# Collatz Problem

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.

## Properties of

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.