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\centerline{\bf Homework 6, MORALLY Due March 11} 
\newcommand{\st}{\mathrel{:}}
\newcommand{\es}{\emptyset}
\newcommand{\implies}{\Rightarrow}
\newcommand{\goes}{\Rightarrow}
\newcommand{\NSQ}{{\rm NSQ}}
\newcommand{\GREEDY}{{\rm GREEDY}}
\newcommand{\OPTIMAL}{{\rm OPTIMAL}}
\newcommand{\OPT}{{\rm OPT}}
\newcommand{\into}{{\rightarrow}}
\newcommand{\lf}{\left\lfloor}
\newcommand{\rf}{\right\rfloor}
\newcommand{\lc}{\left\lceil}
\newcommand{\rc}{\right\rceil}
\newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil}
\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
\newcommand{\Z}{{\sf Z}}
\newcommand{\N}{{\sf N}}
\newcommand{\Q}{{\sf Q}}
\newcommand{\R}{{\sf R}}
\newcommand{\D}{{\sf D}}
\newcommand{\Rpos}{{\sf R}^+}


\begin{enumerate}
\item
(50 points)
In this program we will look at primes of the form $x^2+5y^2$. Send your code to Emily, so that if you get wrong answers, I can give you partial credit. 

\begin{enumerate}
\item 
(0 points but you will need this later)

Write a program that will, given $p$, determines if there exists $x,y$ such that 

$$p=x^2+5y^2.$$

\item
(30 points) 
For all primes $p\in \N$ such that are $7\le p \le 1000$ run the above program.
Produce a table of primes  $p\in\N$, such that $7\le p \le 1000$, of the following form. You must put the table in your pdf. You can either copy and paste it into your latex doc, or take a screenshot.

\[
\begin{array}{|c|c|c|}
\hline
p & \hbox{sum of $x^2 + 5y^2$?} & \hbox{$x^2 + 5y^2$} \cr 
\hline
7 & N                       &                                               \cr
11 & N                      &                                               \cr
13 & N                      &                                               \cr
17 & N                      &                                               \cr
19 & N                      &                                               \cr
23 & N                      &                                               \cr
29 & Y                      & 3^2 + 5\times 2^2                             \cr
\hline
\end{array}
\]


\item
(20 points) 
Give a conjecture of the following form:

{\it Let $p$ be an odd prime such that $p\ge 7$. Then 

$p$ is a sum of the form $x^2 + 5y^2$ iff BLANK. 

}

\end{enumerate} 

\begin{comment} 
\item
(20  points)
Let $\pi(n)$ be the number of primes $p$ such that $7\le p \le n$. 
Let $FIVESQ(n)$ be the number of primes $p$ such that $7\le p\le n$ that can be written in the form $x^2 + 5y^2$. 
Write a program that, given $n$, determines $\pi(n)$, $FIVESQ(n)$, and $\frac{FIVESQ(n)}{\pi(n)}$. 

Using this program draw a graph of the function $n$ goes to $\frac{FIVESQ(n)}{\pi(n)}$. 
Speculate: is it constant? Approaching 0? Approaching 1? something else? 
\end{comment} 

\newpage 

\item 
(50 points)
On the untimed midterm1 you wrote two programs to find, given $n$, 
a way to write $n$ as a sum of squares.  The first one we call GREEDY the second one
we call OPTIMAL.
\begin{enumerate}
\item
(0 points but you need to do this for a later part)
Let $f(n)$ be  

$$\max\{\GREEDY(1), \GREEDY(2), \cdots, \GREEDY(n)\}.$$

Write a program that, given $n$, computes 

$$f(1), \ldots, f(n).$$

\item 
(0 points but you need to do it for the later parts) 
Run this program on $n=1000$. 

\item 
(20 points) 
Make a conjecture about what $f(n)$ looks like. (For example: $f(n)$ is ROUGHLY $\sqrt{n}$.)

\item
(15 points) 
Let $X_1= \{ n \st n\equiv 7 \pmod 8\}$. 
On the untimed midterm1, problem 1g, you probably found that, from your data, 

$$n\in X_1\goes \OPTIMAL(n)=4.$$

Find an infinite set $X_2$ such that $X_1\cap X_2=\es$ and, according to your data, 

$$n\in X_2\goes \OPTIMAL(n)=4.$$

\item 
(15 points) 
Make a conjecture about exactly which numbers $n$ have $\OPTIMAL(n)=4$. 
\end{enumerate}
\end{enumerate}
 

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