Results 61 to 70 of about 509 (137)
Inverse Pairs and the Loop Structure of Collatz-Type Functions
We characterize a necessary structural condition for loops in Collatz-type functions of the form 3x + n. Working with an odd-only reduced map, we define inverse pairs as odd inputs whose net displacements under the map are equal in magnitude and opposite in sign.
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A Functional Approach to the Collatz Conjecture
The Collatz Conjecture, also known as the (3n+1) problem, is a longstanding unsolved problem in number theory. This paper explores a functional and logarithmic measure approach to the conjecture, combining Python-based experimentation, lambda calculus formulation, and log-space analysis.
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Collatz Conjecture and Gamma Function Analysis
Description: This research introduces a novel approach for analyzing natural numbers using a Gamma function, which generalizes and analyzes sequences generated by a Collatz-like iterative process. The Gamma function decomposes natural numbers into sequences of components and allows studying sequence patterns, sums, averages, and other properties.
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A Narrative Review of Speech and EEG Features for Schizophrenia Detection: Progress and Challenges. [PDF]
Bioengineering (Basel), 2023 Teixeira FL +5 more
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Collatz Conjecture Is Analogous to an Inverse Function of Natural Number
We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our
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A Stepwise Relation in the Collatz Function and the Termination of Numbers
This paper presents a formal theorem concerning the behavior of the Collatz function for all even integers. By establishing that for any even integer k=2x, the number of steps required to reach 1 satisfies the relation, steps(k) = steps(2) + steps(x). We provide a recursive structure that reduces the complexity of analyzing Collatz sequences. The proof
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Role of alcohol in urgent ambulance contacts in 15-24-year-olds in Copenhagen: a descriptive population-based study. [PDF]
BMJ Public Health, 2023 Bruun LS +8 more
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Structural Reduction of the Collatz Conjecture to the Dynamics of an Induced Function
This work develops a structural reformulation of the Collatz conjecture based on the decomposition of trajectories into maximal odd segments and the introduction of an induced map acting on their segment-starts. The conjecture is shown to be equivalent to an eventual descent property for this induced function.
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New results on the stopping time behaviour of the Collatz 3x + 1 function
, 2015 Let $σ_n=\lfloor1+n\cdot\log_23\rfloor$. For the Collatz 3x + 1 function exists for each $n\in\mathbb{N}$ a set of different residue classes $(\text{mod}\ 2^{σ_n})$ of starting numbers $s$ with finite stopping time $σ(s)=σ_n$. Let $z_n$ be the number of these residue classes for each $n\geq0$ as listed in the OEIS as A100982. It is conjectured that for
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