Proof of the Collatz Conjecture
Solved 2020: Several solutions to one of the most famous math problems !
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of Fractals
Dipl. El.-Ing. ETH
MAS MTEC ETH
fziegler@fziegler.ch
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of Fractals
Author
Franz ZieglerDipl. El.-Ing. ETH
MAS MTEC ETH
fziegler@fziegler.ch
orcid.org
Paul Erdös said about its difficulty: "Hopeless. Absolutely hopeless !"
Ziegler: It must finally be accepted that the Collatz Conjecture is easy to prove !
Some people asked me, if the German version with the title Beweis der Collatz-Vermutung is the same as Proof of the Collatz Conjecture published at Sept.04.2020. Yes, the English version is a translation of the original version in German (Mars/April 2020). Sorry for the delay over months, but nobody wanted to publish my "simple" ideas !
Hash (MD5): 124f19df36dc5549116ebe676e8809c5 (Timestamp: freetsa.org)
Hash (MD5): 1dc834111342d7c14cedef87f258a993 (Timestamp: freetsa.org)
Additional Tool: Calculated Matchstick List
Please Respekt Copyright Material !
Thanks for your great comments !!
Ziegler: It must finally be accepted that the Collatz Conjecture is easy to prove !
Some people asked me, if the German version with the title Beweis der Collatz-Vermutung is the same as Proof of the Collatz Conjecture published at Sept.04.2020. Yes, the English version is a translation of the original version in German (Mars/April 2020). Sorry for the delay over months, but nobody wanted to publish my "simple" ideas !
Proof of the Collatz Conjecture in 3 Steps
The most simple approach
| Author: Franz Ziegler | Date: Dec.19.2020 | 3 pages, 5 figures | Show PDF |
Abstract
This paper shows the proof of the Collatz conjecture in three simple steps. The main idea of the proof is to represent the Collatz number system as a compilation of loop-free, non-overlapping, single-plug fragments of Collatz numbers series. After showing in the first step, that the Collatz number system is complete, containing every natural number exactly once, the second step shows how the unbranched loop-free and single-plug building blocks can be realized with a few rules added to the standard Collatz rules. In a third step, the set of rays covering the full natural number space is assembled by connecting the rays together, arising no possible loops.Algebra of a Branch
A New Model for the Proof of the Collatz Conjecture
| Author: Franz Ziegler | Date: Dec.15.2020 | 7 pages | Show PDF |
Summary
This paper is focused on the representation of the Collatz number system with loop-free unbranched fragments of Collatz number series. In contrast to the previous works [01] and [02], where the proof of the Collatz conjecture was found by splitting the reduced odd-numbered Collatz tree only, this work used the whole space of natural numbers. To get a loop-free and unbranched representation of the Collatz number system, the model, which initially only consisted of the simple Collatz rules, had been gradually enhanced by additional constraints, until the consistent "Matchstick Model" was found and thus had confirmed the Collatz conjecture again. The found model constraints are corresponding the method proposed in [01] and [02], where the Collatz number tree was cut along the proxy path. Because of it's simplicity, any proof of the Collatz conjecture could lead to the same fundamental numbers and constraints. - With the Matchstick Model the most minimalistic model was found to split the Collatz tree into loop-free, unbranched fragments of Collatz number series.Additional Tool: Calculated Matchstick List
A Proof of the Collatz Conjecture by Synthesis
| Author: Franz Ziegler | Date: Dec.08.2020 | 4 pages | Show PDF |
Summary
All natural odd numbers are contained in the infinite Collatz tree and are defined as members of a LN terminated chain module too. The proposed definition and configuration of such a module has turned out to be an valuable enclosure for the synthesis of the building blocks of the Collatz tree. A logical chain of conditions was used to adapt the Collatz rules and customizing the definition range of the operations in such a way, that the Collatz rules can be uses like bijective functions. The paper not only confirms the Collatz Conjecture but also gives insight into some important key points of theCollatz number structureProof of the Collatz Conjecture
| Author: Franz Ziegler | Date: Nov.02.2020 | Size (A4): 10 | Show PDF |
Summary
The present work contains a proof of the simply formulated mathematical problem known as the Collatz-Syracuse-Ulam problem, which has so far resisted any solution. It is constructed in the concept steps order, reduction, analysis, idea and ends with a proof by synthesis. The basic idea of this proof is to cut the Collatz tree of natural numbers into unique chains of number sequences having all an ending element. All steps are comprehensible without extensive mathematical knowledge and are supplemented with diagrams and tables.Understanding the Proof of the Collatz Conjecture
| Author: Franz Ziegler | Date: Dec.06.2020 | 4 pages | Show PDF |
Summary
With the proxy decomposition method presented in "Proof of the Collatz Conjecture" [01], a valuable model for describing a Collatz tree as a composition of forkfree terminated chains was found. This paper is a short lecture on the same theme and contains a more concentrated elaboration of the basic considerations. For that, the provided analysis-, ordering- and reduction-schemes for the Collatz tree [01] are skipped and the investigations starts directly with the reduced, odd numbered Collatz tree. A new Definition called "evolution chain" will be proposed characterizing the set of conditions needed for possibly any proof of the Collatz Conjecture. The paper confirms the principal correctness of the "Proof of the Collatz conjecture" [01] using slightly modified and terminologically enhanced methods.Please Respekt Copyright Material !
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