
\input amstex


\magnification=1400


\loadmsbm

\input colordvi

\nopagenumbers
\parindent=0pt

\def\cgy{\GreenYellow}     % GreenYellow  Approximate PANTONE 388
\def\cyy{\Yellow}	  % Yellow  Approximate PANTONE YELLOW
\def\cgo{\Goldenrod}	  % Goldenrod  Approximate PANTONE 109
\def\cda{\Dandelion}	  % Dandelion  Approximate PANTONE 123
\def\capr{\Apricot}	  % Apricot  Approximate PANTONE 1565
\def\cpe{\Peach}		  % Peach  Approximate PANTONE 164
\def\cme{\Melon}		  % Melon  Approximate PANTONE 177
\def\cyo{\YellowOrange}	  % YellowOrange  Approximate PANTONE 130
\def\coo{\Orange}	  % Orange  Approximate PANTONE ORANGE-021
\def\cbo{\BurntOrange}	  % BurntOrange  Approximate PANTONE 388
\def\cbs{\Bittersweet}	  % Bittersweet  Approximate PANTONE 167
%\def\creo{\RedOrange}	  % RedOrange  Approximate PANTONE 179
\def\cma{\Mahogany}	  % Mahogany  Approximate PANTONE 484
\def\cmr{\Maroon}	  % Maroon  Approximate PANTONE 201
\def\cbr{\BrickRed}	  % BrickRed  Approximate PANTONE 1805
\def\crr{\Red}		  % Red  VERY-Approx PANTONE RED
\def\cor{\OrangeRed}	  % OrangeRed  No PANTONE match
\def\paru{\RubineRed}	  % RubineRed  Approximate PANTONE RUBINE-RED
\def\cwi{\WildStrawberry}  % WildStrawberry  Approximate PANTONE 206
\def\csa{\Salmon}	  % Salmon  Approximate PANTONE 183
\def\ccp{\CarnationPink}	  % CarnationPink  Approximate PANTONE 218
\def\cmag{\Magenta}	  % Magenta  Approximate PANTONE PROCESS-MAGENTA
\def\cvr{\VioletRed}	  % VioletRed  Approximate PANTONE 219
\def\parh{\Rhodamine}	  % Rhodamine  Approximate PANTONE RHODAMINE-RED
\def\cmu{\Mulberry}	  % Mulberry  Approximate PANTONE 241
\def\parv{\RedViolet}	  % RedViolet  Approximate PANTONE 234
\def\cfu{\Fuchsia}	  % Fuchsia  Approximate PANTONE 248
\def\cla{\Lavender}	  % Lavender  Approximate PANTONE 223
\def\cth{\Thistle}	  % Thistle  Approximate PANTONE 245
\def\corc{\Orchid}	  % Orchid  Approximate PANTONE 252
\def\cdo{\DarkOrchid}	  % DarkOrchid  No PANTONE match
\def\cpu{\Purple}	  % Purple  Approximate PANTONE PURPLE
\def\cpl{\Plum}		  % Plum  VERY-Approx PANTONE 518
\def\cvi{\Violet}	  % Violet  Approximate PANTONE VIOLET
\def\parp{\RoyalPurple}	  % RoyalPurple  Approximate PANTONE 267
\def\cbv{\BlueViolet}	  % BlueViolet  Approximate PANTONE 2755
\def\cpe{\Periwinkle}	  % Periwinkle  Approximate PANTONE 2715
\def\ccb{\CadetBlue}	  % CadetBlue  Approximate PANTONE (534+535)/2
\def\cco{\CornflowerBlue}  % CornflowerBlue  Approximate PANTONE 292
\def\cmb{\MidnightBlue}	  % MidnightBlue  Approximate PANTONE 302
\def\cnb{\NavyBlue}	  % NavyBlue  Approximate PANTONE 293
\def\crb{\RoyalBlue}	  % RoyalBlue  No PANTONE match
%\def\cbb{\Blue}		  % Blue  Approximate PANTONE BLUE-072
\def\cce{\Cerulean}	  % Cerulean  Approximate PANTONE 3005
\def\ccy{\Cyan}		  % Cyan  Approximate PANTONE PROCESS-CYAN
\def\cpb{\ProcessBlue}	  % ProcessBlue  Approximate PANTONE PROCESS-BLUE
\def\csb{\SkyBlue}	  % SkyBlue  Approximate PANTONE 2985
\def\ctu{\Turquoise}	  % Turquoise  Approximate PANTONE (312+313)/2
\def\ctb{\TealBlue}	  % TealBlue  Approximate PANTONE 3145
\def\caq{\Aquamarine}	  % Aquamarine  Approximate PANTONE 3135
\def\cbg{\BlueGreen}	  % BlueGreen  Approximate PANTONE 320
\def\cem{\Emerald}	  % Emerald  No PANTONE match
\def\cjg{\JungleGreen}	  % JungleGreen  Approximate PANTONE 328
\def\csg{\SeaGreen}	  % SeaGreen  Approximate PANTONE 3268
\def\cgg{\Green}	  % Green  VERY-Approx PANTONE GREEN
\def\cfg{\ForestGreen}	  % ForestGreen  Approximate PANTONE 349
\def\cpg{\PineGreen}	  % PineGreen  Approximate PANTONE 323
\def\clg{\LimeGreen}	  % LimeGreen  No PANTONE match
\def\cyg{\YellowGreen}	  % YellowGreen  Approximate PANTONE 375
\def\cspg{\SpringGreen}	  % SpringGreen  Approximate PANTONE 381
\def\cog{\OliveGreen}	  % OliveGreen  Approximate PANTONE 582
\def\pars{\RawSienna}	  % RawSienna  Approximate PANTONE 154
\def\cse{\Sepia}		  % Sepia  Approximate PANTONE 161
\def\cbr{\Brown}		  % Brown  Approximate PANTONE 1615
\def\cta{\Tan}		  % Tan  No PANTONE match
\def\cgr{\Gray}		  % Gray  Approximate PANTONE COOL-GRAY-8
\def\cbl{\Black}		  % Black  Approximate PANTONE PROCESS-BLACK
\def\cwh{\White}		  % White  No PANTONE match


