$C(n) = \frac{n}{2}$ if $n$ is even, or $3n + 1$ if $n$ is odd This function is normally applied to constant numbers, but it can be used on linear polynomials $ax + b$. There are 3 possibilities of what $ax + b$ can be, it can be always even, it can be always odd, or it can vary depending on the ...

The proof works as follows. If n \in N diverges to infinity, it must do so under an aperiodic pattern. We know from the Collatz function that it is not allowed to have two consecutive odd numbers in a Collatz sequence, however, we prove using a result from Mercas that an infinite aperiodic binary...