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Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3
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)
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. Preprints 2023, 2023040093. https://doi.org/10.20944/preprints202304.0093.v3 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.v3
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 ., where z, k, C1.Suppose the integer n violates Collatz conjecture by reappearing, then the equation modifies to . The article takes an elementary approach to this problem by calculating the maximum value of . Correspondingly, an upper limit on the integer n is placed that re-appears in the sequence. The limit is found to be . Next, it shown that the integer n repeats in the sequence if . 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.
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Commenter: Gaurav Goyal
Commenter's Conflict of Interests: Author
Secotions have been renamed.
A few changes are done to the abstract.