Tagged Questions

Dynamics of the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2$ if $n$ is even. Generalizations to $n \to 3n-1$ or $n \to 5n+1$ or even to $n \to pn+q$ . Other names are "$3x+1$-problem","syracuse problem". If you have a question, please be specific to your detail. MSE is not ...

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Does this extension of the Collatz sequence converge for n=550?

The Collatz function, or $3n + 1$ function is well known. A heuristic argument that most inputs should converge with repeated application of the function is as follows. With probability 1/2 an input ...
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Is it possible to describe the Collatz function in one formula?

This is related to Collatz sequence, which is that $$C(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ Is it possible to describe ...
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Could this odd insight help explain part of the difficulty in proving the Collatz Conjecture?

Background: Here's a crash course on the Collatz Conjecture. Basically, you take a number and if it is even you divide it by two. If a number is odd, you multiply it by three and then add one. You ...
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Is the Collatz function piecewise linear?

I read somewhere that the Collatz function $\mathbb Z \rightarrow \mathbb Z$: $$\text{Collatz}(x) = \begin{cases} x/2 &&x \; \mathrm{even} \\ 3x+1 &&x \; \mathrm{odd}\end{cases}$$ is ...
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Besides the $3x + 1$ problem, for which similar problems are still unresolved regarding trayectory?

Generalize the $3x + 1$ problem as $cx \pm 1$, where $c$ is a positive odd integer and $x$ is a positive integer iterated through the function as far as possible to discover a cycle. If $x$ is even, ...
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Twin prime conjecture (Goldbach-Collatz remix)

Assuming Goldbach's conjecture, let's denote $r_{0}(n)$ for any integer $n$ greater than $1$ the smallest non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Let $f$ be the map ...
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Simon Letherman, Dierk Schleicher, and Reg Wood 1999 paper on the 3n+1 problem, further results

I recently read with great interest the 1999 paper : "The 3n+1-Problem and Holomorphic Dynamics Simon Letherman, Dierk Schleicher, and Reg Wood" Experimental Mathematics 8: 3 (1999), 241–252. I ...
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How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
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arithmetic sequence $8n+1$ and the collatz conjecture

Is it a known result that if for all $n$ the collatz sequence of $8n+1$ lead to $1$, all natural numbers will?
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It's sufficient to prove collatz conjecture for $3+6k, k \geq 0$?

Thinking about this problem, I saw two interesting properties of Collatz graph. Firstly, if we consider that every even number $e$ can be represented (on a single way) as $e = o 2^n$, where $o$ is an ...
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For how many consecutive numbers Collatz conjecture was checked?

I heard here that Collatz conjecture was checked at least for every first $5 \cdot 10^{18}$ natural numbers, but I cannot find any source or actual information about this. Can anyone help to find out ...
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Examples of “eventually reaches y under iteration” other than the Collatz problem

The Collatz conjecture states that iteratively applying the map $$f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ to any ...
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Repeating cycles in the $3n-1$ problem

While tracking sequences beginning with 1-to-3 digit integers, I have found 3 different repeating cycles in the $3n-1$ problem (similar to the Collatz Conjecture). They are 1, 2, 1..., 5, 14, 7, 20, ...
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Collatz cycle necessary condition.

Has it been established that a nontrivial m-cycle of the Collatz conjecture on the positive integers would require two consecutive raises (i.e., if $\{x_1, x_2, \ldots x_n\}$ is the odd positive ...
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Are there any explanations for these patterns in the Collatz sequences?

I've been messing around with the Collatz sequences a bit, and have come across a few patterns - I was wondering if there are any known explanations for these patterns. The first is the plot of ...
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My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
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Proof of Terras' method for finding T(k)(x) of Collatz sequence.

Terras (via Lagarias) has a method explained here (2.2) for finding the kth term of any Collatz sequence (that is) $$T(n) = \frac{3n+1}{2}$$ if n is odd and $$=\frac n2$$ if n is even. The lambda ...
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Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.

For what values of $m,n$ natural, do $2^n - 1$ is divisible by $2^{m+n} - 3^m$? Thank you very much.
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Diophantine equation $3^{m}x+3^{m}-3^{m-i} 2^{i} +3^{m-i} 2^{i+s} -2^{m+s}=x 2^{m+s}$

As part of my research and my calculations I got the following diophantine equation. I do not have much experience with the diophantine equation. Is there any known method to solve it? Any help is ...
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Let $a_1$ be an integer. Then we assume $$a_{n+1} = \begin{cases} 3a_n+1,&\text{a_n is odd}\\ \frac{a_n}{2},&\text{a_n is even} \end{cases}$$ Now we prove that for any ...
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Where can I find out more on Collatz-conjecture like sequences?

I'm interested in Collatz-conjecture (the 3n+1 problem) like sequences. I'm interested in any literature that contains information about problems that are divided into similar cases. I'm ...
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What is the simplest collatz like problem that is undecidable?

I have read that problems resemblings collatz have been shown to be undecidable. Conway proved that apparantly but Im not sure if the proof was constructive. So I wonder : What is the simplest ...
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Collatz conjecture and related problems - mathematical machinery

Collatz conjecture stands as an open problem. That leads me to believe that the conjecture cannot be resolved by elementary means. Which brings me to my question: What techniques/machinery from ...
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collatz conjecture $\mod 2^n$ stopping distance

an interesting book Old and new unsolved problems in plane geometry and number theory by Klee/Wagon (1991) includes the Collatz conjecture. on p225 they consider iterates $\mod 2^n$ and state that ...
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I worked on the Collatz conjecture extensively for fun and practise about a year ago (I'm a CS student, not mathematician). Today, I was browsing the Project Euler webpage, which has a question ...
Why is $3$ the multiplicative coefficient in the Collatz conjecture?
What's the importance of multiplying an odd number by $3$ and adding $1$, instead of just adding $1$? After all, if you add $1$ to an odd number then it turns into an even number. Here is a example ...