Questions tagged [collatz-conjecture]

For questions about the iterated map $n \to 3n+1$ if $n$ is odd and $n \to \frac n2 $ if $n$ is even, and its generalizations.

The Collatz conjecture asserts that every positive integer, when iterated over the function:

$$ f(n) = \begin{cases} \frac n2 & \text{if $n$ is even} \\\ 3n+1 & \text{if $n$ is odd} \end{cases} $$

will eventually be transformed to the cycle $1 \to 4 \to 2 \to 1$.

For example, $7 \to 22 \to 11\ \to 34 \to 17 \to 52 \to 26 \to 13 \to 40 \to \dots \to 5 \to 16 \to \dots \to 1$.

The Collatz conjecture has been verified for $n\le 19\cdot 2^{58}$ [Mathworld].

It may be generalized in multiple ways:

  • One way is to increase the domain on which it is defined, for example to the integers or real numbers. In the former case, it is conjectured that it eventually reaches one of $4$ cycles:

    1. $1 \to 4 \to 2 \to 1$,
    2. $-1 \to -2 \to -1$,
    3. $-5 \to -14 \to -7 \to -20 \to -10 \to -5$,
    4. $−17 \to −50 \to −25 \to −74 \to −37 \to −110 \to −55 \to −164 \to −82 \to −41 \to −122 \to −61 \to −182 \to −91 \to −272 \to −136 \to −68 \to −34 → −17, $

    This is sometimes called the generalized Collatz conjecture.

  • Another way is to change the definition to something of the form $$ f(n) = \begin{cases} \frac n2 & \text{if $n$ is even} \\\ an+b & \text{if $n$ is odd} \end{cases} $$ for fixed constants $a$ and $b$.

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Ways of disproving this "proof" of the Collatz Conjecture?

I jokingly suggested for someone to prove the Collatz Conjecture, and they came up with their own proof. I have no idea how to disprove proofs, so can anyone tell me either what is wrong with this proof or how I could have determined that…
Waffle
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Is there a new cycle for $f(n)$ in a "$1 x+3$"-variation of the Collatz-problem?

Let $f$ be defined as follows. $$ f(n) := \begin{cases} n+3 & \text {if $n$ is odd,} \\ \frac{n}{2} & \text {if $n$ is even.} \end{cases}$$ If we start at $n=15$, we get the following sequence by successive applications of…
Dumb user
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Can there be integer solutions (please PROVE)

With the following equation- $\frac{(3^x-2^x)}{(2^{(x+y)}-3^x)} = z $ What $x,y,z$ (all integers) satisfy this? The trivial solutions are- $X=1,y=1,z=1 ,X=2,y=1,z=-5 ,X=1,y=0,z=-1 $ This is my own “restatement” of the impossibility of the…
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Polynomial extension of Collatz graph... does it converge finitely?

Let $\Bbb Z[2^{-1},2^{-1/2},2^{-1/4,\ldots}]$ be the ring of dyadic rationals extended to include dyadic powers of $2$. Then let $2^{\nu_2(x)}$ extend the 2-adic valuation to dyadic powers of $2$ (using the rule given below). Let…
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Preserving historical information of the Collatz function?

In some sense this two equations are the same, namely $f_2$ preserves the historical information of $f_1^n$, where the exponent is function composition, but I am not sure how to show this rigorously. $f_1$ is well known, $ f_1(s)= \begin{cases} …
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Finding an equation for fixed points

as part of a project that I'm doing on fractals I was watching this video, in which it was stated that the fixed points of the function $$z\mapsto f(z)=\frac{(7z+2)-e^{i\pi z} (5z+2)}{4}$$ were given by $$ \lim_{n \rightarrow \infty } (z_n)…
Tom H
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Is it possible to find such a $l$,which that the known counter-example is only $l$?

Is it possible to find such a $l$,which that the known counter-example is only $l$, for any $k$, $f^k(n)≠1$? [Is there a modified Collatz rule 3n+$l$ where there is only one loop or "counter-example" and is not 3n+1?] $$f(n) = \begin{cases} n/2…
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Collatz Conjecture - Programmed in Python with Inf Result

Having written the following program in Python: #Collatz Conjecture #if n is odd, 3n+1 #if n is even, n/2 while True: print("**********") n=input("Which number would you like to test? ") i=1 print("**********") while n!=1: …
Nathan
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On the Collatz Conjecture with odd multiplier of five and with the starting value, n_0 = 7.

In the Collatz Conjecture, if the odd multiplier is changed from three to five, what is the outcome for a starting value of seven? Does the Collatz sequence converge to one or diverge? (Reference link: Collatz Data/Proof of Collatz Conjecture) We…
Dave
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About Collatz Conjecture

Please explain to me, what do Terence Tao and other mathematicians who write books about the Collatz Conjecture, what do they want to do in general? Do they want to prove this hypothesis, or disprove it? What can we generally conclude from these…
Student
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Looking for a verification or refutation my attempted proof of why the Collatz conjecture is probably false.

Most people think that the Collatz conjecture is true, but I think that I can prove the opposite. Let's make two functions, $f(x)$ and $g(x)$. $f(x) = $ The amount of numbers that can be solved in x steps. $g(x) = $ The amount of numbers that can be…
Mastrem
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