# Questions tagged [collatz-conjecture]

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

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### Collatz Conjecture and Hailstone Sequence [closed]

Is it proved that the function in collatz conjecture always produce a hailstone sequence?
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### A possibly easy question about latent geometry in Collatz sequences

I have a question about some (seemingly unimportant) behavior I noticed in Collatz sequences, which I haven't been able to find a general answer to upon rough scan of the literature (please be aware ...
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### The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd

This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here. We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
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### Two reasons why the Collatz conjecture could fail

Let $\mathbb{N}$ denote the set of positive integers. The Collatz function $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we ...
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### For which primes does this iterated function act transitively? (Sort of a finite analogue of Collatz conjecture.)

Background: I was trying to prove something having to do with cyclic group actions on matroids and was able to show that what I want holds if a particular elementary-looking number-theoretic property ...
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### Is there a known Turing machine which halts if and only if the Collatz conjecture has a counterexample?

Some of the simplest and most interesting unproved conjectures in mathematics are Goldbach's conjecture, the Riemann hypothesis, and the Collatz conjecture. Goldbach's conjecture asserts that every ...
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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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### Collatz stopping-time and Poisson distribution, and connection to other problems?

I read many threads about Collatz here - so don't worry, this is no attempt to any proof, just asking about a curious fact: This graph gives the stopping-time of Collatz sequences up to $n=10^8$ (...
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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 cycles (of some length $m$) in Collatz-type problems: typical ratio between largest and smallest element?

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. ...