On the Fundamentals of Collatz Conjecture

2018 ◽  
Author(s):  
Joseph Olloh
Keyword(s):  
Collatz Conjecture

scholarly journals Orbits Theory. A Complete Proof of the Collatz Conjecture

2021 ◽  
Author(s):  
Jorge Crespo Alvarez
Keyword(s):  
Natural Number ◽  
Complete Proof ◽  
Collatz Conjecture

In this work a complete proof of the Collatz Conjecture is presented. The solution assumes as hypothesis that Collatz's Conjecture is a consequence. We found that every natural number n_i∈N can be calculated starting from 1, using the function n_i=((2^(i-Ω)-C))⁄3^Ω , where: i≥0 represents the number of steps (operations of multiplications by two subtractions of one and divisions by three) needed to get from 1 to n_i, Ω≥0 represents the number of multiplications by three required and 0≤C≤2^(i-⌊i/3⌋ )-2^((i mod 3)) 3^⌊i/3⌋ is an accumulative constant that takes into account the order in which the operations of multiplication and division have been performed. Reversing the inversion, we have obtained the function: ((3^Ω n_i+C))⁄2^(i-Ω)=1 that proves that Collatz Conjecture it’s a consequence of the above and also proofs that Collatz Conjecture it’s true since ((3^Ω n_i+C))⁄2^(i-Ω) is the recursive form of the Collatz’s function.

Xmark: Dynamic Software Watermarking Using Collatz Conjecture

2019 ◽  
Vol 14 (11) ◽  
pp. 2859-2874
Author(s):  
Haoyu Ma ◽  
Chunfu Jia ◽  
Shijia Li ◽  
Wantong Zheng ◽  
Dinghao Wu
Keyword(s):  
Collatz Conjecture ◽  
Dynamic Software ◽  
Software Watermarking

High-uncertainty audio signal encryption based on the Collatz conjecture

2019 ◽  
Vol 46 ◽  
pp. 62-69 ◽  
Author(s):  
Diego Renza ◽  
Sebastián Mendoza ◽  
Dora M. Ballesteros L.
Keyword(s):  
Audio Signal ◽  
Collatz Conjecture ◽  
Signal Encryption ◽  
High Uncertainty

About the Collatz conjecture

1998 ◽  
Vol 35 (2) ◽  
pp. 167-179 ◽  
Author(s):  
Ştefan Andrei ◽  
Cristian Masalagiu
Keyword(s):  
Collatz Conjecture

scholarly journals Divisions by Two in Collatz Sequences: A Data Science Approach

2020 ◽  
Vol 21 ◽  
pp. 1-13
Author(s):  
Christian Koch ◽  
Eldar Sultanow ◽  
Sean Cox
Keyword(s):  
Number Theory ◽  
Data Science ◽  
Theory Problem ◽  
Mathematical Methods ◽  
Collatz Conjecture ◽  
Science Approach ◽  
Specific Length

The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture.

scholarly journals Collatz conjecture revisited: an elementary generalization

2020 ◽  
Vol 12 (1) ◽  
pp. 112-127
Author(s):  
Amauri Gutierrez
Keyword(s):  
Natural Number ◽  
Collatz Conjecture

AbstractCollatz conjecture states that iterating the map that takes even natural number n to {n \over 2} and odd natural number n to 3n + 1, will eventually obtain 1. In this paper a new generalization of the Collatz conjecture is analyzed and some interesting results are obtained. Since Collatz conjecture can be seen as a particular case of the generalization introduced in this articule, several more general conjectures are also presented.

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