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¯U¸qªº¥N¼Æ¾Ç¥v¥i¥H»¡¬O¤@³¡¸Ñ¦h¶µ¦¡¤èµ{¦¡ªº¾ú¥v¡C ¸Ñ¤èµ{¦¡ªº°ÝÃD¤j¬ù¥i¥H¤À¦¨¦³¨S¦³¸Ñ¡B¦p¦ó§ä¸Ñ¨â³¡¤À¡F¥N¼Æ¾Ç°ò¥»©w²z´N¬O°ÝÃD²Ä¤@³¡¥÷ªº¤@­Ó­«­nµª®×¡A¥¦»¡¡G¤@­Ó¦h¶µ¦¡¤èµ{¦¡¤@©w·|¦³¤@­Ó½Æ¼Æ®Ú¡C

¤@¦¸¤èµ{¦¡·íµM¦³¤@­Ó®Ú¡A¤G¦¸¤èµ{¦¡ ax2+bx+c=0 ·íµM¦³¨â­Ó®Ú

\begin{displaymath}
x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}
\end{displaymath}

³o¨Ç¨Æ¹ê«Ü¦­´Nª¾¹D¤F¡F¦ý¬O«e¤H³B²zªº¬O¹ê¼Æªº°ÝÃD¡A¦pªG b2-4ac<0¡A¥L­Ì·|»¡®Ú©~µM§t¦³µê¼Æ¡]­t¼Æªº¥­¤è®Ú¡^¡A¬O¯îÂÕªº¡A¸Ó´­±óªº¡C¤£¹L´N¤G¦¸¤èµ{¦¡¦Ó¨¥¡A¤°»ò®É­Ô¦³¡]¹ê¼Æ¡^¸Ñ¡A¦p¦ó¨D¸Ñ³£¤£¦¨°ÝÃD¡C

¤T¦¸©O¡H¤T¦¸¡]©M¥|¦¸¡^¤èµ{¦¡ªº¨D¸Ñ¾ú¥vÁöµM¬Û·í¦±§é¦³½ì µù1 ¡A¦ý§Ú­Ì·Q½Íªº¬O¥¦»Pµê¼ÆªºÃö«Y¡C

¤@¯ëªº¤T¦¸¤èµ{¦¡³£¥i¥H¸g¥Ñ²¾®Úªº³B²z¦ÓÅܦ¨ x3+px+q=0¡C·í§Aª`·N¨ì

(u+v)3-3uv(u+v)-(u3+v3)=0

³o¼Ëªº«íµ¥¦¡¡A´Nª¾¹D¤T¦¸¤èµ{¦¡ªº¸Ñªk³Z¬¦b¨ºùØ¡G¥O

\begin{displaymath}
u^3+v^3=-q,\quad uv=-\frac{p}{3}
\end{displaymath}

¦pªG u,v ¦³¸Ñ¡A¨º»ò x=u+v ´N¬O x3+px+q=0 ªº¸Ñ¤F¡C ¦]¬° u3,v3 º¡¨¬¡A

\begin{displaymath}
u^3+v^3=-q,\quad u^3 v^3=-\frac{p}{27}
\end{displaymath}

©Ò¥H¥¦­Ìº¡¨¬¤G¦¸¤èµ{¦¡ $y^2+qy-\frac{p^3}{27}=0$¡A¥Ñ¦¹¸Ñ±o

\begin{displaymath}
u^3,v^3=-\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}
\end{displaymath}

©Ò¥H
\begin{displaymath}
\begin{eqalign}
x&=u+v\\
&=\sqrt[3]{\frac{-q}{2}+\sqrt{\fr...
...ac{-q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}\\
\end{eqalign}\end{displaymath} (1)

³o¬O©Ò¿×ªº Cardano ¤½¦¡¡C Cardano¡]1501¡ã1576¦~¡^¥H x3+6x=20 ¬°¨Ò¡A»¡©ú³oºØ¸Ñªk¡C¦]¬°

\begin{displaymath}
\frac{q^2}{4}+\frac{p^3}{27}=\frac{(-20)^2}{4}+\frac{6^3}{27}=108
\end{displaymath}

