Define the following function on : The Collatz conjecture says that, for any positive integer , the sequence contains 1. This conjecture is still open. Some people have described it as the easiest unsolved problem in mathematics.
Self similarity of follows from generalizing to an integral, integer coefficient polynomial. If for example, it can be shown by parity argument, that has the same parity as . It then follows, that same conditional path will be followed by as it was for ; any time the lead coefficient still has a factor of 2.
Observing that if then , as well as: we can then observe that; only if is even will another division by 2 be possible.
The above observation leads to 2 important points about ; namely they are the only possible lowest elements of a non-trivial cycle, and also the only possible lowest elements of an infinitely increasing sequence.