On the solution of the Collatz problem

TL;DR

This paper proves that all Collatz sequences eventually reach 1 by explicitly expressing all odd numbers and demonstrating the sequence's convergence, thus claiming to solve the longstanding Collatz problem.

Contribution

The paper introduces a unique sequence representation for all odd numbers and proves that every Collatz sequence ultimately reaches 1, claiming a solution to the problem.

Findings

All Collatz sequences return to 1.

Explicit formula for all odd numbers in Collatz sequences.

Proof that the Collatz conjecture is true.

Abstract

In this paper, we first prove that given a nonnegative integer $m$ and an odd number $t$ not divisible by $3$, there exists a unique Collatz's Sequence \[ S_{c}(m,t)=\{n_{0}(m,t),n_{1}(m,t),n_{2}(m,t),\ldots,n_{m}(m,t),n_{m+1}(m,t)\} \] produced by a function $n_{i+1}(m,t)=(3n_{i}(m,t)+1)/2$ for $i=0,1,2,\ldots,m$ and ended by an even number $n_{m+1}(m,t)$ where $n_{i}(m,t)=2^{m+1-i}\times3^{i}t-1$ for $i=0,1,2,\ldots,m+1$, by which all odd numbers can be expressed. Then we discuss the Collatz problem in two ways and prove that each Collatz's Sequence always returns to 1, i.e., the Collatz problem is solved.