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\homework{10: Infinite Descent}{Due Thursday, Week 5, at the start of class.}

Solve \textbf{one} of the following \textbf{three} problems.  As always, prove your claims!

These problems are all centered around the concept of \textbf{infinite descent}.  Specifically, notice that the natural numbers have the following property:
\begin{property}
There is no infinite decreasing sequence of natural numbers $\{a_n\}_{n=0}^\infty$.
\end{property}
This is not hard to see: if we start at some value $a_0 \in \mathbb{N}$ and we're always decreasing, we have at most $a_0$-more terms in our sequence before we get to 0, at which point we can no longer decrease!  

This is a rather simple observation that is used in a number of clever problems; we list three here!
\begin{enumerate}
\item[0.] Solve any un-signed-up-for problems from HW\#9!
\item Prove that there are no three integers $a,b,c > 0$ such that
\begin{align*}
a^2  -b^2 = 2abc.
\end{align*}
(Hint: look at $ab$ and its factors!)
\item A sequence $\{a_n\}_{n=0}^\infty$ is called an $k$-\textbf{arithmetic progression} if the difference between consecutive terms $a_{n+1} - a_n$ is equal to $k$.  For example,
\begin{align*}
1,4,7,10,13, 16, \ldots
\end{align*}
is an 3-arithmetic progression.

Prove that for $k \neq 0$, there is no $k$-arithmetic progression $\{a_n\}_{n=0}^\infty$ made entirely out of perfect squares

\item Find all of the prime numbers $p$ such that the following holds: there are positive integers $a, b$ and $n$ such that
\begin{align*}
p^n = x^3 + y^3.
\end{align*}
(Hint: factor $x^3 + y^3$, and use the fact that $p$ is a prime.)
\end{enumerate}





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