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

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 $(n, k, z, p, q) = (1, 1, 2, 0, 0)$