The Collatz conjecture, introduced by Lothar Collatz in 1937, a longstanding unsolved problem, centers on a simple yet a puzzling question: Does repeatedly applying a specific rule to any positive integer always lead to the number 1 ? This paper explores the Collatz conjecture from two directions. First, we propose a novel approach to understanding the conjecture’s behavior, which involves constructing a framework where all Collatz sequences converge upwards (meaning their terms get larger) and define a function, g, for odd numbers. This function, however, generates sequences that diverge downwards (terms get smaller) within the structure. Notably, for each term in the divergent sequence produced by g, there exists a corresponding convergent sequence generated by the Collatz function itself. The paper then demonstrates that g diverges for any positive odd integer. These findings provide strong evidence in favor of the Collatz conjecture holding true for all positive integers. The second part of the paper delves into cryptography proposing a novel cryptosystem based on the chaotic nature of the Collatz sequence. In this system, a secret key pair is shared between the sender and the receiver. The message is encrypted using a large, odd number (part of the secret key) and a specific number of iterations of the Collatz sequence applied to that number. The chaotic property of the sequence ensures that even slight changes in the starting number significantly alter the resulting sequence, making the encryption process secure. This cryptosystem boasts advantages in its simplicity – it relies solely on multiplication and addition – and its efficiency. Furthermore, its potential lies in building even more secure cryptosystems through repeated encryption or integration with other encryption methods.