If you are still looking for an answer, take a look at my code.
\documentclass[a4paper,12pt,twoside,leqno]{article}
\usepackage[marginratio={4:6, 5:7}, textwidth=121mm, noheadfoot]{geometry}
\usepackage{amsmath}
%\usepackage{amssymb}
%\usepackage{mathtools}
%\usepackage{mathrsfs}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{mathabx}
\usepackage{mathspec}
\defaultfontfeatures{Mapping=tex}
\defaultfontfeatures{Numbers=Proportional, WordSpace =1.6}
\setmainfont{Century Old Style}
\setmathsfont(Digits){Century Old Style}
\setmathsfont(Latin){Century Old Style}
%\setmathsfont(Greek){Century Old Style}
\newfontfamily{\Times}{Times}
\newfontfamily{\CenturyOldStyle}{Century Old Style}
\newfontfamily{\CenturyOldStyleStd}{Century Old Style Std}
\newfontfamily{\MinionPro}{Minion Pro}
\newfontfamily{\OldStandardTT}{Old Standard TT}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsthm}
\usepackage{thmtools}
\declaretheoremstyle[%
spaceabove=\topsep,
spacebelow=\topsep,
headfont={\scshape\MinionPro},
bodyfont=\itshape,
notefont=\normalfont,
notebraces={(}{)},
headformat=\NAME,
headindent=\parindent
]{theorem}
% Theorem
\declaretheorem[style=theorem,name=Theorem,numberwithin=section]{theorem}
% Corollary
\declaretheorem[style=theorem,name=Corollary,sibling=theorem]{corollary}
% Lemma
\declaretheorem[style=theorem,name=Lemma,sibling=theorem]{lemma}
% Definition
\declaretheorem[style=theorem,name=Definition,sibling=theorem]{definition}
% Proposition
\declaretheorem[style=theorem,name=Proposition,sibling=theorem]{proposition}
% Property
\declaretheorem[style=theorem,name=Property,sibling=theorem]{property}
\let\proof\relax
\let\endproof\relax
\declaretheoremstyle[%
spaceabove=0pt,
spacebelow=\lineskip,
headfont={\scshape\MinionPro},
bodyfont=\normalfont,
notefont=\normalfont,
notebraces={(}{)},
headpunct={.},
headformat=\NAME,
headindent=\parindent
%qed={\raisebox{-\baselineskip}{\llap{Q.e.d.}}}%
]{proof}
% Proof
\declaretheorem[style=proof,name=Proof]{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{A PRIME-REPRESENTING FUNCTION}
\author{W. H. MILLS}
\date{}
\begin{document}
\maketitle
A function $f(x)$ is said to be a prime-representing function if $f(x)$ is a prime number for all positive integral values of $x$. It will be shown that there exists a real number $A$ such that $[A^{8^{x}}]$ is a prime-representing function, where $[R]$ denotes the greatest integer less than or equal to $R$.
Let $p_n$ denote the $n$th prime number. A. E. Ingham\textsuperscript{1} has shown that
\begin{equation}
p_{n+1} - p_n < Kp_{n}^{5/8}
\end{equation}
where $K$ is a fixed positive integer.
\begin{lemma}
If $N$ is an integer greater than $K^8$ there exists a prime $p$ such that $N^8 < p < (N+1)^3 - 1$.
\end{lemma}
\begin{proof}
Let $p_n$ be the greatest prime less than $N^3$. Then
\begin{equation}
\begin{split}
N^3 < p_{n+1} &< p_n + Kp_{n}^{5/8} < N^3 + KN^{15/8} < N^3 + N^2 \\
&< (N + 1)^3 -1
\end{split}
\end{equation}
\end{proof}
Let $P_0$ be a prime greater than $K^8$. Then by lemma we can construct an infinite sequence of primes, $P_0, P_1, P_2, \dots ,$ such that $P_{n}^{3} < P_{n+1} < (P_n + 1)^8 -1$. Let
\begin{equation}
u_n = P^{3 - n}_{n}, \qquad v_n = (P_n + 1)^{3-n}.
\end{equation}
Then
\begin{equation}
v_n > u_n, \qquad u_{n+1} = P_{n+1}^{3-n-1} > P_{n}^{3-n} = u_n,
\end{equation}
\vspace{-0.8cm}
\begin{equation}
v_{n+1} = (P_{n+1} + 1)^{3-n-1} < (P_n + 1)^{3-n} = v_n .
\end{equation}
It follows at once that the $u_n$ form a bounded monotone increasing sequence. Let $A = \lim_{n \to \infty} u_n$.
\begin{theorem}
$[A^{3^{n}}]$ is a prime-representing function.
\end{theorem}
\begin{proof}
From (4) and (5) it follows that $u_n < A < v_n$, or $P_n < A^{3^{n}} < P_n +1$.
Therefore $[A^{3^{n}}] = P_n$ and $[A^{3^{x}}]$ is a prime-representing function.
\end{proof}
\end{document}
Some Remarks.
1.- Since it uses mathspec XeLaTeXcompilation is needed.
2.- You need to have installed Century Old Style and Minion Pro fonts. You will have the latest if you have Adobe.
3.- Finally, I can't get small caps for Century Old Style, so I have used Minion Pro.
I hope you like it.
