Collatz Conjecture

The Collatz Conjecture, also known as the 3n + 1 conjecture, is a mathematical problem that has intrigued mathematicians for decades. The conjecture is named after the German mathematician Lothar Collatz, who first proposed it in 1937. The problem is deceptively simple, yet its resolution remains an unsolved mystery.

The conjecture involves a sequence of numbers derived from a specific set of rules applied to a starting positive integer \(n\):

1. If \(n\) is even, divide it by 2 (\(n/2\)).
2. If \(n\) is odd, multiply it by 3 and add 1 (\(3n + 1\)).

The process is then repeated with the resulting number. The conjecture suggests that, regardless of the starting positive integer, the sequence will eventually reach the value 1 and then continue in a loop of 4, 2, 1. This loop is often referred to as the "hailstone sequence" because of its erratic behavior.

Despite its simplicity, the Collatz Conjecture poses a profound challenge. While the conjecture has been verified for a vast range of starting values through extensive computer simulations, a general proof confirming that the sequence always converges to 1 remains elusive. The Collatz Conjecture is an example of a problem in number theory that is easy to state but notoriously difficult to prove or disprove. Its unsolved status adds to its fascination within the mathematical community.