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\noindent{\Large{\textsc{Number Theory \hfill Assignment 3.2  \\ Fall 2023 \hfill Due September 13}}}

\large

\vspace{1cm}

\begin{exercise}
Let $n \in \N$, $n \ge 2$.  Suppose $n$ has the property that whenever $n|ab$ for some $a, b \in \N$, then $n|a$ or $n|b$.  Prove that $n$ is prime.  [{\em Suggestion.}  If $n=ab$, then $n|ab$.]
\end{exercise}

\begin{exercise}
Use the Fundamental Theorem of Arithmetic to prove that if $p$ is prime, then $\sqrt{p}$ is irrational.
\end{exercise}

\begin{exercise}
Show that every odd prime is of the form $4k \pm 1$.
\end{exercise}

\begin{exercise}
Prove that there are infinitely many primes of the form $4k-1$ as follows.

\begin{itemize}

\item[\bf a.]  Show that if $n_1, n_2, \ldots, n_r \in \Z$ all have the form $4k+1$, then so does $n_1 n_2 \cdots n_r$.  [{\em Suggestion.}  Use induction.]

\item[\bf b.]  Suppose that $n \in \N$ has the form $4k-1$.  Prove that $n$ must have a prime factor also of the form $4k-1$.

\item[\bf c.]  Suppose $p_1, p_2, \ldots , p_r$ are primes of the form $4k-1$ and let $N = 4p_1 p_2 \cdots p_r -1$.  Prove that $N$ has a prime factor of the form $4k-1$ that is different from $p_1, p_2, \ldots, p_r$.   

\item[\bf d.]  Conclude that there are infinitely many primes of the form $4k-1$.

\end{itemize}

\end{exercise}

\vspace{0.5cm}

\noindent{\bf Remarks.}

\begin{itemize}

\item[\bf 1.]  The label ``prime'' for those $n \ge 2$ with the property that $n=ab$ implies $n=a$ or $n=b$ is entirely standard and completely mysterious to me.  More meaningful terms might be ``irreducible'' or ``atomic,''  since these more naturally suggest that the prime numbers cannot be factored nontrivially.  Indeed, this is the terminology preferred in ring theory.

\item[\bf 2.]  Exercise 1 shows that an integer $n$ is prime {\em if and only if} $n$ has the property that for any $a,b \in \Z$, $n|ab$ implies $n|a$ or $n|b$.  In more general rings the latter property is what one calls ``prime.''  While it is still true that every prime is irreducible in an arbitrary ring, the converse need not hold.  The simplest example is the ring 
\[
\Z[\sqrt{-5}] = \left\{ a + b \sqrt{-5}\, | \, a, b \in \Z   \right\},
\]
although the proof is nontrivial.

\item[\bf 3.]  One can modify the argument in Exercise 4 to prove that there are also infinitely many primes of the form $4k+1$, but this requires the theory of quadratic residues.

\end{itemize}

\end{document}
