\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Number Theory\hfill Assignment 4.1  \\ Spring 2022 \hfill Due March 16}}}

\large

\vspace{1cm}

\begin{exercise}
Without appealing to the existence of irreducible/prime factorizations in $\N$, use strong induction to prove that every $a>1$ is divisible by an irreducible/prime.  This weaker result is sufficient to prove the infinitude of the set of primes using Euclid's familiar argument.
\end{exercise}

\begin{exercise}
Let $k \in \N$ and $a>1$.  Let $a = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r}$ be the canonical prime factorization of $a$.  Prove that $a$ is a $k$th power in $\N$ if and only if $k|e_j$ for all $j$.
\end{exercise}

\begin{exercise}
Given $a, b > 1$, explain why it is always possible to assume that the canonical prime factorizations of $a$ and $b$ involve exactly the same primes, provided we allow some of the exponents to be 0.
\end{exercise}

\begin{exercise}
Given $a, b > 1$, let
\[
\begin{aligned}
a &= p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r},\\[5pt]
b &= p_1^{f_1} p_2^{f_2} \cdots p_r^{f_r},\\[5pt]
\end{aligned}
\]
be the canonical prime factorizations of $a$ and $b$, as in the preceding exercise.  Prove that
\[
(a,b) = p_1^{\min \{e_1,f_1\}} p_2^{\min \{e_2, f_2 \}} \cdots p_r^{\min \{e_r, f_r \}}.
\]

\end{exercise}

\begin{exercise}
Choose an enumeration $p_1, p_2, p_3, \ldots$ of the primes.  Given $a \in \N$ write
\[
a = \prod_{i=1}^{\infty} p_i^{e_i(a)}
\]
for the canonical prime factorization of $a$, where $e_i(a) \ne 0$ for only finitely many $i$.  Prove that the ``unzipping'' function $u:\N \rightarrow \N \times \N$ defined by
\[
a \mapsto  \left( \prod_{i=1}^{\infty} p_i^{e_{2i}(a)} , \prod_{i=1}^{\infty} p_{i}^{e_{2i-1}(a)} \right)
\]
is a bijection.
\end{exercise}

\end{document}