The Collatz Conjecture, also known as the 3n+1 conjecture, is a famous open problem named after Lothar Collatz.

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Collatz property implying infinite “fall below” trajectories, is it known?

(this was discovered analyzing Collatz empirically.) a key aspect of resolving Collatz involves looking at the number of iterations for trajectories to "fall below" the initial value. consider ...
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3n+1 problem and cycles

Just to make sure I am up to date with this problem. I know (or I think I do) that it is not yet proven that there are no non-trivial cycles for the collatz sequence (please correct me if I am wrong). ...
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Largest permutation groups without “non-mixing” subgroups

We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 ...
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Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
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Group with 2 orbits on the nonnegative integers — description of the orbits

Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition ...
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A curious sequence of rationals: finite or infinite?

Consider the following function repeatedly applied to a rational $r = a/b$ in lowest terms: $f(a/b) = (a b) / (a + b - 1)$. So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$. I am wondering if it is ...
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Point me to an attempt to Proove Collatz Conjecture by Substitution and Factor analysis? [closed]

Summary of Question: Where can I find a discussion of attempting to prove the Collatz Conjecture via substitution and abstract examination? I've done a lot of reading on the problem, including ...
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Implication for m-cycles in Collatz-type problems.

Background Consider Collatz-type problems of the form $an + 1$, where $a > 2$ is a positive, odd integer (e.g., $3n + 1$, $5n +1 $, $7n + 1$, etc.). For convenience, automatically divide by two. ...
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Larger cycle than 4, 2, 1 in Collatz iteration?

(Here I discuss the Collatz problem only for positive integers.) It is possible, by computation, to find all cycles in the Collatz iteration of a fixed length. It is clear that an increase must be ...
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Collatz related question [closed]

Howdy, Not sure this will be entirely clear, but when considering the relationship between a start value n in the Collatz algorithm and the length of the sequence generated by n, is there a function ...