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\title{A Note on Prime Numbers}
\author{Jane Mathematician}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This short note presents a simple result about prime numbers and their distribution.
\end{abstract}

\section{Introduction}

Prime numbers have fascinated mathematicians for centuries. In this note, we present a simple result about prime numbers.

\section{Main Results}

\begin{theorem}
There are infinitely many prime numbers.
\end{theorem}

\begin{proof}
Suppose there are only finitely many primes $p_1, p_2, \ldots, p_n$. Consider the number
\[
N = p_1 \cdot p_2 \cdot \ldots \cdot p_n + 1
\]
This number is not divisible by any of the primes $p_i$, since division would leave a remainder of 1. Therefore, either $N$ is itself prime, or it has a prime factor different from all $p_i$. Either way, we have a contradiction.
\end{proof}

\begin{lemma}
If $p > 3$ is prime, then $p^2 \equiv 1 \pmod{24}$.
\end{lemma}

\begin{proof}
If $p > 3$ is prime, then $p$ is of the form $6k \pm 1$ for some integer $k$. Thus $p^2 = (6k \pm 1)^2 = 36k^2 \pm 12k + 1 = 24(\frac{3k^2 \pm k}{2}) + 1$, which is congruent to 1 modulo 24.
\end{proof}

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\title{Introduction to Number Theory}
\author{John Smith}

\begin{document}

\frontmatter
\maketitle

\chapter{Preface}
\lipsum[1]

\tableofcontents

\mainmatter
\chapter{Divisibility and Prime Numbers}

\section{Basic Definitions}

\begin{definition}
An integer $p > 1$ is called a \emph{prime number} if its only positive divisors are 1 and $p$.
\end{definition}

\begin{theorem}[Fundamental Theorem of Arithmetic]
Every integer greater than 1 can be expressed as a product of primes in a unique way, up to the order of the factors.
\end{theorem}

\begin{proof}
(Sketch) The proof proceeds in two parts: existence and uniqueness...
\end{proof}

\begin{remark}
This theorem forms the foundation of number theory and has numerous applications.
\end{remark}

\section{Properties of Prime Numbers}
\lipsum[2]

\chapter{Congruences}
\lipsum[3]

\backmatter
\chapter{Bibliography}

\begin{thebibliography}{9}
\bibitem{Hardy} G.H. Hardy and E.M. Wright, \emph{An Introduction to the Theory of Numbers}, Oxford University Press, 2008.
\bibitem{Ireland} K. Ireland and M. Rosen, \emph{A Classical Introduction to Modern Number Theory}, Springer, 1990.
\end{thebibliography}

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\begin{document}

\title{Recent Advances in Algebraic Topology}

\author{Alice Johnson}
\address{Department of Mathematics, University of Example, Exampleville, EX 12345}
\email{ajohnson@example.edu}

\author{Bob Smith}
\address{Institute for Advanced Study, Theoremville, TH 54321}
\email{bsmith@theory.edu}

\subjclass[2020]{Primary 55N15; Secondary 55P42, 57R19}
\keywords{Cohomology operations, homotopy theory, spectral sequences}

\begin{abstract}
We present new results on the computation of stable homotopy groups using novel spectral sequence techniques. Our approach simplifies several existing proofs and leads to new insights in algebraic topology.
\end{abstract}

\maketitle

\section{Introduction}

The computation of stable homotopy groups of spheres remains one of the central problems in algebraic topology. In this paper, we present a new approach based on...

\section{Preliminaries}

We begin by recalling some basic definitions and results from algebraic topology.

\begin{theorem}
For any CW complex $X$, there is a natural isomorphism
\[
H^n(X; G) \cong [X, K(G,n)]
\]
where $K(G,n)$ is an Eilenberg-MacLane space and $[X, K(G,n)]$ denotes the set of homotopy classes of maps from $X$ to $K(G,n)$.
\end{theorem}

\section{Main Results}

Our main contribution is the following result...

\begin{thebibliography}{10}
\bibitem{Adams} J.F. Adams, \emph{Stable Homotopy and Generalised Homology}, University of Chicago Press, 1974.
\bibitem{Hatcher} A. Hatcher, \emph{Algebraic Topology}, Cambridge University Press, 2002.
\end{thebibliography}

\end{document}