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\def\ctln{\centerline}
\def\u{\underbar}
\def\ssk{\smallskip}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\hsk{\hskip.1in}
\def\hhsk{\hskip.2in}
\def\dsl{\displaystyle}
\def\hskp{\hskip1.5in}

\def\lra{$\Leftrightarrow$ }


\ctln{\bf Math 445 Number Theory}

\ssk

\ctln{November 15, 2004}

\msk

{\bf Sums of four squares.}

\msk

For every $n\in{\Bbb N}$, there are $x,y,z,w\in{\Bbb Z}$ so that $x^2+y^2+z^2+w^2=n$.

\ssk

Elements of the proof:

\ssk

$(x_1^2+y_1^2+z_1^2+w_1^2)(x_2^2+y_2^2+z_2^2+w_2^2)$ = 

\hhsk $(x_1x_2+y_1y_2+z_1z_2+w_1w_2)^2+(x_1y_2-x_2y_1+z_2w_1-z_1w_2)^2$ 

\hhsk \hhsk $+$ $(x_1z_2-x_2z_1+y_1w_2-w_1y_2)^2+(x_1w_2-x_2w_1+y_2z_1-y_1z_2)^2$

\ssk

So we may focus on primes $p$. $p=2 = 1^2+1^2+0^2+0^2$, so focus on odd primes. Then

\ssk

{\it Proposition:} $0\leq x,y\leq (p-1)/2$ and $x\neq y$ implies $x^2\not\equiv y^2$ (mod $p$). 
This is because
$p|x^2-y^2 = (x-y)(x+y)$ implies $p|x-y$ and $-(p-1)/2\leq x-y\leq (p-1)/2$  so $x=y$, 
or $p|x+y$ and $0\leq x+y\leq p-1$ so $x+y=0$ so $x=y=0$ . Then

\ssk

{\it Proposition:} For any
$a$, $x^2$ and $a-y^2$, with $0\leq x,y\leq (p-1)/2$ must have a value, mod $p$, 
in common. For otherwise, since $x^2$ and $a-y^2$ each take on 
$(p+1)/2$ different values, $x^2+y^2-a=x^2-(a-y^2)$ would take on $p+1$ different values, 
mod $p$. So in particular, $x^2+y^2\equiv -1$ (mod $p$) has a solution. 

\ssk

Then
$x^2+y^2+1^2+0^2=Mp$ for some $M$; with the restrictions on $x,y$ above, we have $M<p$.
Choose the smallest positive $M$ with $Mp=x^2+y^2+z^2+w^2$. We claim: $M=1$ (so
$p=x^2+y^2+z^2+w^2$ is a sum of 4 squares). 

\ssk

First, $M$ is odd, since if $M$ is even, then $x^2+y^2+z^2+w^2$ is even, so
an even number of $x,y,z,w$ are even.
After renaming the variables to group them by parity, we have

\ssk

$\displaystyle{{M}\over{2}}p = ({{x-y}\over{2}})^2+({{x+y}\over{2}})^2+({{z-w}\over{2}})^2+({{z+w}\over{2}})^2$
\hhsk where each of the numbers on the right are integers.
If $M>1$ is odd, then choose 
$\displaystyle -{{M}\over{2}}\leq x_1,y_1,z_1,w_1\leq{{M}\over{2}}$ with
$x\equiv x_1$ (mod $M$), etc. Then $x_1^2+y_1^2+z_1^2+w_1^2\equiv x^2+y^2+z^2+w^2\equiv 0$ (mod $M$), so 
$x_1^2+y_1^2+z_1^2+w_1^2=NM$ ; since $\dsl |x_1|,|y_1|,|z_1|,|w_1|<{{M}\over{2}}$ , 
$x_1^2+y_1^2+z_1^2+w_1^2<M^2$, so $N<M$. Note also that $N>0$, since otherwise $x_1=y_1=z_1=w_1=0$,
so $M|x,y,z,w$, so $M^2|x^2+y^2+z^2+w^2=Mp$, so $p|M$, contradicting $M<p$ . Then

\msk

$NM^2p$ = $(x_1^2+y_1^2+z_1^2+w_1^2)(x^2+y^2+z^2+w^2)$ = 

\hhsk $(x_1x+y_1y+z_1z+w_1w)^2+(x_1y-xy_1+zw_1-z_1w)^2$

\hhsk \hhsk $+(x_1z-xz_1+y_1w-w_1y)^2+(x_1w-xw_1+yz_1-y_1z)^2 = a^2+b^2+c^2+d^2$

\ssk

and we can check that, mod $M$, 

\hhsk $a=x_1x+y_1y+z_1z+w_1w \equiv x^2+y^2+z^2+w^2\equiv 0$,

\hhsk $b=x_1y-xy_1+zw_1-z_1w\equiv xy-xy+zw-zw\equiv 0$, 

\hhsk $c=x_1z-xz_1+y_1w-w_1y\equiv xz-xz+yw-yw\equiv0$, 

and $d=x_1w-xw_1+yz_1-y_1z\equiv xw-xw+yz-yz\equiv 0$ . 

So $a=MA,b=MB,c=MC,d=MD$ and 
$NM^2p = M^2(A^2+B^2+C^2+D^2)$ so $A^2+B^2+C^2+D^2=Np$ with $0<N<M$, a 
contradiction. So $M=1$, as desired.


\vfill
\end

