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.v1
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.v1
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.v1
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.v1
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.v1
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 re-appearing, 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 $z > 2k$ that helps obtain the relations $3^k/2^z = 3/4 - p$ and $2^z - 3^k = 2^z/4 + q$, where $p, q$ are some variables. The values of $p, q$ are determined by substitution in the $2^zn = 3^kn + C$, and the solution found is $(n, k, z, p, q) = (1, 1, 2, 0, 0)$
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.
