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\noindent{\Large{\textsc{Number Theory \hfill Assignment 4.1  \\ Fall 2023 \hfill Due September 20}}}

\large

\vspace{1cm}

\begin{exercise}
Let $n \in \N$.  If $\sqrt{n} \in \Q$, then we can write $\sqrt{n} = \frac{a}{b}$ with $a,b \in \N$.  Squaring both sides of this equation and clearing denominators yields $nb^2 = a^2$.    
\begin{itemize}

\item[\bf a.]  Let $\{ p_1, p_2, \ldots , p_r \}$ be the set of prime factors of $abn$.  Express $a$, $b$ and $n$ in modified canonical form using these primes and conclude that $n$ is a perfect square.

\item[\bf b.]  Part {\bf a} shows that if $\sqrt{n}$ is rational, then $n$ is a perfect square.  Formulate the (logically equivalent) contrapositive of this statement.  

\end{itemize}
\end{exercise}

\begin{exercise}
Textbook exercise 3.1.15.  Generalize ``square'' to ``$n^{\text{th}}$ power,'' for arbitrary $n \ge 2$. 
\end{exercise}

\begin{exercise}
Textbook exercise 3.1.16.  Be sure to also prove that the factors in the product  of part (b) are unique.  
\end{exercise}



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