What is the Collatz Conjecture in mathematics?

The Collatz Conjecture, also known as the 3n+1 conjecture, is an unsolved mathematical problem and one of the most famous and accessible unsolved problems in mathematics. It was first introduced by German mathematician Lothar Collatz in 1937 and is remarkably simple to describe:

Start with any positive integer n.
If n is even, divide it by 2 (n/2).
If n is odd, multiply it by 3 and add 1 (3n+1).
Repeat the process indefinitely.
The conjecture posits that, regardless of the starting value of n, this sequence of operations will eventually reach the number 1, and from there, it will continue in a loop of 4-2-1. In other words, the sequence will always terminate with the number 1, no matter what positive integer you start with.

For example, if you start with n = 6, the sequence would go: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

The Collatz Conjecture remains unproven, despite extensive computational testing for very large numbers. It has puzzled mathematicians for decades due to its deceptively simple yet elusive nature. The conjecture's apparent simplicity contrasts with the complexity of the behavior exhibited by the sequence for different values of n.

The Collatz Conjecture is not just a curiosity; it has connections to various areas of mathematics, including number theory and dynamical systems theory. Many mathematicians have attempted to prove or disprove it, but as of my last knowledge update in September 2021, the conjecture remains an open problem. Its unsolved status continues to captivate mathematicians and enthusiasts alike, making it a popular topic in recreational mathematics and computer programming for generating intriguing number sequences.