Formal Proof of the Collatz Conjecture Using Graph Theory

The Collatz conjecture, also known as the 3n + 1 problem, is one of the most famous unsolved problems in mathematics. Proposed by Lothar Collatz in 1937, the conjecture describes a sequence of simple operations applied to any positive integer. Despite its apparent simplicity, it has puzzled mathematicians for more than eight decades.

The process is as follows: starting with any positive integer n, if n is even, it is divided by 2; if it is odd, it is multiplied by 3 and 1 is added. This process is repeated with the resulting number. The Collatz conjecture asserts that, regardless of the initial number, the sequence will eventually reach the number 1.

Since its formulation, the conjecture has been the subject of intense research. Numerous mathematicians have proven it to hold for vast quantities of numbers, but a general proof that confirms the conjecture for all positive integers remains elusive. The difficulty lies in the unpredictable and chaotic nature of the generated sequences, which seem to defy any attempt at generalization.

In 1952, Bryan Thwaites independently posed the same conjecture, and since then it has been referred to by various names, including the Thwaites problem and the Syracuse problem. Over the years, many prominent mathematicians, such as Paul Erdős and John Conway, have contributed to its study, suggesting that its solution might require new mathematical tools yet to be discovered.

In the computing era, algorithms have verified the conjecture for numbers as large as 2^68, which amounts to more than 295 quintillion numbers. Despite this, the definitive mathematical proof remains elusive, making it a fascinating enigma that continues to inspire new generations of mathematicians.

The Collatz conjecture is not only a mathematical challenge but also a reminder of the mystery and inherent beauty of mathematics. Its history is a testament to the power of human curiosity and persistence in the pursuit of knowledge, waiting for the day when a definitive answer to this simple yet baffling problem is found.

In this article, we will show that it is possible to provide a formal proof of the Collatz Conjecture using graph theory applied to the Collatz sequence.

"Every positive integer has an odd parent using the representation of even and odd numbers in the Collatz sequence, and those relationships are unique and bidirectional between them." ~ Manuel Núñez.