The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence. It is named after mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate. It is also known as the problem, the 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. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered US$500 for its solution. Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". Statement of the problem Consider the following operation on an arbitrary positive integer: If the number is even, divide it by two. If the number is odd, triple it and add one. In modular arithmetic notation, define the function as follows: Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next. In notation: (that is: is the value of applied to recursively times; ). The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. The smallest such that is called the stopping time of . Similarly, the smallest