\documentclass[12pt, reqno]{amsart}

\textwidth=7in
\textheight=9.5in
\topmargin=0in
\headheight=0in
\headsep=.5in
\hoffset=-.85in

\newcommand{\ndiv}{\hspace{-4pt}\not|\hspace{2pt}}

\begin{document}

\title[Math 250 Handout 3 - proof by contradiction]{Math 250 Handout 3 - proof by contradiction}\maketitle

\begin{enumerate}

\item Prove that if $x + y > 5$, then $x > 2$ or $y > 3$.

\item Let $0 < \alpha <$1. Prove that $\sqrt{\alpha} > \alpha$.

\item Prove that there are no integer solutions to the equation $x^2 = 4y + 2$

\item Prove that there are no positive integer solutions to the equation $x^2 - y^2 = 10$.

\item Prove that there is no smallest positive real number.

\item Let $b_1, b_2, b_3,b_4$ be positive integers such that
 $$\frac{1}{b_1} + \frac{1}{b_2} + \frac{1}{b_3} + \frac{1}{b_4} = 1.$$ 
Prove that at least one of the $b_k$'s is even. Hint: clear the denominators.



\item Show that if $a$ is rational and $b$ is irrational, then $a + b$ is irrational.



\item Prove that $\sqrt{3}$ is irrational.

\item Prove that if $r^3 + r + 1 = 0$ then $r$ is irrational.

\item Let $a, b, c$ be integers satisfying $a^2 + b^2 = c^2$. Show that $abc$ must be even. (Harder problem: show that $a$ or $b$ must be even.)

\item If $a, b, c$ are odd integers, prove that $ax^2 + bx + c = 0$ does not have a solution $x$ such that $x$ is a rational number.

  
 \item Prove that if $3 \mid (a^2 + b^2)$, then $3 \mid a$ and $3 \mid b$. Hint: If $3 \ndiv a$ and $3 \ndiv b$, what are the possible remainders of $a$, $b$, $a^2$, and $b^2$ upon division by 3?

% \item Prove that there do not exist positive integers $a, b, c$
% and $n$ such that $a^2 + b^2 + c^2 = 2nabc$.

% \item Show that the equation $b^2 +b+1=a^2$ has no positive integer solutions $a,b \in \mathbf{Z}_{\geq 1}$.


\item Prove that $\log_{10} 7$ is irrational.
\vspace{4pt}

\item Let $b \in \mathbf{Z}_{\geq 1}$. Prove that $\log_b 3 / \log_b 2$ is irrational. 
\vspace{4pt}



\item Prove that $\sqrt[5]{5}$ is irrational.
\vspace{4pt}

\item Prove that the equation 
\[
(x^2 - y^2)(x^2 - 4y^2) = 7
\]
has no solutions with $x,y \in \mathbf{Z}$.


\item Prove that there are infinitely many primes of the form $6n+1$ or there are infinitely many primes of the form $6n+5$. 
\item Prove that there are infinitely many primes of the form $6n+5$.
\item Try to prove that there are infinitely many primes of the form $6n+1$. What goes wrong in the argument from the previous problem?


\item Prove that if n $\geq$ 2, then $\sqrt[n]{n}$ is
 irrational. Hint: use that if $n > 2$, then $2^n > n$.
\vspace{4pt}

\vspace{4pt}

% \item Suppose that $p$ is an odd prime and that 
% $$\frac{a}{b}= 1 + \frac{1}{2}+ \frac{1}{3} + \cdots + \frac{1}{p-1}$$
% Show that $p \mid a$. 

\end{enumerate}

\end{document}