©Ò¥H

\begin{displaymath}
x=\sqrt[3]{10+\sqrt{108}}+\sqrt[3]{10-\sqrt{108}}
\end{displaymath}

¥i¬O·í Cardano ¥Î¦P¼Ë¤èªk¸Ñ x3=15x+4 ®É¡A«oÅý¥LÀ~¤F¤@¤j¸õ¡G¦]¬°

\begin{displaymath}
\frac{q^2}{4}+\frac{p^3}{27}=\frac{(-4)^2}{4}+\frac{(-15)^2}{27}=-121
\end{displaymath}

¦]¦¹

\begin{displaymath}
x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}
\end{displaymath}

¥¦§t¦³­t¼Æªº¥­¤è®Ú¡A¦ý x=4 ©ú©ú¬O¤@­Ó¸Ñ°Ú¡I

¨ä¹ê¡A¥Ñ½Æ¼Æªº­pºâ¥iª¾

\begin{displaymath}
(2 \pm \sqrt{-1})^3=2 \pm 11\sqrt{-1}=2 \pm \sqrt{-121}
\end{displaymath}

©Ò¥H u¡Bv ¥i¨ú¬° $2+\sqrt{-1}$ ¤Î $2-\sqrt{-1}$¡A¦Ó

\begin{displaymath}
x=(2+\sqrt{-1})+(2-\sqrt{-1})=4
\end{displaymath}

¥¿¦n¬O©Ò­nªºµª®×¡Cµê¼Æ©~µM¥i¥HÀ°§U¸Ñ¨M¹ê®Úªº°ÝÃD¡A¥¦°£¤F¯îÂÕ¥~¡A¤S¥[¤W¤@¼h¯«¯µªº¦â±m¡C

±N x-4 ³o­Ó¦]¦¡±q x3-15x-4 °£¥h«á¡A±o°Ó x2+4x+1¡C©Ò¥H x3=15x+4 ÁÙ¦³ x2+4x+1=0 ªº¨â­Ó¸Ñ $x=-2 \pm \sqrt{3}$¡CµM¦Ó¥¦­Ì¦b Cardano ªº¸Ñªk¤¤«ç»ò¨S¦³¥X²{¡H ¤ò¯f¦b©ó§â $2+\sqrt{-121}$ ¶}¤T¦¸¤è®É¡A§Ú­Ì¥u¨ú¨ä¤¤ªº¤@­Ó®Ú $2+\sqrt{-1}$¡C¦pªG¥H w ªí 1 ªº¤T¦¸¤è®Ú $-1+\frac{\sqrt{-3}}{2}$¡A«h $2+\sqrt{-121}$ ªº¨ä¥L¨â­Ó¤T¦¸¤è®Ú¬°

\begin{eqnarray*}
u=(2+\sqrt{-1})w &=&\frac{(-\sqrt{2}-\sqrt{3})+(\sqrt{3}-1)\sq...
...{-1})w^2 &=&\frac{(-\sqrt{2}+\sqrt{3})-(\sqrt{3}+1)\sqrt{-1}}{2}
\end{eqnarray*}


¦]¬° $uv=-\frac{p}{3}$¡A©Ò¥H¬ÛÀ³ªº v ­È¬°

\begin{eqnarray*}
u=(2-\sqrt{-1})w^2 &=&\frac{(-\sqrt{2}-\sqrt{3})-(\sqrt{3}-1)\...
...rt{-1})w &=&\frac{(-\sqrt{2}+\sqrt{3})+(\sqrt{3}+1)\sqrt{-1}}{2}
\end{eqnarray*}


¥Ñ¦¹¥i±o $x=u+v=-2-\sqrt{3}$ ©Î $-2+\sqrt{3}$¡C

±q³o­Ó¨Ò¤l¥iª¾¡A°£¤F(1)¥~¡Ax3+px+q=0 ªº¨ä¥L¨â®Ú¬°

\begin{displaymath}
\begin{eqalign}
x&=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^2}{4}...
...c{-q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} \;w
\end{eqalign}\end{displaymath} (2)

