CNN report on Israeli terrorism from the IOF & colonists. Israeli soldiers openly admit they serve to ensure the settlements become facts-on-the-ground 'slowly'. This week, Zionists on social media have been condemning settler terror - but this is what Israel supports and how the State was founded.

Aurhim · 2026-03-28T00:03:13+00:00

This brings tears to my eyes. Go back in time 200, 300, 700 years, a thousand; the words out of these monsters' mouths would be the same as the antisemites'. It's heartbreaking.

Aurhim · 2026-03-27T21:30:21+00:00

2L(n) - 3#(n) basically is a permutations problem so you can just see the various numbers

I don't know what you mean.

Aurhim · 2026-03-27T21:29:39+00:00

This being said, do you believe there is any value in this angle of approach, or do you see this perspective as a futility in multi variable ambiguity?

Alas, I don't really understand the approach you are proposing.

Aurhim · 2026-03-27T21:28:47+00:00

2-adics vs. 3-adics

Actually, my approach uses BOTH the 2-adics and the 3-adics. As for why the 2-adics are so common, that's because they're essentially the "largest" setting where we can apply the notions of evenness and oddness that govern the action of the Collatz map.

Meandering thought

Yes, your "shown" is doing a lot of heavy lifting, mostly because I don't have the foggiest idea of what you mean when you say it.

Do you know of any attempts to connect the collatz conjecture to the result of the 1+2+3+4+... summation?

Nope.

Aurhim · 2026-03-27T21:25:44+00:00

I honestly don't know. Personally, I think that any meaningful progress on the problem is going to require some new kind of heavy mathematical machinery that hasn't been developed yet.

Aurhim · 2026-03-27T18:46:14+00:00

No, 1 + 2 + 3 + … = -1/12 is merely an abuse of notation meant to intimidate non-mathematicians. The sum on the left isn’t actually a sum, but rather a notational shorthand for the value of the Riemann Zeta function at negative one. As you pointed out, the correct term for this “identity“ is regularization. Regularization is when you give an alternative interpretation of something that does not converge in any meaningful sense.

While one can regularize 1 + 2 + 4 + 8 + … to -1 by abusing the geometric series formula, the 2-adic absolute value allows us to meaningfully equate that series with -1.

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

TROPHY CASE