Collatz conjecture
Statement of the problem

Consider the following operation on an arbitrary positive integer:

In modular arithmetic notation, define the function f as follows: f (n) = { n / 2 if n ≡ 0 (mod 2), 3 n + 1 if n ≡ 1 (mod 2). {\displaystyle f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}}

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: a i = { n for i = 0, f (a i − 1) for i > 0 {\displaystyle a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}} (that is: a is the value of f applied to n recursively i times; a = f (n)).

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each n {\displaystyle n}, there is some i {\displaystyle i} with a i = 1 {\displaystyle a_{i}=1}.

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 i such that a < a is called the stopping time of n. Similarly, the smallest k such that a = 1 is called the total stopping time of n. If one of the indexes i or k doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite.

The Collatz conjecture asserts that the total stopping time of every n is finite. It is also equivalent to saying that every n ≥ 2 has a finite stopping time.

Since 3 n + 1 is even whenever n is odd, one may instead use the "shortcut" form of the Collatz function: f (n) = { n 2 if n ≡ 0 (mod 2), 3 n + 1 2 if n ≡ 1 (mod 2). {\displaystyle f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}} This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.
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