Collatz conjecture

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The Collatz conjecture is an unsolved conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem;[1] the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers,[2] or as wondrous numbers.[3]

Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO[4]) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.[5]

Paul Erdős said about the Collatz conjecture: "Mathematics is not yet ready for such problems." He offered $500 for its solution.[6]

In 2006, researchers Kurtz and Simon, building on earlier work by J.H. Conway in the 1970s,[7] proved that a natural generalization of the Collatz problem is undecidable.[8] However, as this proof depends upon the generalization, it cannot be applied to the original Collatz problem.

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