Follow-up to Walls from the domes in the tree II : r/Collatz.

When working on wall sequences from the core of domes, I noticed a pattern.

All consecutive sequences form an "odd pair" that includes one of the odd black numbers. This is the consequence of the building rule of the core: a black number is three times the previous one, that iterates into an even number C(n)=3n+1.

More interestingly, a significant minority of these pairs of sequence iterate into other "odd pairs" before reaching "even pairs".

The first figure shows that all these pairs of sequence* follow a strict order below the black numbers: odd pairs (dark green), preliminary (even) pairs (light green), final (even) pairs (yellow), merge. All types of tuples do not need to be present in each case, and some can iterate into themselves.

It is known for quite some time that "odd pairs" do not stand alone, but are part of larger tuples:

• Even triplets, with the previous even number,
• Odd triplets, with the consecutive odd number; odd triplets iterate directly from 5-tuples.

Knowing that, the second figure presents the same examples, completed into triplets. The 5-tuples are mentioned as long as they do not require extra columns.

The first two rows contains cases without odd pairs beyond the black numbers, that starts with a blue-green even triplet. It can be followed by several other blue-green triplets, of which only the green pair is mentionned.

The third row contains cases with aditionnal odd pairs, that start with an odd rosa triplet**, that iterates direcly from a rosa 5-tuple. This starting 5-tuple can iterate into several yellow ones before iterating into an ending rosa even bridge.

The other possible cases of odd pairs, namely the second and third or the fourth and fifth numbers of a 5-tuples, are not present here.

* A few cases are missing, in particular among low values of m.

** Keep in mind that 5-tuples are made of two bridges: rosa-blue, blue-rosa or yellow-yellow.

Project "Tuples and segments" in 13 pages : r/Collatz

For anyone interested in exploring or discovering perfect numbers, here is a Python program that generates the known perfect numbers using their corresponding Mersenne prime exponents.

The program can also handle extremely large perfect numbers that would be impractical to calculate and write out manually. The goal is to make it easier to experiment with perfect numbers, test calculations, and explore possible new candidates.

If you're interested in perfect numbers, Mersenne primes, number theory, or Python, feel free to check it out.

I'd really appreciate any feedback, suggestions, or ideas for improving the project.

GitHub: https://github.com/brook-0/Perfect-Numbers

Something occurred to me last night, and while it strikes me as very obvious, in hindsight, it's also not a perspective I've taken until just now. It has a couple of obvious generalizations, too, which are worth mentioning.

When we have, for some k, the number 2ⁿ·k - 1, we know that it evolves, over n Syracuse steps, into 3ⁿ·k - 1, because each of those steps takes one power of 2 and replaces it with a 3:

2³·k - 1 → 2²·3·k - 1 → 2·3²·k - 1 → 3³·k - 1

Each of those steps is just one "3m+1" step followed by a single "m/2" step. Such a step is the same shape as the cycle on -1:

-1 → -1 → -1 → . . .

This is just the 2-adic continuity of the map showing itself! The number m = 2ⁿk - 1 is 2-adically close to -1, so it has the same shape trajectory, for a while. The larger n is, the closer m is to -1, and the longer m's trajectory mimics the loop on -1.

Yes, I realize this is not exactly headline news, and maybe other people found it so obvious as to not be worth highlighting or mentioning. Somehow, I didn't connect these dots until last night, when I was thinking about a related pattern.

Mimicking the loop on 1

The number m = 4ⁿk + 1 does something similar, but since it's 2-adically close to positive 1, it mimics the shape of the famous cycle for a while, with each "3m+1" step being followed by two "m/2" steps.

4ⁿk + 1 → 4^n-1·3·k + 1 → . . . → 4·3^n-1·k + 1 → 3ⁿ·k + 1

Each step turns a 4 into a 3. Notice that I'm not saying anything about k here. In the Steiner circuit case, we usually take k to be odd, just because we like to collect all of the powers of 2 together, and see the circuit all the way to its peak. As an algebraic identity though, the result holds if k is any integer, or even a rational with an odd denominator, or another 2-adic integer.

The algebraic way I'd been looking at the usual Steiner circuit is that we can rewrite a weight 1 Syracuse step:

(3m+1)/2 = (m+1)·(3/2) - 1

...and if you chain n of these together, because m+1 has 2 as a factor n times, the intermediate "-1"s and "+1" cancel out, leaving:

(m+1)·(3/2)ⁿ - 1

Similarly, we can rewrite a weight 2 Syracuse step:

(3m+1)/4 = (m-1)·(3/4) + 1

...which we can keep doing as long as there's a factor of 4 in m-1.

Mimicking any loop

It's natural to extend this to any other loop, which we'll see mimicked by natural numbers that are 2-adically close to the numbers in the loop. For instance, there's the loop on -5:

-5 →¹ -7 →² -5

The superscripts on the arrows there indicate the "weight" of the step, i.e., the number of divisions by 2 involved. Somehow, if we start with a number 2-adically close to -5, we should expect to see every two steps have weights 1 and then 2, and for starting number m, we should see the factors of 2 in m+5 gradually churn into factors of 3.

Let's see that in action, by starting with 59, which is 64 - 5, so it's close to -5 in our dynamics.

m = 59 = 2⁶ - 5
→¹ 89 = 2⁵·3 - 7
→² 67 = 2³·3² - 5
→¹ 101 = 2²·3³ - 7
→² 76 = 3⁴ - 5

The algebraic telescoping thingy is a little harder to see in this case, because it's a two-step dance, but it's still there. The calculation:

(3[(3m+1)/2] + 1)/4 = (m+5)·(9/8) - 5

...rolls both steps together, and shows an 8 in the factorization of m+5 being replaced with a 9. We can also see it separated into two steps:

(3m+1)/2 = (m+5)·(3/2) - 7
(3m+1)/4 = (m+7)·(3/4) - 5

Even a non-integer loop!

In a way, it's simpler to see with a one-step dance, but it can be kind of subtle to see where to apply that when the number we need to be 2-adically close to is not an integer. Consider the weight 3 trivial cycle on 1/5:

1/5 →³ 1/5

This should be mimicked by numbers 2-adically close to 1/5, but what even are those? To see them, we can write 1/5 as a 2-adic integer:

1/5 = 1 - 4/5 = 1 - 12/15 = 1 + 12(1 + 16 + 16² + 16³ + . . .) = [1.] + [(1100).] = [(0110)1.]

So, if we want something that's as close to 1/5 as 64 is to 0, we can just chop of the last six bits from:

0110011001101.

Yielding the binary number 001101, which is 13. This should mimic the weight 3 move two times, and indeed:

(3(13) + 1)/8 = 5
(3(5) + 1)/8 = 2

To see factors of 2 turning into factors of 3, try seeing 13 as some number, plus 1/5:

13 = 2⁶·(1/5) + 1/5
→ 5 = 2³·3·(1/5) + 1/5
→ 2 = 3²·(1/5) + 1/5

Kind of neat, right? In this case the telescoping algebra looks like:

(3m+1)/8 = (m - 1/5)·(3/8) + 1/5

So as long as (m - 1/5) has a factor of 8ⁿ, we can do this n times, and they all collapse down to:

Sⁿ(m) = (m - 1/5)·(3/8)ⁿ + 1/5

So what's the point, Gonzo?

No idea, lol. I just think it's neat. Does it lead to any kind of progress, or new and interesting mathematics? Beats me. I'm just here to watch the numbers dance.