On the Convergence of the Collatz Conjecture Sequence

Abstract

The Collatz conjecture, also known as the 3n + 1 problem, is a famous unsolved problem in mathematics. This work converts the conjecture dynamics into its corresponding difference equation. To determine the boundedness of the sequence, we commence with a boundedness analysis. This process allows us to identify a necessary and sufficient condition for the sequence to be bounded. Following this, we employ standard mathematical techniques to demonstrate conclusively that the 4-2-1 cycle is the only one. Additionally, we show that it is impossible for the sequence to diverge given any positive starting point. Ultimately, we demonstrate that the sequence invariably converges to 1.