Version 1
: Received: 6 April 2023 / Approved: 6 April 2023 / Online: 6 April 2023 (11:21:24 CEST)
Version 2
: Received: 7 April 2023 / Approved: 10 April 2023 / Online: 10 April 2023 (08:40:06 CEST)
Version 3
: Received: 10 April 2023 / Approved: 11 April 2023 / Online: 11 April 2023 (10:05:24 CEST)
Version 4
: Received: 20 April 2023 / Approved: 21 April 2023 / Online: 21 April 2023 (09:25:33 CEST)
Version 5
: Received: 4 May 2023 / Approved: 5 May 2023 / Online: 5 May 2023 (10:18:23 CEST)
Version 6
: Received: 6 May 2023 / Approved: 9 May 2023 / Online: 9 May 2023 (04:15:37 CEST)
How to cite:
Goyal, G. Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3. Preprints2023, 2023040093. https://doi.org/10.20944/preprints202304.0093.v2
Goyal, G. Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3. Preprints 2023, 2023040093. https://doi.org/10.20944/preprints202304.0093.v2
Goyal, G. Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3. Preprints2023, 2023040093. https://doi.org/10.20944/preprints202304.0093.v2
APA Style
Goyal, G. (2023). Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3. Preprints. https://doi.org/10.20944/preprints202304.0093.v2
Chicago/Turabian Style
Goyal, G. 2023 "Resolution of the $3n+1$ Problem Using Inequality Relation Between Indices of 2 and 3" Preprints. https://doi.org/10.20944/preprints202304.0093.v2
Abstract
Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, it is written as $2^z = 3^kn + C$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by reappearing, then the equation modifies to $2^z n =3^kn +C$. The article takes an elementary approach to this problem by stating that the inequality $2^z > 3^k$ must hold for $n$ to violate the Collatz conjecture. It leads to the inequality $z > 3k/2$ that helps obtain bounds on the value of $3^k/2^z$ and $2^z - 3^k . It is found that the $3n+1$ series loops for $1$ and negative integers. Finally, it is proved that the $3n+1$ series shows pseudo-divergence but eventually arrives at an integer less than the starting integer.
Keywords
Collatz conjecture; 3n+1; inequality relations
Subject
Computer Science and Mathematics, Algebra and Number Theory
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.
Commenter: Gaurav Goyal
Commenter's Conflict of Interests: Author
Section on divergence of the 3n+1 is added.