Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Gaurav Goyal

doi:10.20944/preprints202304.0093.v3

Preprint Article Version 3 Preserved in Portico This version is not peer-reviewed

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)

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^{k} n + C}{2^{z}}

, where z, k, C

\geq

1.Suppose the integer n violates Collatz conjecture by reappearing, then the equation modifies to

n = \frac{3^{k}}{2^{z}} n + \frac{C}{2^{z}}

. The article takes an elementary approach to this problem by calculating the maximum value of

\frac{C}{2^{Z}}

. Correspondingly, an upper limit on the integer n is placed that re-appears in the sequence. The limit is found to be

n < 3.5

. Next, it shown that the integer n repeats in the sequence if

n < 0

. Finally, it is shown that no integer chain exists that does not lead to 1.

Collatz conjecture; 3n+1; inequality relations

Computer Science and Mathematics, Algebra and Number Theory

Comment 1

Received: 11 April 2023
Commenter: Gaurav Goyal
Commenter's Conflict of Interests: Author

Comment: The confusion caused by using terms like converging and diverging is removed.
Secotions have been renamed.
A few changes are done to the abstract.

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Resolution of the 3n+1 Problem Using Inequality Relation Between Indices of 2 and 3

Abstract

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