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.v4
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.v4
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.v4
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.v4
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.v4
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^z2^i}\right)^{-1} \frac{C}{2^z2^i}$. The article takes an elementary approach to this problem by calculating the bounds on the values of $\frac{C}{2^z2^i}$ and $\frac{3^k}{2^z2^i}$. Correspondingly, an upper limit on the integer $n$ is placed that can re-appear in the sequence. The integer $n$ is found to lie in the $(-\infty, 7]$ range. 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