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diff --git a/entries/markusde/collatz/collatz.pdf b/entries/markusde/collatz/collatz.pdf Binary files differnew file mode 100644 index 0000000..efec35b --- /dev/null +++ b/entries/markusde/collatz/collatz.pdf diff --git a/entries/markusde/collatz/collatz.png b/entries/markusde/collatz/collatz.png Binary files differnew file mode 100644 index 0000000..afc8d8f --- /dev/null +++ b/entries/markusde/collatz/collatz.png diff --git a/entries/markusde/collatz/collatz.tex b/entries/markusde/collatz/collatz.tex new file mode 100644 index 0000000..8e18088 --- /dev/null +++ b/entries/markusde/collatz/collatz.tex @@ -0,0 +1,21 @@ +\documentclass[12pt]{article} +\usepackage[landscape]{geometry} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amsthm} +\renewcommand{\phi}{\varphi} +\begin{document} + +\noindent The least fixed point of +\begin{align*} + h & : (\mathbb{N}_\bot \to \mathbb{Z}_\bot) \to (\mathbb{N}_\bot \to \mathbb{Z}_\bot) \\ + h(F)(n) & = + \begin{cases} + 1 & n = 1 \\ + (\phi^{n/2} + (-\phi)^{-n/2})\, F_{n/2} & n \textrm{ is even} \\ + \sqrt[3]{\frac{\phi^{3n+1}-\phi F_{3n+1}}{10} + \frac{\sqrt{25 \phi^{3n+1} - 25\phi F_{3n+1} - 20}}{50}} + \sqrt[3]{\frac{\phi^{3n+1}-\phi F_{3n+1}}{10} - \frac{\sqrt{25 \phi^{3n+1} - 25\phi F_{3n+1} - 20}}{50}} & \textrm{otherwise} \\ + \end{cases} +\end{align*} +is the Fibonacci sequence if and only if the Collatz conjecture is true. + +\end{document} |
