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The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3''n'' + 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;〔According to Lagarias (1985, p. 4), the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to Syracuse University.〕 the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud),〔(【引用サイトリンク】url=http://mathworld.wolfram.com/HailstoneNumber.html )〕 or as wondrous numbers. 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 3''n'' + 1. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) 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. Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems."〔Guy (2004) p. 330〕 He also offered $500 for its solution.〔R. K. Guy: Don't try to solve these problems, Amer. Math. Monthly, 90 (1983), 35–41. By this Erdos means that there aren't powerful tools for manipulating such objects.〕 ==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 ''f'' 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.'' That smallest ''i'' such that ''a''''i'' = 1 is called the total stopping time of ''n''.〔Lagarias 1985.〕 The conjecture asserts that every ''n'' has a well-defined total stopping time. If, for some ''n'', such an ''i'' doesn't exist, we say that ''n'' has infinite total stopping time and the conjecture is false. 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 might enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Collatz conjecture」の詳細全文を読む スポンサード リンク
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