Replace individual page numbers with mathematical expressions

Question

I'd like to replace a few individual page numbers in a document with mathematical expressions, like $a^0$ for page 1, $2(2)^1$ for 4, etc. I don't have a ToC or index, so I'm not concerned about the actual page numbers changing; just what's printed at the bottom of the page.

I'm not sure it's helpful, but here's a MWE. Instead of a 1 at the bottom of the page, I want $a^0$.

\documentclass{article}

\usepackage{amsmath}

\begin{document}
    Body of page, which is about exponential functions
\end{document}

Answer 1

Macro \thepage can be redefined to give the desired representation of the page number:

\documentclass{article}
\renewcommand*{\thepage}{%
  \ifcase\value{page}%
    0%
  \or
    $a^0$%
  \or
    $2^1$%
  \or
    3%
  \or
    $2^2$%
  \else
    \ifnum\value{page}=9 %
      $3^3$
    \else
     \arabic{page}%
    \fi
  \fi
}
\begin{document}
\section{My document}
\label{sec:doc}
Body of page, which is about exponential functions.\\
Reference to section \ref{sec:doc} at page \pageref{sec:doc}.
\end{document}