All Questions
Tagged with collatz-conjecture recreational-mathematics
19 questions
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Convergence of infinite series generated by inverse Syracuse functions
NOTE: While this question was inspired by me playing with the Collatz Conjecture, it is not related to the Collatz Conjecture directly. If I should remove the tag, please let me know.
Define: $N = \{n ...
- 10.5k
3
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1
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An argument that the value of m in a Collatz conjecture 2-cycle is subject to the same constraints as for a 1-cycle.
Question: Is the reasoning of the following argument valid?
Objective: To derive a constraint on the permissible number of total divisions by 2, here referred to as $m$, for a Collatz 2-cycle ...
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8
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Is there a way to predict periodic behavior in Collatz conjecture 1-cycles?
This is a recreational math question.
If I use the equation for the smallest element in a 1-cycle,$$X_1 = \frac{3^n-2^n}{2^m-3^n}$$ with $n*ln_23 < m < n*ln_23 +1$,
peak values of this function ...
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Collatz-like problem involving prime factors
Unfortunately I am not well-versed in LaTeX so I will try my best to keep this looking presentable.
As an overview, I was investigating a variation of the Collatz conjecture:
Define $f(1) = 1$
Then, ...
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2
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A question about mathematical reasoning regarding observations in Collatz conjecture 1-cycles
This is a question about the use of empirical observations to guide strategies in constructing mathematical proofs. It uses prior discussions of 1-cycles in the Collatz conjecture as a model.
In a ...
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Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?
Let's consider the following variant of Collatz $(3n+1) : $
If $n$ is odd then $n \to n^2-1.$
$1\to 0.$
$3\to 8\to 1\to 0.$
$5\to 24\to 3\to 0.$
$7\to 48\to 3\to 0.$
$9\to 80\to 5\to 0.$
$11\to 120\to ...
- 22.6k
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Collatz Related? Are there any generalized rules for the following?
Following up on my last question, Last Question.
I have also noticed the following:
If $x \bmod 5=0$ : execute $x/5$, or elseIf $x$ ends in $1$ : execute $(x⋅2)+3$, or elseif $x$ ends in $3$ : execute ...
- 1,425
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290
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Jacobsthal numbers occurring in reduced cobweb plot for the Collatz Problem
I have been working on the Collatz problem for a while now, and have made this efficient cobweb plot function for it, where it automatically does all the dividing by two and always returns an odd ...
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Where is the flaw in my logic in this collatz conjecture idea?
Using this article from math stack exchange ( open problem - What does proving the Collatz Conjecture entail? - Mathematics Stack Exchange fourth answer down that starts "I think, for the ...
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Generating (almost?) all odd numbers for the 3n + 1 problem [closed]
The 3n + 1 problem
The $3n + 1$ problem can be described as a set of simple rules. For any positive integer apply the following two rules:
If the number is even: divide by 2
If the number is odd: ...
- 249
1
vote
1
answer
232
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These equations can generate integers that have the same total stopping time in the Collatz Conjecture. Has this been discovered?
Conjecture #1:
$a_(x,y)=(1/3)(2^{(2y+7)-2x)}(5(4^{x})-2)$
Generates positive even integers with a total stopping time S where $S=2y+13$, y is the set of all natural numbers and x is the set of all ...
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Are there specific total stopping times that are finite for the collatz conjecture? [closed]
For example, are there only certain groups of numbers that have a total stopping time of 6 or 30?
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Finding the stopping times or steps for positive integers in the Collatz conjecture using a formula
Is it possible to find a closed-form $a_n$ for some positive integers and use another formula to find the number of steps for all of those integers? If so, can you find multiple closed forms? For ...
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2
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229
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Smallest element of cycle of length $k$ in Collatz 3x+1 map?
In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $...
- 1,164
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Smallest number of a hypothetical second Collatz Cycle
Update/warning: The explaination below contains some obvious errors.
These errors have been worked on and turned this research into: Generating (almost?) all odd numbers for the 3n + 1 problem
Most ...
- 249
2
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1
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75
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(More efficiently) solving for residue classes $\pmod {2^A}$?
This is in context to a detail for the generalized Collatz-problem (generalized to various multipliers, like 3x+1, 5x+1, 7x+1, ... ,mx+1, ...)
I am currently looking at the 11x+1 problem , and as a ...
- 35.2k
1
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2
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937
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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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$5n+1$, $3n-1$ problem, smallest repeating cycle and Collatz conjecture
Among the Collatz conjecture we have other "similar" problems that are solved and have repeating cycles.
$5n+1$ has the repeating cycle $13, 66, 33, 166, 83, 416, 208, 104, 52, 26$, with a length of $...
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$N \equiv 3 (\textrm{mod } 4)$ and Collatz conjecture
Can the Collatz conjecture also be interpreted as behaviour, transformation of number of form $N\equiv 3(\textrm{mod }4)$ to the form of $N\equiv 1(\textrm{mod }4)$
Because integers of the form $N\...
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