In 1937 Lothar Collatz proposed the 3n+1 conjecture (known by a long list of aliases), is stated as follows.
First, we define the function on the set of positive integers:
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:
, and . Believe it or not, this is the restriction to the positive integers of the complex-valued map .
The 3n+1 conjecture is: The sequence
will eventually reach the number 1, regardless of which positive integer is chosen initially.
This is still unsolved, though a lot of people have worked on it. For a recent survey of results, see the paper by Chamberland.