Asian Research Journal of Mathematics

  • About
    • About the Journal
    • Submissions & Author Guideline
    • Accepted Papers
    • Editorial Policy
    • Editorial Board Members
    • Reviewers
    • Propose a Special Issue
    • Reprints
    • Subscription
    • Membership
    • Publication Ethics and Malpractice Statement
    • Digital Archiving Policy
    • Contact
  • Archives
  • Indexing
  • Publication Charge
  • Submission
  • Testimonials
  • Announcements
Advanced Search
  1. Home
  2. Archives
  3. 2019 - Volume 14 [Issue 2]
  4. Original Research Article

Submit Manuscript


Subscription



  • Home Page
  • Author Guidelines
  • Editorial Board Member
  • Editorial Policy
  • Propose a Special Issue
  • Membership

Proof of Collatz Conjecture

  • R. Deloin

Asian Research Journal of Mathematics, Page 1-18
DOI: 10.9734/arjom/2019/v14i230123
Published: 21 June 2019

  • View Article
  • Download
  • Cite
  • Statistics
  • Share

Abstract


Collatz conjecture (stated in 1937 by Collatz and also named Thwaites conjecture, or Syracuse, 3n+1 or oneness problem) can be described as follows:
Take any positive whole number N. If N is even, divide it by 2. If it is odd, multiply it by 3 and add 1. Repeat this process to the result over and over again. Collatz conjecture is the supposition that for any positive integer N, the sequence will invariably reach the value 1. The main contribution of this paper is to present a new approach to Collatz conjecture. The key idea of this new approach is to clearly differentiate the role of the division by two and the role of what we will name here the jump: a = 3n + 1. With this approach, the proof of the conjecture is given as well as informations on generalizations for jumps of the form qn + r and for jumps being polynomials of degree m >1. The proof of Collatz algorithm necessitates only 5 steps:


1- to differentiate the main function and the jumps;
2- to differentiate branches as well as their rst and last terms a and n;


3- to identify that left and irregular right shifts in branches can be replaced by regular shifts in 2m-type columns;
4- to identify the key equation ai = 3ni−1 + 1 = 2m as well as its solutions;
5- to reduce the problem to compare the number of jumps to the number of divisions in a trajectory.


Keywords:
  • Collatz
  • 3n 1
  • Syracuse
  • Thwaites
  • oneness
  • conjecture
  • even
  • odd
  • jumps
  • integer
  • Full Article - PDF
  • Review History

How to Cite

Deloin, R. (2019). Proof of Collatz Conjecture. Asian Research Journal of Mathematics, 14(2), 1-18. https://doi.org/10.9734/arjom/2019/v14i230123
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
  • Abstract View: 3260 times
    PDF Download: 2041 times

Download Statistics

Downloads

Download data is not yet available.
  • Linkedin
  • Twitter
  • Facebook
  • WhatsApp
  • Telegram
Make a Submission / Login
Information
  • For Readers
  • For Authors
  • For Librarians
Current Issue
  • Atom logo
  • RSS2 logo
  • RSS1 logo


© Copyright 2010-Till Date, Asian Research Journal of Mathematics. All rights reserved.