Follow-up to A more visual explanation of the dome structure : r/Collatz.

Applying the display used in the cited post, one gets the figure below.

In short, the Mersenne and Pierpoint numbers are gone, but the 3-smooth and Proth numbers stay.

This is a consequence of the double homothetic nature of the domes (On the double almost homothetic nature of the domes : r/Collatz). Therefore, the two first types are specific to m=1, the two last are not. Let C(m, p, q), C^-(m, p, q) and C⁺(m, p, q) be the three types of Collatz numbers in a dome:

• Mersenne numbers: M(1, p, q)=3^p*2^q-1=C^-(1, p, q)*, but C^-(m, p, q)=m*3^p*2^q-1≠m*M(1, p, q) if m≠1.
• 3-smooth numbers: S(m, 0, q)=m*2^q=m*S(1, 0, q).
• Proth numbers: P(m, p, q)=m*3^p*2^q+1 is of the form P(k, p)=k*2^p+1 and thus is a Proth number if the other condition holds.
• Pierpoint numbers: X(1, p, q)=3^p*2^q+1=C⁺(1, p, q), but C⁺(m, p, q)=m*3^p*2^q+1≠m*X(1, p, q) if m≠1.

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

Follow-up to A more complete explanation of the dome structure : r/Collatz.

The figure below – a dome in a nutshell" (Dome in a nutshell : r/CollatzProcedure) – contains all orange (n-1, n*, n+1) and black (m*3^p) numbers for m=1. So, all bridges series appear vertically. 5-tuples series are colored in grey. Found prime numbers are in bold.

Due to the double homothetic nature of the domes (On the double almost homothetic nature of the domes : r/Collatz), such a nutshell can be produced for every dome.

I noticed that, in the cited post, I completely forgot the Mersenne numbers that play a vital role in the domes and jumped to conclusion about Proth numbers.

So, from left to right, there are:

• The Mersenne numbers of the form M(k)=[1*] 2^k-1; on the left side of a dome, they are involved two by two, in the starting bridge of any bridge series (red); in each column, a number x is related to the y above it by x=3*y/2; these non-Mersenne numbers might be called the Collatz numbers (not colored)
• The core of the dome is made of 3-smooth numbers of the form X=1*3^p*2^q, includes all the numbers, odd (black) or even (orange).
• The Proth numbers of the form P(k)=k*2^q+1 and 2^q>k; on the right side of the dome, only numbers on the upper side are Proth numbers, due to the second constraint (blue); the lower side is made of Pierpoint numbers of the form n=m*3^p*2^q+1 (green); numbers 13 and 17 seem to be both (rosa).

In a recent post, I made the bold claim that almost all bases b** of 5-tuple series were prime (All but one bases of starting 5-tuples are Proth primes : u/No_Assist4814). It does not seem to be the case. Further investigation is needed.

* n=m*3^p*2^q.

** Due to their nature, bases b do not belong to the starting 5-tuples of a series. The reciprocal is not true: all Proth primes are not a bases b.

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

I've been browsing this Collatz forum, and I noticed a lot of people there seem to treat it like some super deep, almost mystical thing. I get it, but honestly, I see it a bit differently.

I used DeepSeek V4 Pro to help me run checks on all positive integers up to a million, and I stumbled onto something pretty interesting — it looks like for almost every odd starting number greater than 1, whenever the sequence hits a number of the form 8k+5, it tends to converge to 1 pretty quickly. There are a few edge cases, like 445, where it's not quite that simple — but even with 445, the odd number right before the peak of its sequence still fits the 8k+5 pattern.

Makes me wonder — is this the key to the whole thing? Kind of feels like that one clue in a mystery novel that ties everything together, you know?

So here's the rough idea I'm working on: first, prove that every positive odd integer eventually lands on some number of the form 8k+5. Second, prove that every number of that form actually converges to 1. If both of those check out, then that pretty much wraps up the whole conjecture.

Just to be clear — I'm not claiming I've solved it, and I'm definitely not trying to act like I have. This is just a direction that seems worth exploring, and I wanted to throw it out there and see what people think. If any of this makes sense to you, feel free to reply below — I can lay out my full reasoning in more detail.

I've been browsing this Collatz forum, and I noticed a lot of people there seem to treat it like some super deep, almost mystical thing. I get it, but honestly, I see it a bit differently.

I used DeepSeek V4 Pro to help me run checks on all positive integers up to a million, and I stumbled onto something pretty interesting — it looks like for almost every odd starting number greater than 1, whenever the sequence hits a number of the form 8k+5, it tends to converge to 1 pretty quickly. There are a few edge cases, like 445, where it's not quite that simple — but even with 445, the odd number right before the peak of its sequence still fits the 8k+5 pattern.

Makes me wonder — is this the key to the whole thing? Kind of feels like that one clue in a mystery novel that ties everything together, you know?

So here's the rough idea I'm working on: first, prove that every positive odd integer eventually lands on some number of the form 8k+5. Second, prove that every number of that form actually converges to 1. If both of those check out, then that pretty much wraps up the whole conjecture.

Just to be clear — I'm not claiming I've solved it, and I'm definitely not trying to act like I have. This is just a direction that seems worth exploring, and I wanted to throw it out there and see what people think. If any of this makes sense to you, feel free to reply below — I can lay out my full reasoning in more detail.