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\homework{18: More Elliptic Curves}{Due Friday at 11:30am, Finals Week}

Solve \textbf{three} of the following \textbf{six} problems.  Also, this set is \textbf{extra-credit!}  This set can be submitted by email if you can't turn it in to my office.  Have fun!


\begin{enumerate}
\item Consider the following problem: you have a large collection of tangerines.  You, being really bored, want to stack them in a pyramid!  Specifically, suppose you are stacking them in a square pyramid:\ that is, your first layer has one tangerine, your second layer has four tangerines, your third layer has nine tangerines, and so on/so forth.
\begin{center}
\includegraphics[width=3in]{spherepacking.png}
\end{center}
If you have a square number of tangerines that is strictly greater than 1, is it possible that you can stack all of them in a single pyramid?  Or is this impossible for any square?  (That is; you cannot stack 9 tangerines in such a pyramid, because our pyramid sizes go 1,5,14,30\ldots  Our question is the following: is there some $n$ such that $n^2$ is a ``pyramidal'' number?)
\item In class, we said that a curve like $y^2 = x^3$ is not an elliptic curve, because its derivative at $(0,0)$ was undefined.  

A thing you might hope would work:\ what happens if you just delete the point $(0,0)$?  To make this a concrete problem, take the collection of all points $(x,y)$ with $y^2 = x^3$ and $(x,y) \neq (0,0)$ in $\mathbb{Z}/7\mathbb{Z}$.  
\begin{enumerate}
\item Plot all of the points that are on this curve and not equal to (0,0).
\item Add in the point at infinity, and try to make a group table.  Do you get a group, and/or is our operation $+$ well-defined on these points?  Or when you go to add points together, do you sometimes get (0,0)?  
\end{enumerate}
\item Consider the elliptic curve $y^2 = x^3 + 7$.  Prove that it has no integer solutions. (Hint: look at things mod 4 or 8!)
\item Like the above: consider the elliptic curve $y^2 = x^3  - 6$.  Prove that it has no integer solutions. 
\item Show that the only integral point on the elliptic curve $y^2 = x^3 - 1$ is $(1,0)$.  (Hint: work in $\mathbb{Z}[i]$, and write $x^3 = y^2 + 1 = (y-i)(y+i)$.  Show that $y+i, y-i$ are relatively prime, and work from there!)
\item In class, we saw that sometimes an elliptic curve over a finite field could have no points other than $\mathcal{O}$!  For example, consider $y^2 = x^3 + 2x + 2$ over $\mathbb{Z}/3\mathbb{Z}$.  We know that the only squares are $0^2 = 0, 1^2 = 1, 2^2 \equiv 1 \mod 3$.  Therefore, 
\begin{itemize}
\item at $x = 0$ the equation $0^3 + 2 \cdot 0 + 2 = 2 = y^2$ has no solutions, 
\item at $x = 1$ the equation $1^3 + 2 \cdot 1 + 2 = 5 \equiv 2 = y^2$ has no solutions, and
\item at $x = 2$ the equation $2^3 + 2 \cdot 2 + 2 = 14 \equiv 2 = y^2$ has no solutions.
\end{itemize}
So our curve has no points other than $\mathcal{O}$!

Find the largest value of $p$ for which this can happen.
\end{enumerate}
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