Yuki Wang '23
Collatz Conjecture
Named after Lothar Collatz, the Collatz conjecture is a conjecture that defines a sequence: starting with any positive integer n, 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. No matter what the value of n is, the sequence will always end up being 1. It can be written in arithmetic notation as:
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Form a sequence by performing this operation repeatedly, beginning with any positive integer, we can write:
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That is: a_i is the value of f applied to n recursively i times; a_i = f^i (n)The smallest i such that a_i < a_0 is called the stopping time of n. Similarly, the smallest k such that a_k = 1 is called the total stopping time of n. If one of the indexes i or k doesn't exist, the stopping time or the total stopping time is infinite. According to the conjecture, the total stopping time of every n is finite, which can also be said that every n>=2 has a finite stopping time. However, the conjecture has not been verified for every number; instead, only numbers less than 2^68 have been proved to follow this sequence.