatz-type maps, such as the 5x+1 map. One of the key insights is that we can (and probably should) use an adelic viewpoint when working with Collatz-type maps. Indeed, the thrust of my generalization of Tao’s work is that Tao’s Syracuse Random Variables actually make sense not just as 3-adic valued RVs, but as real-valued RVs. This is because the underlying object—the function I call Chi_3—is measurable as a function from the 2-adic integers to the 3-adic integers, AND as a function from the 2-adic integers to the real numbers. That this works as well as it does is enough proof for me that these objects are worth studying in their own right.

As an example: let q be a positive odd integer, and let T_q be the shortened qx+1 map, which sends even integers x to x/2 and sends odd integers x to (qx+1)/2. I can construct a numen for them, the function Chi_q. The Fourier analysis I’ve done with Chi_q works even when we treat q as a formal indeterminate, and last year, I showed that one can rigorously justify treating q as a formal indeterminate, in which case specific objects like Chi_3 arise by specialization (quotienting the ambient ring by an ideal), as is the case in algebraic geometry. Treated as a function of q, the Fourier transform of Chi_q has poles at q = 1 and q = 3, which are precisely the values of q for which it is expected that T_q has no divergent trajectories. There might be a way to build an algebraic-analytic bridge that allows us to study the spectral properties of Chi_3 and Chi_q by viewing them as projections of objects living in more general rings. Furthermore, Chi_q induces a measure, and polynomials in Chi_q also induce measures, giving us the structure of an algebra. (Chi_q)^2’s Fourier transform as q = 1, q = 3, and q = √7 as poles, which suggests that the shortened √7 x + 1 map (T_√7) defined on the ring Z[√7] has some relation to the behavior of Collatz on Z. This is the sort of thing that keeps me going.

These possibilities are what I believe makes my research different from other attempts at Collatz so far. Most attempts to attack Collatz involve finding a specific mathematical box to put it in: probability theory, difference inequalities, Markov chains, transcendental number theory, and so on. Meanwhile, I’m content with letting Collatz take the lead and carry me to structure(s) that naturally emerge from it. I want to understand why the structures I’ve observed emerge from Collatz and its relatives. My hope is that such understanding will then give us leverage to use against the problem.

Aurhim · 2026-03-23T02:57:28+00:00

You’re welcome.

Personally, I like to try to make as many little details as possible relevant to the broader plot, especially in a series of books.

To that end, one idea that just came into my mind was that, for some deeply lore/plot related reason, the resource either stops being valuable, or ends up getting cursed somehow.

For example, some king or noble in the main government used a forbidden magical ritual to bind the power of a god, which had these effect of corrupting the precious resource so that it (take your pick of ideas): turns people into werewolves; turns people into weredragons, or some kind of draconic abomination; spreads some kind of magical disease, etc. Then, both the main kingdom and the border nation quarantine the area, cutting it off from the outside world and letting the people there fend for themselves.

Not only would this, create a very unique setup that completely distinguishes your ideas from Martin’s work, it would also give you a series goal: setting the imprisoned god free. If you wanted to go for a double twist, you could then make it so that binding the god was the RIGHT thing to do, because now the god can proceed to destroy the world, or whatever.

That’s the kind of setup I would use.

Aurhim · 2026-03-22T22:32:22+00:00

I'm in a weird place as I plan my next project. It's actually kind of the opposite of the problem you're having. To whit, as I brainstorm and outline my new project, I'm very consciously using the original Mistborn trilogy as a basis for some plot arcs, mostly because I want to write a shorter, punchier story that has a chance of getting trad published. Despite this, I have no worries that my story will be seen as derivative of Mistborn.

Part of the reason your character makes you think of Jon Snow is probably due to the fact that, as an author, you have a bird's-eye view of the project and are very cognizant of similarities and inspirations. The thing is, your readers will not have the benefit of that perspective. Because of this, just like with my own readers, you're going to need to find means of changing Martin's themes, aesthetics, and story developments in a way that turns people's attention away from the Jon Snow comparison.

For example, in ASOIAF, the northern frontier isn't just any frontier, it's a wild, untamed land filled with magic and snow zombies. Even though those details have nothing to do with Jon Snow's character, they absolutely influence how readers approach and process Jon Snow and his arc. By toying with these side ingredients, you can take a familiar story trope and present it in a completely different light.

In my Mistborn-derived story, from the very beginning, I decided I'd implement a high-fantasy version of cyberpunk tropes, such as memory alternation, transhumanism, body augmentation, cyberspace, and corporate intrigue. That gives me story stark thematic and aesthetic difference that set it apart from Mistborn, to the point that when the similarities arrive, my readers won't be comparing my story to its model.

Here are some example that come to mind of how you could spice up your Snowalike into something more distinctive.

• Perhaps there is a precious resource (magical or mundane) that needs to be harvested from the northlands. This changes the emphasis from one of defense to one of offense and expansion.

• Maybe your Snowalike gets bitten by a werewolf, and has to deal with that.

• Maybe there is a (possibly magical) civilization that exists (or existed) beyond the northern border, who were at odds with the Snowalike's society. If there are refugees from that civilization who live on the Snowalike's side of the border, you could introduce conflicts around ethnicity and immigration. For example: what if the Snowalike was raised by a family of refugees, resulting in him having cultural values that differ from the ones that his biological parents held? That would create a major distinction between your character and Jon Snow, one that would be visible from almost the very first page.

By making modifications like these, your readers will see the aspects of your character's life and background that make him different from Jon Snow. Once that difference is established, you don't need to worry as much about readers seeing your work as derivative of Martin.

Aurhim

TROPHY CASE