
\input amstex


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\input colordvi

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\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}
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\def\lra{$\Leftrightarrow$ }


\ctln{\bf Math 445 Number Theory}

\smallskip

\ctln{October 15, 2004}

\medskip

Some computations: our basic facts are

\ssk

$\Big({{p}\over{q}}\Big)\Big({{q}\over{p}}\Big) = 
(-1)^{({{p-1}\over{2}})({{q-1}\over{2}})}$ if $p,q$ distinct odd primes, 
$\Big({{2}\over{p}}\Big) = (-1)^{{p^2-1}\over{8}}$,
$\Big({{-1}\over{p}}\Big) = (-1)^{{p-1}\over{2}}$,
$\Big({{ab}\over{p}}\Big) = \Big({{a}\over{p}}\Big)\Big({{b}\over{p}}\Big)$,
and 
$\Big({{a+pk}\over{p}}\Big) = \Big({{a}\over{p}}\Big)$ .


\ssk

With these we can, in principle, decide for any prime $p$ and $(a,p)=1$ if $x^2\equiv a\pmod{p}$ has
a solution. And the fun part is that, in the last of these facts, we can choose something
equivalent to $a$ mod $p$ in \underbar{any way we want}, which can lead to some very
different computations of the same result!

\msk

$\Big({{619}\over{1229}}\Big)$ : \hskip.1in 
$\Big({{619}\over{1229}}\Big)\Big({{1229}\over{619}}\Big)=(-1)^{{{619-1}\over{2}}{{1229-1}\over{2}}}
=(-1)^{309\cdot 614}=1$ , so $\Big({{619}\over{1229}}\Big)=\Big({{1229}\over{619}}\Big)=
\Big({{1238-9}\over{619}}\Big)=\Big({{-9+619\cdot 2}\over{619}}\Big)
=\Big({{-9}\over{619}}\Big)=\Big({{(-1)(3)^2}\over{619}}\Big)
=\Big({{-1}\over{619}}\Big)(\Big({{3}\over{619}}\Big))^2=\Big({{-1}\over{619}}\Big)
=(-1)^{309}=-1$
\hskip.1in so $\Big({{619}\over{1229}}\Big)=-1$, so $x^2\equiv 619\pmod{1229}$ has no solutions.

\bsk

$\Big({{617}\over{1229}}\Big)$ : \hskip.1in 
$\Big({{617}\over{1229}}\Big)\Big({{1229}\over{617}}\Big)=
(-1)^{{{617-1}\over{2}}{{1229-1}\over{2}}}
=(-1)^{308\cdot 614}=1$ , so $\Big({{617}\over{1229}}\Big)=\Big({{1229}\over{617}}\Big)=
\Big({{1234-5}\over{617}}\Big)=\Big({{-5+617\cdot 2}\over{617}}\Big)=
=\Big({{-5}\over{617}}\Big)=\Big({{(-1)(5)}\over{617}}\Big)
=\Big({{-1}\over{617}}\Big)\Big({{5}\over{617}}\Big)$ . Now,
$\Big({{-1}\over{617}}\Big) = (-1)^{{617-1}\over{2}} =(-1)^{308}=1$ , so 
$\Big({{617}\over{1229}}\Big)=\Big({{1229}\over{617}}\Big)=\Big({{5}\over{617}}\Big)$

\ssk

$\Big({{5}\over{617}}\Big)\Big({{617}\over{5}}\Big)=
(-1)^{308\cdot 2}=1$ , so $\Big({{5}\over{617}}\Big) = \Big({{617}\over{5}}\Big)=
\Big({{5\cdot 123 +2}\over{5}}\Big)=\Big({{2}\over{5}}\Big)\equiv 2^{{5-1}\over{2}}=
2^2=4\equiv -1\pmod{5}$ , so $\Big({{617}\over{1229}}\Big) = \Big({{5}\over{617}}\Big)
=\Big({{2}\over{5}}\Big) = -1$ , so $x^2\equiv 617\pmod{1229}$ also has no solutons.

