This question is about the Collatz conjecture. Let $\Bbb{N}$ include $0$. The Collatz conjecture function is given by: $$ f: \Bbb{N} \to \Bbb{N}, \\ f(n) = \begin{cases} \dfrac{n}{2}, \text{ if } n = 0 \pmod 2,\\ \dfrac{3n + 1}{2}, \text{ if } n = 1\pmod 2 \text{ and } n \gt 1\\ \end{cases} $$ N...

Currently, iterates $f^i(\cdot)$ of the Collatz 3n+1 function form a monoid. Was wanting to do $f^{-i}(\cdot)$ but as you can imagine that is impossible because from for example $f(x) = f(y) = \dfrac{3x + 1}{2} = \dfrac{y}{2}$ we can't track whether it was the $x$ or the $y$ that when $f$ was ap...

I'm trying to see if I can find a fast factoring algorithm. I probably will not succeed, but it's a good exercise to at least try to find one with your current knowledge and to then find where it falls apart. I have noticed that you can easily use the quadratic formula to factor any given quadrat...

The odd integers and $0$ form a ringed quasigroup. Clearly, since any $2^ix \in \Bbb{Z}$ when passed into the Collatz function $f$ will iterate down to $f(x)$ since the function works first by dividing out $2$ if the number is even. Thus, it's clear that the Collatz conjecture is equivalent to $f...