Is this Recursive Pattern in the Collatz Sequence Known?

[MrHellmet]

TL;DR

I am empirically studying the transitions between successive odd numbers in the Collatz sequence, focusing on the change in the 2-adic valuation (Δv) between steps. I've observed that integers with a constant Δv form arithmetic progressions.

Hello everyone,I've been empirically analyzing the transitions between successive odd numbers in the Collatz sequence ($$m_{new} = (3m_{old}+1)/2^N$$).I've observed that odd integers that produce a specific "valuation jump", $$\Delta v = v_2(3m_{new}+1) - v_2(3m_{old}+1)$$, seem to fall into arithmetic progressions. I've been studying the 'seed' of these progressions, defined as the smallest positive odd integer $$m$$ satisfying the conditions for a given family.For the family of integers with an initial valuation $$v_2(3m+1) = 2$$, the seeds for positive valuation jumps ($$\Delta v = k \geq 1$$) appear to follow this recursive formula:

$a(k) = a(k-1) + C_{pos}(2,k) \cdot 2^{k+3}$

Here, $$a(k)$$ is the seed for a jump of $$\Delta v = k$$, and the starting seed is $$a(1)=17$$. The coefficient $$C_{pos}(2,k)$$ is periodic depending on

$$k \pmod 6$$:$C_{pos}(2,k) = \begin{cases} 3 & \text{if } k \equiv 3 \pmod 6 \\ -1 & \text{if } k \equiv 0 \pmod 6 \\ 1 & \text{otherwise} \end{cases}$

For example, this correctly predicts the seed for $$\Delta v = 2$$:$a(2) = a(1) + C_{pos}(2,2) \cdot 2^{2+3} = 17 + (1) \cdot 32 = 49.$My questions are:1. Is this specific recursive structure, or the periodic nature of its coefficient, a known result in the literature on the 3n+1 problem?2. If it is not a known result, does anyone see a potential path or a related mathematical structure that could help in proving it?I have verified this and similar patterns for other families extensively, but I haven't been able to find a formal proof. Any references or insights would be greatly appreciated.

 

[jedishrfu]

If anyone has seen this it is likely terence tao

Heres a paper he published on the conjecture

https://arxiv.org/pdf/1909.03562

In general, mathematicians play with the conjecture for a short time to understand the depth and quirkiness of it but don’t devote their lives to it.

Mathematicians will tell their students to steer clear of it citing Erdos who famously said math is not yet ready for this kind of problem.

Personally, I have played with it using modulo arithmetic. The idea was to show inductively that for any seed number and any modulo up to that number evaluated to 1 after iterating through the sequence but i ran into a few roadblocks where I’d get a zero and then things fell apart unless I replaced the zero with the modulo number ala clock arithmetic.

 

[MrHellmet]

If anyone has seen this it is likely terence tao

Heres a paper he published on the conjecture

https://arxiv.org/pdf/1909.03562

In general, mathematicians play with the conjecture for a short time to understand the depth and quirkiness of it but don’t devote their lives to it.

Mathematicians will tell their students to steer clear of it citing Erdos who famously said math is not yet ready for this kind of problem.

Thank you for this very thoughtful and insightful response. I really appreciate it.Thank you especially for the link to Tao's paper. I am familiar with his work, and I agree his approach is very deep. My framework is quite different as it focuses more on the local, modular structure of the 2-adic valuation jumps from one odd number to the next.I completely understand your point about the problem's reputation and Erdos's famous quote. It is indeed a notoriously challenging area, which is why my goal was not to attempt a full proof, but to rigorously classify the local dynamics and see if any predictable structure emerged. and i obtained quite interesting patterns that seems to mantain for quite large ranges.
Your personal experience with modulo arithmetic is very interesting. The roadblock you mention is a classic challenge. It seems my approach is related but distinct, as I'm focused on the 'valuation jump' as the primary classifier.
Thanks again for taking the time to engage seriously with the question, in other pages people werent so polite.

 

[jedishrfu]

Thank you, but I do need to remind you that we don't discuss personal research here on PF.

Once your research is peer reviewed and published in a reputable math journal we could if we have the expertise to do so.

However while doing your research you had a question on some step that confounded you we might be able to look at it.