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\centerline{\bf Homework 7, MORALLY Due March 25} 
\centerline{\bf Recall: Spring Break is March 18-22}
\newcommand{\st}{\mathrel{:}}
\newcommand{\goes}{\Rightarrow}
\newcommand{\NSQ}{{\rm NSQ}}
\newcommand{\BLANK}{{\rm BLANK}}
\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
(30 points-10 points each)
Write the following sequences in closed form:
For example, if the sequence was

$$1,1,2,2,3,3,\ldots$$

(So $a_1=1$, $a_2=1$, $a_3=2$, $a_4=2$, $\ldots$.) 

then the answer is

\begin{equation}
a_n=
\begin{cases}
(n+1)/2  & \text{if $n\equiv 1 \pmod 2$} \\
n/2      & \text{if $n\equiv 0 \pmod 2$}\\
\end{cases}
\end{equation}

\begin{enumerate}

\item
$2,5,10,17,26,\ldots$. 


\item
$1,1,1,2,2,2,3,3,3,\ldots$

\item
$1,-4,9,-16,\ldots$.

\end{enumerate}

\newpage

\item
(25 points)
We define a sequence as follows:

$a_1=1$. For all $n\ge 2$, $a_{n} = a_{n-1} +  a_{\floor{n/2}}$. 

\begin{enumerate}
\item
(0 points but you will need it for the next part.) 
Write a program that will, given $n$,  find the following:

\begin{enumerate}
\item 

\begin{itemize}
\item For $1\le i\le n$, $$\frac{\textrm{The number of } a_i\equiv 0 \pmod 2}{n}$$ 
\item 
For $1\le i\le n$, $$\frac{\textrm{The number of } a_i\equiv 1 \pmod 2}{n}$$ 
\end{itemize} 


\bigskip
\bigskip

\item 

\begin{itemize} 
\item 
For $1\le i\le n$, $$\frac{\textrm{The number of } a_i\equiv 0 \pmod 3}{n}$$ 
\item 
For $1\le i\le n$, $$\frac{\textrm{The number of } a_i\equiv 1 \pmod 3}{n}$$ 
\item 
For $1\le i\le n$, $$\frac{\textrm{The number of } a_i\equiv 2 \pmod 3}{n}$$ 
\end{itemize}

\bigskip
\bigskip

\item
{\bf Similar} for mod 5,7,11,13,17,19.
\end{enumerate}

\bigskip
\bigskip

(There is more to this problem on the next page) 

\newpage

\item
(15 points) 
Run your program with n=1000. Report the results neatly.


\item 
(10 points)
Based on your data give conjectures about the following

\begin{enumerate}
\item
Is $a_n\equiv 0 \pmod 2$ infinitely often?
Do the cases $a_n\equiv 0 \pmod 2$ and 
Do the cases $a_n\equiv 1 \pmod 2$ occur about the same amount of time? 


\item
{\bf Similar} for Mod 3,5,7,11,13,17,19.
\end{enumerate} 
\end{enumerate} 

\newpage

\item
(25 points) In this problem we guide you through a proof that number of the
form $4^n(8k+7)$ cannot be written as the sum of 3 squares. 

\begin{enumerate}
\item
(0 points but you will need this for later)
Find the following set 

$$X = \{ x^2 \pmod 8 \st x\in \{0,1,2,3,4,5,6,7\}\}.$$

\item
(7 points) 
Show that if $x\equiv 7 \pmod 8$ then $x$ is NOT the sum of three squares. 

({\it Hint:} Show that any three elements of $X$ from Part a can never sum to 7 mod 8.) 


\item
(8 points) 
Prove that if $x^2+y^2+z^2 \equiv 0 \pmod 4$ then $x,y,z$ are all even. 
({\it Hint:} Look at what happens when:

1 of $\{x,y,z\}$ is odd, 

2 of $\{x,y,z\}$ is odd,

3 of $\{x,y,z\}$ is odd. 

)

\item
(9 points) 
Prove the following by induction on $n$. 

{\bf Theorem} Let $n\ge 0$. Let $k\in\N$. Show that $4^n(8k+7)$ cannot be written
as the sum of 3 squares. 

\end{enumerate}

\newpage

\item
(20 points) 
On HW 4 problem 3 you were asked for a conjecture of the form

$$(\forall a\in \Z)[a^m \equiv a \pmod m] \hbox{ iff } \BLANK(m).$$

You based your conjecture on a program that you wrote and ran up to 200.

ALL of your had the answer $\BLANK(m)$ is {\it $m$ is prime}. 

Run your program up to 1000. 

Did you find any value $m$ such that:

\begin{itemize}
\item 
$(\forall a\in \Z)[a^m \equiv a \pmod m]$
\item
$m$ is NOT prime.
\end{itemize}

If you DID then output the smallest such $m$.

If you DID NOT then state
a Theorem in Mathematics, it can be a little one, that 
proves your conjecture. 

\end{enumerate}

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