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.v6
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.v6

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.v6

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.v6

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.v6

Abstract

Collatz conjecture states that an integer $n$ reduces to $1$ when certain simple operations are applied to it. Mathematically, the Collatz function is written as $f^k(n) = \frac{3^kn + C}{2^{z}}$, where $z, k, C \geq 1$. Suppose the integer $n$ violates Collatz conjecture by reappearing as $2^in$, where $i \geq 1$, then the equation modifies to $n \left(1 - \frac{3^k}{2^{z}2^i}\right) = \frac{C}{2^{z}2^i}$. The article takes an elementary approach to this problem by calculating the bounds on the values of $\frac{C}{2^{z}2^i}$ and $1 - \frac{3^k}{2^{z}2^i}$. Correspondingly, an upper limit on the integer $n$ is placed that can re-appear in the sequence. The integer $n$ lies in the $(-\infty, 5)$ range, and the limit on the number of odd steps is $k < 3$. Finally, it is shown that no integer chain exists that does not lead to 1.

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