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Article

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

This version is not peer-reviewed.

Submitted:

07 April 2023

Posted:

10 April 2023

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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 reappearing, 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 $z > 3k/2$ that helps obtain bounds on the value of $3^k/2^z$ and $2^z - 3^k . It is found that the $3n+1$ series loops for $1$ and negative integers. Finally, it is proved that the $3n+1$ series shows pseudo-divergence but eventually arrives at an integer less than the starting integer.
Keywords: 
Collatz conjecture; 3n+1; inequality relations
Subject: 
Computer Science and Mathematics  -   Algebra and Number Theory
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.

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