On the solution of the Collatz problem
Shan-Guang Tan
TL;DR
The paper addresses the Collatz conjecture by introducing parametric Collatz sequences $S_c(m,t)$ with closed-form odd terms $n_i(m,t)=2^{m+1-i}3^{i}t-1$. It provides two proofs that every sequence reaches 1: a matrix-determinant approach that rules out nontrivial cycles in finite series and a range-based argument that propagates convergence through successive intervals $(N,\rho N]$ with $\rho=1+2^{-\kappa}$. Together these results claim to establish that all Collatz sequences terminate at 1, effectively solving the problem. The work thus combines exact algebraic representations of sequence elements with structural and probabilistic analyses to assert universal convergence to 1 for all starting values.
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.