
\input amstex


\magnification=1200


\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\moda{\medspace {\underset a\to \equiv} \medspace}
\def\modb{\medspace {\underset b\to \equiv} \medspace}
\def\modc{\medspace {\underset c\to \equiv} \medspace}

\def\lra{$\Leftrightarrow$ }


\ctln{\bf Math 445 Number Theory}

\ssk

\ctln{November 22, 2004}

\msk

{\bf Rational points on curves:}

\msk

The geometric process we have developed works for any equation 
$ax^2+by^2+cz^2=0$ , i.e., $aX^2+bY^2+c=0$ , for which we
know a single rational solution $(X_0,Y_0)$, to find all rational solutions to $aX^2+bY^2+c=0$ 
(and hence all integer solutions to $ax^2+by^2+cz^2=0$). Looking at lines with rational
slope through our known solution, we can find its
points of intersection with the ellipse/hyperbola $aX^2+bY^2+c=0$ by
finding the roots of a quadratic equation 

\ssk

$aX^2+b(r(X-X_0)+y_0)^2 +c=0$

\ssk

with rational coefficients, for which we already know one root, $X_0$. The other root 
(which depends on the variable $r$) is then also rational.

\msk

For example, knowing that 
$2x^2+3y^2=5$ has solution $(x_0,y_0)=(1,1)$ we find, using the line 
$y=r(x-1)+1$ with rational slope $r$ through $(1,1)$, that 

\ssk

$0=2x^2+3y^2-5 = 2x^2+3(rx-r+1)^2-5 = 2x^2+3(r^2x^2-2r^2x+2rx+r^2-2r+1)-5 =
(2+3r^2)x^2+(6r-6r^2)x+(3r^2-6r-2) = ((x-1)((2+3r^2)x-(3r^2-6r-2))$ , so $x=1$ or
$\dsl x={{3r^2-6r-2}\over{3r^2+2}}$ , giving $\dsl y=rx-r+1 = r{{3r^2-6r-2}\over{3r^2+2}}-r+1 = 
-{{3r^2+4r-2}\over{3r^2+2}}$ . Setting $\dsl r={{u}\over{v}}$ , we get, as before,

$2(3u^2-6uv-2v^2)^2+3(3u^2+4uv-2v^2)^2=5(3u^2+2v^2)$ . For example, setting
$u=11,v=4$, we have $2(67)^2+3(507)^2=5(395)^2$.

\msk

Or, starting from $2^2+5^2=29$, we find, after solving 

$x^2+(r(x-2)+5)^2-29=0
=(r^2+1)x^2+(10r-4r^2)x+(4r^2-20r-4) = (x-2)((r^2+1)-(2r^2-10r-2))$, that
$\dsl x={{2r^2-10r-2}\over{r^2+1}}$ and $\dsl y=r({{2r^2-10r-2}\over{r^2+1}}-2)+5 = -{{5r^2+4r-5}\over{r^2+1}}$ .
So setting $\dsl r={{u}\over{v}}$ , we get $(2u^2-10uv-2v^2)^2+(5u^2+4uv-5v^2)^2 = 29(u^2+v^2)^2$ . 
This gives (after dividing these three terms by their common g.c.d., and then multplying by a common factor) all integer
solutions to $x^2+y^2=29z^2$. For example, setting $u=11, v=28$, we have $(1754)^2+(4547)^2=29(905)^2$ .

\msk

The hard part: finding the first solution! For the special situation $x^2+y^2=nz^2$, we know from 
a homework problem awhile back that if $X^2+Y^2=n$ has a rational solution, then it 
has an {\it integer} solution, which we can look for by an exhaustive search among
$0\leq X,Y\leq \sqrt{n}$ ! Note that our newest result gives us a quick criterion to decide
if $x^2+y^2=n$ has an integer solution, when $n$ is square-free; 
we need $\dsl\Big({{-1}\over{n}}\Big) = 1$ . The interested reader can check that this 
really is equivalent to the (slightly more long-winded) answer we found before...


\vfill
\end



























