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\title{Math 412  Homework 2}
\date{Due date: Sept 11, 2015}
\author{your name}


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\maketitle

Solve the following problems. Please remember to use complete sentences
and good grammar. Each problem is 4 points.



\begin{enumerate}

\item Show that if $p_k$ is the $k$-th prime, where $k$ is a positive integer, then $p_n\le p_1p_2\ldots p_{n-1}+1$ for all integers $n$ with $n\ge 3$. 

\item Show that the integer $Q_n=n!+1$, where $n$ is a positive integer, has a prime divisor greater than $n$.  Conclude that there are infinitely many primes.  


\item Show that there are infinite many primes of the form $3k-1$. 

\item Let $S=\{\log_{10} p:  p \text{ prime}\}$. Prove that the elements of $S$ are linearly independent over $Q$, the rationals. 


\item Show that $\sqrt{3}-\sqrt[3]{2}$ and $\log_{10} 5432$ are irrational. 

\item Find the prime factorization of $32!$.  How many zeros are at the end of $32!$?

\item  Find the last two digits of $9^{9^9}$.

\item (extra credit) Show that for any integer $n\ge 2$, $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$ is not an integer. 

\item (extra credit) Show that $\sin \frac{\pi}{7}$ is irrational. 


\end{enumerate}

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