\bsk

$\Big({{223}\over{617}}\Big)$ : \hskip.1in Check that they are prime! Then:
$\Big({{223}\over{617}}\Big)\Big({{617}\over{223}}\Big)=(-1)^{{{617-1}\over{2}}{{223-1}\over{2}}}
=(-1)^{308\cdot 111}=1$ , so 
$\Big({{223}\over{617}}\Big) = \Big({{617}\over{223}}\Big) = \Big({{669-52}\over{23}}\Big)=
\Big({{223\cdot 3 -52}\over{223}}\Big) = \Big({{-52}\over{223}}\Big) = 
\Big({{(-1)(2)^2(13)}\over{223}}\Big) = 
\Big({{-1}\over{223}}\Big)(\Big({{2}\over{223}}\Big) )^2\Big({{13}\over{223}}\Big)
=\Big({{-1}\over{223}}\Big)\Big({{13}\over{223}}\Big)$ . 
\hskip.1in But 
$\Big({{-1}\over{223}}\Big) = (-1)^{{223-1}\over{2}}=(-1)^{111}=-1$ , and
$\Big({{13}\over{223}}\Big)\Big({{223}\over{13}}\Big) = (-1)^{{{13-1}\over{2}}{{223-1}\over{2}}}
= (-1)^{6\cdot 111} = 1$ , so $\Big({{13}\over{223}}\Big) = \Big({{223}\over{13}}\Big) =
\Big({{260-37}\over{13}}\Big) = \Big({{-37}\over{13}}\Big) 
= \Big({{-37+39}\over{13}}\Big) = \Big({{2}\over{13}}\Big) = (-1)^{{13^2-1}\over{8}} =
(-1)^{21} = -1$ , so 
$\Big({{223}\over{617}}\Big) = \Big({{-52}\over{223}}\Big) =(-1)\Big({{13}\over{223}}\Big)
=(-1)\Big({{2}\over{13}}\Big) = (-1)(-1) =1$ , so 
$x^2\equiv 223\pmod{617}$ has a solution.

\bsk

$\Big({{555}\over{1663}}\Big)$ : $555=5\cdot 111 = 5\cdot 3\cdot 37$, so 
$\Big({{555}\over{1663}}\Big)=$
$\Big({{5}\over{1663}}\Big)\Big({{3}\over{1663}}\Big)\Big({{37}\over{1663}}\Big)$ 
And we can compute: \hsk
$\Big({{5}\over{1663}}\Big) = 
(-1)^{{{1663-1}\over{2}}{{5-1}\over{2}}}\Big({{1663}\over{5}}\Big)
=(-1)^{831\cdot 2}\Big({{1663}\over{5}}\Big) = 
\Big({{1663}\over{5}}\Big) = \Big({{3}\over{5}}\Big) \equiv 3^{{5-1}\over{2}}
=3^2=9\equiv -1\pmod{5}$ , so $\Big({{5}\over{1663}}\Big) =-1$ .
\hsk
$\Big({{3}\over{1663}}\Big) = 
(-1)^{{{1663-1}\over{2}}{{3-1}\over{2}}}\Big({{1663}\over{3}}\Big)
=(-1)^{831\cdot 1}\Big({{1663}\over{3}}\Big) = 
(-1)\Big({{1663}\over{3}}\Big) = (-1)\Big({{3\cdot 554+1}\over{3}}\Big)
=(-1)\Big({{1}\over{3}}\Big) = (-1)(1)^{{3-1}\over{2}} = -1$
\hsk
$\Big({{37}\over{1663}}\Big) = 
(-1)^{831\cdot 18}\Big({{1663}\over{37}}\Big) = 
\Big({{1663}\over{37}}\Big) = \Big({{1663-1350}\over{37}}\Big)
= \Big({{313}\over{37}}\Big) = \Big({{313-370}\over{37}}\Big)
= \Big({{-57}\over{37}}\Big) = \Big({{-57+74}\over{37}}\Big)
= \Big({{17}\over{37}}\Big) = (-1)^{{{17-1}\over{2}}{{37-1}\over{2}}}\Big({{37}\over{17}}\Big)
= (-1)^{8\cdot 18}\Big({{37}\over{17}}\Big) = \Big({{37}\over{17}}\Big)
= \Big({{37-34}\over{17}}\Big) = \Big({{3}\over{17}}\Big)
= (-1)^{1\cdot 8}\Big({{17}\over{3}}\Big) = \Big({{17}\over{3}}\Big)
= \Big({{17-18}\over{3}}\Big) = \Big({{-1}\over{3}}\Big) = 
(-1){{3-1}\over{2}} = (-1)^1 = -1$ .

\ssk

So, putting them together, 
$\Big({{555}\over{1663}}\Big) = 
\Big({{5}\over{1663}}\Big)\Big({{3}\over{1663}}\Big)\Big({{37}\over{1663}}\Big)
= (-1)(-1)(-1) = -1$, so $x^2\equiv 555\pmod{1663}$ has no solutions.






\vfill\end







