Version 1
: Received: 18 January 2024 / Approved: 19 January 2024 / Online: 19 January 2024 (13:14:04 CET)
Version 2
: Received: 30 January 2024 / Approved: 30 January 2024 / Online: 30 January 2024 (12:51:11 CET)
How to cite:
Khoshakhlagh Varnosfaderani, M. A Proof of Collatz Conjecture Using Pyramid Fractions. Preprints2024, 2024011488. https://doi.org/10.20944/preprints202401.1488.v2
Khoshakhlagh Varnosfaderani, M. A Proof of Collatz Conjecture Using Pyramid Fractions. Preprints 2024, 2024011488. https://doi.org/10.20944/preprints202401.1488.v2
Khoshakhlagh Varnosfaderani, M. A Proof of Collatz Conjecture Using Pyramid Fractions. Preprints2024, 2024011488. https://doi.org/10.20944/preprints202401.1488.v2
APA Style
Khoshakhlagh Varnosfaderani, M. (2024). A Proof of Collatz Conjecture Using Pyramid Fractions. Preprints. https://doi.org/10.20944/preprints202401.1488.v2
Chicago/Turabian Style
Khoshakhlagh Varnosfaderani, M. 2024 "A Proof of Collatz Conjecture Using Pyramid Fractions" Preprints. https://doi.org/10.20944/preprints202401.1488.v2
Abstract
In this paper, I introduce a new concept of representing numbers in base ; in other words, I have found new series for any number similar to series that could be written according to the collatz sequence which is called zi(n) in this article. These series need to end with 1. Then, we use two sets of rules to make a diagram which then proves the existence of such series for any number.In this diagram, by branching numbers into different branches in accordance to the modularity of 4 and continuing branching to the point that their numbers have enough common terms in their collatz series, we could reduce zi(n) to zi(k) so that k<n in any branch by Rule Number One. This diagram shows that zi(n) exists for every number, in other words, this proves the theorem (A) or zi-existence theorem. The proof of zi-existence theorem leads to a proof of the collatz conjecture because the collatz series could be written as a linear combination of such series that all of them end with 1, so the collatz series for any number ends with 1.
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
