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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131 views

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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1answer
28 views

Collatz sequences meet at a point

I was solving the problem,in which we are given two starting values of collatz sequence and our task is to say after how many steps their sequences “meet” for the first time. For ex - a= 7 , b= 8 a ...
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1answer
30 views

Given a natural number $a$ find its index in a set of structural descriptions

Looking at orbits of the collatz-like $(5x+1)/2^X$ - map I come to a useful structural description for all odd integers $a$. If I write $$ {5a+1 \over 2^A} \to b \qquad \qquad \text{for odd positive } ...
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1answer
117 views

Outline approach to Collatz 3n+1 conjecture / Criticism needed

This is a sketch of an approach to proving the Collatz 3n+1 conjecture true along the following lines. Instead of trying to show there are no loops and no sequences that increase without bounds, ...
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0answers
68 views

Is this a valid statement that would imply the Collatz Conjecture?

Let $f$ denote the Collatz transformation: $f(x) = \left\{ \begin{array}{ll} {x\over 2} & \quad x\equiv 0 \mod 2 \\ 3x+1 & \quad x \equiv 1\mod 2 ...
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0answers
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Is this first order version of the Collatz conjecture decidable in peano arithmetic?

Let $\phi(x)$ be a first order formula in the language of arithmetic with one free variable $x$. Consider the sentence $\psi_\phi$, defined as: $$\phi(0)\wedge \phi(1) \wedge (\forall x \phi(x) \to ...
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0answers
43 views

Alternate proofs for Collatz 1-Cycles

To follow up on my comment here, I present my proofs of the Collatz 1-Cycles. I have asked to make this cw so others can edit as they wish. Here we show that the differences are equal to $t-1:$ ...
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58 views

”Mehrstellenverfahren” of Collatz

I can do Taylor expansion on the Left hand side but I would like to know how to do Taylor expansion on the right side. Can anyone help me with finding the Taylor expansion for the double deivatives ...