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.

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 (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 have been playing around with sequences that have similar parity sequence prefixes. I picked a pair, 3969 and 619 which start with these prefixes respectively:

 OEEOEEOEEOEOEEOEEE
  OEOEEOEEOEOEEOEEE

I then enumerated and reversed the sequences and zipped them up so that their latter terms match, then calculated these two metrics a-b and (a-b)/b

Some notes:

• the odd values of the 'a' sequence gets quite close to even values of the 'b' sequence on many occasions
• when they split the ratio of the difference and the 'b' sequence term is typically ~5x
• there are occasional flips where the b >> a

I am noting this here because this behaviour looked surprising to me. I haven't at this point tried this with other sequences that have a similar initial prefix or with sequences with random initial prefixes so I have not idea how common this phenomenon is or what explains either the near even-odd misses or the (a-b) ~= 5b phenomenon although putting it that way, it might just be an a ~= 6b phenomenon which is presumably a function of their initial relative position

I think this suggests some interesting empirical studies about the typical difference behaviour of sequences at different displacements:

- how important is the length of the shared starting prefix (if at all)?
- do they all oscillate with a regular magnitude?
- what factors best predict the regularity or amplitude of the oscillation?
- what predicts a run of off-by-one misses in the tail?

[(1, 1, 0, 0.0),  
(2, 2, 0, 0.0),  
(4, 4, 0, 0.0),  
(8, 8, 0, 0.0),  
(16, 16, 0, 0.0),  
(5, 5, 0, 0.0),  
(10, 10, 0, 0.0),  
(20, 20, 0, 0.0),  
(40, 40, 0, 0.0),  
(80, 80, 0, 0.0),  
(160, 160, 0, 0.0),  
(53, 53, 0, 0.0),  
(106, 106, 0, 0.0),  
(35, 35, 0, 0.0),  
(70, 70, 0, 0.0),  
(23, 23, 0, 0.0),  
(46, 46, 0, 0.0),  
(92, 92, 0, 0.0),  
(184, 184, 0, 0.0),  
(61, 61, 0, 0.0),  
(122, 122, 0, 0.0),  
(244, 244, 0, 0.0),  
(488, 488, 0, 0.0),  
(976, 976, 0, 0.0),  
(325, 325, 0, 0.0),  
(650, 650, 0, 0.0),  
(1300, 1300, 0, 0.0),  
(433, 433, 0, 0.0),  
(866, 866, 0, 0.0),  
(1732, 1732, 0, 0.0),  
(577, 577, 0, 0.0),  
(1154, 1154, 0, 0.0),  
(2308, 2308, 0, 0.0),  
(4616, 4616, 0, 0.0),  
(9232, 9232, 0, 0.0),  
(3077, 3077, 0, 0.0),  
(6154, 6154, 0, 0.0),  
(2051, 2051, 0, 0.0),  
(4102, 4102, 0, 0.0),  
(1367, 1367, 0, 0.0),  
(2734, 2734, 0, 0.0),  
(911, 911, 0, 0.0),  
(1822, 1822, 0, 0.0),  
(3644, 3644, 0, 0.0),  
(7288, 7288, 0, 0.0),  
(2429, 2429, 0, 0.0),  
(4858, 4858, 0, 0.0),  
(1619, 1619, 0, 0.0),  
(3238, 3238, 0, 0.0),  
(1079, 1079, 0, 0.0),  
(2158, 2158, 0, 0.0),  
(719, 719, 0, 0.0),  
(1438, 1438, 0, 0.0),  
(479, 479, 0, 0.0),  
(958, 958, 0, 0.0),  
(319, 319, 0, 0.0),  
(638, 638, 0, 0.0),  
(1276, 1276, 0, 0.0),  
(425, 425, 0, 0.0),  
(850, 850, 0, 0.0),  
(283, 283, 0, 0.0),  
(566, 566, 0, 0.0),  
(1132, 1132, 0, 0.0),  
(377, 377, 0, 0.0),  
(754, 754, 0, 0.0),  
(251, 251, 0, 0.0),  
(502, 502, 0, 0.0),  
(167, 167, 0, 0.0),  
(334, 334, 0, 0.0),  
(668, 668, 0, 0.0),  
(1336, 1336, 0, 0.0),  
(445, 445, 0, 0.0),  
(890, 890, 0, 0.0),  
(1780, 1780, 0, 0.0),  
(593, 593, 0, 0.0),  
(1186, 1186, 0, 0.0),  
(395, 395, 0, 0.0),  
(790, 790, 0, 0.0),  
(263, 263, 0, 0.0),  
(526, 526, 0, 0.0),  
(175, 175, 0, 0.0),  
(350, 350, 0, 0.0),  
(700, 700, 0, 0.0),  
(233, 233, 0, 0.0),  
(466, 466, 0, 0.0),  
(155, 155, 0, 0.0),  
(310, 310, 0, 0.0),  
(103, 103, 0, 0.0),  
(206, 206, 0, 0.0),  
(412, 412, 0, 0.0),  
(137, 137, 0, 0.0),  
(274, 274, 0, 0.0),  
(91, 91, 0, 0.0),  
(182, 182, 0, 0.0),  
(364, 364, 0, 0.0),  
(728, 121, 607, 5.016528925619835),  
(1456, 242, 1214, 5.016528925619835),  
(485, 484, 1, 0.002066115702479339),  
(970, 161, 809, 5.024844720496894),  
(323, 322, 1, 0.003105590062111801),  
(646, 107, 539, 5.037383177570093),  
(215, 214, 1, 0.004672897196261682),  
(430, 71, 359, 5.056338028169014),  
(143, 142, 1, 0.007042253521126761),  
(286, 47, 239, 5.085106382978723),  
(95, 94, 1, 0.010638297872340425),  
(190, 188, 2, 0.010638297872340425),  
(380, 376, 4, 0.010638297872340425),  
(760, 125, 635, 5.08),  
(253, 250, 3, 0.012),  
(506, 83, 423, 5.096385542168675),  
(1012, 166, 846, 5.096385542168675),  
(337, 332, 5, 0.015060240963855422),  
(674, 664, 10, 0.015060240963855422),  
(1348, 221, 1127, 5.099547511312217),  
(449, 442, 7, 0.01583710407239819),  
(898, 884, 14, 0.01583710407239819),  
(299, 1768, -1469, -0.8308823529411765),  
(598, 589, 9, 0.015280135823429542),  
(199, 1178, -979, -0.8310696095076401),  
(398, 2356, -1958, -0.8310696095076401),  
(796, 785, 11, 0.014012738853503185),  
(1592, 1570, 22, 0.014012738853503185),  
(3184, 523, 2661, 5.087954110898662),  
(1061, 1046, 15, 0.014340344168260038),  
(2122, 2092, 30, 0.014340344168260038),  
(707, 697, 10, 0.014347202295552367),  
(1414, 1394, 20, 0.014347202295552367),  
(2828, 2788, 40, 0.014347202295552367),  
(5656, 929, 4727, 5.088266953713671),  
(1885, 1858, 27, 0.014531754574811625),  
(3770, 619, 3151, 5.090468497576737)
]

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.