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If we look at any initial condition or any configuration in a collatz-sequence, do we find patterns to primes? I just found out the first 16 possible configurations adds up to the primes, but I need to check if this is true for numbers up to a hundred or a thousand to be sure this is true.

My research-notes have gone over 30 pages so far. I guess thats not a big deal for those who have written dozens of papers. I am probably the only one who understands my own work, and need to practise more on writing it in a formal matter. So there's no point for me to share my notes so far. Since I am looking at the Collatz-problem from a simple perspective, I just wonder, are there any simple basic research in this area, or does it involve complicated equations?

To reiterate my question, it goes like this: Is there any patterns to primes in relations to an initial condition or any configuration in a Collatz sequence?

And if someone have an answer, please answer in simple terms, I would appreciate that.

Thanks.

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    Tha answer to your question is "almost surely not". The collatz problem has been well studied though not solved. It's unlikely that you will make progress on it with elementary methods, looking for primes up to the hundreds of thousands. I applaud your interest and effort, but you probably won't get any help on this site. – Ethan Bolker Mar 11 '18 at 13:54
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    I completely agree with @EthanBolker, but I want to stress that you should still try if you enjoy it. I've tried to crack this particular theorem, as well as many other unproved theorems (for example, the Riemann Hypothesis), and I know it's (extremely) unlikely for me to solve them, but it's still good practice, and most of all: fun. Do what you enjoy doing, but stay realistic. –  Mar 11 '18 at 14:04
  • To the close voters: please leave a comment as to what you think is missing. If you don't give feedback, the OP can't improve on their mistake(s). –  Mar 11 '18 at 14:06
  • Thank you for your kind answers to such a difficult problem. It is good practice indeed. While studying this simple to understand but difficult to crack problem, all sorts of sequences and series popped up. A friend gave me a book about the Riemann Hypothesis that I read through, but after all these years I have not found a particular plan to approach that problem. The Collatz seem more accessible for me at the moment. –  Mar 11 '18 at 14:34
  • Actually, I just found out it didnt have any relations to the primes, but rather the Jacobsthal sequence. –  Mar 11 '18 at 15:50
  • Not that I would vote to close this question (unless it was a duplicate), but what exactly do you mean by "the first 16 possible configurations"? Can you give us an example at least of the very first possible configuration? – Robert Soupe Mar 12 '18 at 04:36
  • Let $a_n$ be any positive odd integer. Shortcut the Collatz map by $a_{n+1} = \frac{3a_n+1}{2}$. if (initial configuration) $a_n = 33$, then (the first possible configuration is) $a_{n+1} = 17$. In base 2 we can map the msb of $a_n$ to the amphichiral rule 219 of an L-tetromino to the odd $a_{n-1}$. We are left with 1 of 16 possible configurations mapped from the L-tetromino. In my example 17 was dependent on left boundary conditions of the previous configuration of 33: Simply $a_{n-1}$ is dependent on $a_n$'s left-boundary conditions. –  Mar 12 '18 at 07:26
  • I have literally just been posting in recent days about the Jacobsthal sequence such as here: https://math.stackexchange.com/questions/2683706/ It may not be immediately obvious but the left and right transforms I define generate this sequence. Is it coincidence that you post on this same subject so soon after? – it's a hire car baby Mar 12 '18 at 20:19
  • That is a coincidence. Im looking into local behaviour, so I have let alone the Jacobstal sequence for now. That sequence thought, was easy to see in the binary computations I did. But it's a global feature imho, and that's not what I am studying at the moment. Im more into studying the details of how the numbers are actually generated. I would also like to add that the terms you used are a bit out of my league. For example what did you mean about Prufer 2-group to be Noetherian? What is Prufer 2-group and Noetherian? –  Mar 13 '18 at 00:07

2 Answers2

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Prime number "patterns" in the Collatz Conjecture are most likely coincidence, unless the configuration requires finding primes in the first place. Since a lot of Collatz research focuses on odd numbers, one is more likely to come across prime numbers simply by eliminating the even numbers. When you introduce the "short cut" map of the Conjecture where if $x$ is odd, then $x$ = $(3x+1)/2$, this reduces the amount of time it takes to get another odd number, thus increasing the chance of finding more prime numbers.

There is research on the connection between the Collatz Conjecture and the Mersenne primes. More can be found on that here. As for the Jacobsthal-Lucas numbers, I believe it is also a coincidence since they are similar to the Mersenne primes.

As for other related Collatz pattern formulas, Gottfried Helms explains it better here.

Aside from that, I can only speak from experience that I have unsuccessfully found a connection between prime numbers and the Collatz Conjecture.

Griffon Theorist697
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  • Yes, and in your formula for odd numbers, one could also consider the denominator to be $2^{g(x)}$ where $g(x)$ is The Ruler/Thomae's Function for positive integers, without 1 added. I'll check out your links. –  Mar 12 '18 at 14:14
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One of two operations in Collatz Conjecture is:

  • if the number is $odd$, triple it and add $1$.

One of the polynomial form for all prime numbers is:

  • if the number is $odd$, triple it and add $2$ or $4$.
LAAE
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  • The second part seems to be nonsense. – Tobias Kildetoft Mar 12 '18 at 20:11
  • In what way...? – LAAE Mar 12 '18 at 20:19
  • This is potentially much more closely related to Collatz than you realise. But I don't know what you mean by "One of the polynomial form for all prime numbers is..." What precisely does this mean? – it's a hire car baby Mar 12 '18 at 20:21
  • In the way that your proposed "form" gives non-primes as well. – Tobias Kildetoft Mar 12 '18 at 20:21
  • You are talking about: 25,35,49,55,65,77,85... and so on - this is natural how they get into composition using the above form - the key is how to separate them from prime numbers. – LAAE Mar 12 '18 at 20:25
  • ...on the face of it this could be related to the Goldbach conjecture because if we permit even numbers then $3x+2^{v_2(x)}$ generates collatz-equivalent sequences shifted by powers of $2$. Some rigorous algebraic relation between primes and $3x+(2\text{ or }4)$ has potential to relate certain primes to multiples of $2$. – it's a hire car baby Mar 12 '18 at 20:26
  • What I have done is I have reduced all coefficients by 3 from the form 6n+5 and 6n+7 and changed indeterminate to odd. It also works for indeterminate as even as well. – LAAE Mar 12 '18 at 20:31