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\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{Z}_\bot \to \mathbb{Z}_\bot) \to (\mathbb{Z}_\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}
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