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\noindent{\Large{\textsc{Number Theory\hfill Assignment 11.2  \\ Fall 2020 \hfill Due November 18}}}

\large

\vspace{1cm}

\begin{exercise}
Let $a, b, n \in \N$.  Prove that if $a^n = b^n$, then $a=b$.  [{\em Suggestion:}  You can avoid using the FTA by observing that $a^n - b^n = (a-b)(a^{n-1} + a^{n-2} b + \cdots + a b^{n-2} + b^{n-1})$.]
\end{exercise}

\begin{exercise}
Let $m \in \R$.  Prove that 
\[
\left( \frac{-2m}{m^2+1}, \frac{1-m^2}{m^2+1} \right)
\]
is a rational point if and only if $m \in \Q$.
\end{exercise}

\begin{exercise}
Let
\[
C(\Q) = \{(X,Y) \, | \, X, Y \in \Q, X^2 + Y^2 = 1 \}.
\]
Prove that the function $\pi : \Q \to C(\Q)$ given by
\[
\pi(m) = \left( \frac{-2m}{m^2+1}, \frac{1-m^2}{m^2+1} \right)
\]
is a bijection.  [{\em Suggestion:}  Show that, away from $(0,1)$, the inverse is given by $(X,Y) \mapsto \frac{Y-1}{X}$.]
\end{exercise}

\begin{exercise}
Let $r,s \in \Z$ with $(r,s) =1$, $s \ne 0$, and $r \equiv s \equiv 1 \pmod{2}$.  Write $r=2k+1$ and $s=2 \ell+1$.  Let $u = k+ \ell +1$ and $v = \ell -k$.  
\[
\left( \frac{-2rs}{r^2 + s^2}, \frac{s^2 - r^2}{r^2 + s^2} \right) = \left( \frac{v^2 - u^2}{u^2 + v^2}, \frac{2uv}{u^2 + v^2} \right),
\]
that the coordinates on the right hand side are reduced, and that $\gcd(u,v)=1$ and $u \not \equiv v \pmod{2}$.
\end{exercise}

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