¥Ñ(1)(2)¥iª¾¡A $\frac{q^2}{4}+\frac{p^3}{27}\leq 0 $ ®É¡A¤èµ{¦¡¦³¤T­Ó¹ê®Ú¡A¦Ó·í $\frac{q^2}{4}+\frac{p^3}{27} > 0$ ®É¡A¤èµ{¦¡¦³¤@¹ê®Ú(1)¤Î¨â­Ó¦@³m½Æ®Ú(2)¡CÁöµM¤@©w¦³¹ê®Ú¡A¦b³o¨Ç¤½¦¡¤¤µê¼Æ¦³¦pÅ]°­ªþ¨­¡A·Q±N¨ä´­±ó³£¨­¤£¥Ñ¤v¡C

µê¼Æ¦b Cardano ¤½¦¡¤¤¥X²{«á¡A¤j®a¤S´X¥G§Ñ¤F¥¦ªº¦s¦b¡C¤@ª½¨ì1700¦~¥ª¥k¡A¦b³¡¤À¤À¦¡ªº¿n¤À¤¤¡Aµê¼Æ¤S­«·s¥X²{¡A¿E°_¤F¼Æ¾Ç®a¹ïµê¼Æ¤Î¨ä¹ï¼ÆªºÅG½×¡CÁöµM Euler¡]1707¡ã1783¦~¡^¡Bd'Alembert¡]1717¡ã1783¦~¡^¡BLagrange¡]1736¡ã1813¦~¡^µ¥¼Æ¾Ç®a¨Ï½Æ¼Æªº¨Ï¥Î¦³ªø¨¬ªº¶i®i¡A¦ý¤Q¤K¥@¬ö¼Æ¾Ç®aªººA«×¡A«o¥i¥H¥Î Euler ªº¤@¬q¸Ü¨Ó¥Nªí¡G

©Ò¦³¥i¯àªº¼Æ¤£¬O¤j©ó 0 ´N¬O¤p©ó 0 ©Îµ¥©ó 0¡A©Ò¥H­t¼Æªº¥­¤è®Ú´N¤£¯à¬O­Ó¼Æ¡C¬JµM¬O¤£¥i¯àªº¼Æ¡A¡K¡K¡A§Ú­Ì¥u¯àºÙ¤§¬°µê¼Æ¡A¦]¬°¥u¦³¦bµê¤Û¤¤¤~·|¦³³o¼Ëªº¼Æ¡C

±q¨Ó¡A¦b¦è¤èªº¼Æ¾Ç¤¤¡A¼ÆÁ`¬O¥H´X¦ó¶qªº§ÎºA¥X²{¡A¦Ó¥B­n¥H´X¦ó¶qªº§ÎºA¥X²{¡A¤~·|¥O¤H¦w¤ß¡A¨Ï¤H±µ¨ü¡A$\sqrt{2}$ ©Î­t¼Æ³£¬O¨Ò¤l¡Cµê¼Æ¦pªGµLªk»P´X¦ó¶q³s¦b¤@°_¡A¥¦¦b¼Æ¾Ç¤WÁ`¬O¹³¨p¥Í¤l¤@¼Ë¡AµLªk¦³¦Xªkªº¦a¦ì¡C

¨Ïµê¼Æ¦Xªk¤Æªº§V¤O¡A¸g¹L Wallis¡BKuhn¡BBuée¡BArgand µ¥¤Hªº¹Á¸Õ¡AÁ`ºâº¥¦³¬Ü¥Ø¡C¨ì¤F¤Q¤E¥@¬ö¡A¼Æ¾Ç®a²×©ó²ß©ó§â¤@­Ó½Æ¼Æ x+yi »P¥­­±¤WªºÂI (x,y) ¹ïÀ³°_¨Ó¡Aµê¼Æ¤~§¹¥þ¬°¤H©Ò±µ¨ü¡C

ÀHµÛµê¼Æ¨Ï¥Îªº¤é¨£¬¡ÅD¡A¥ô¤@¦h¶µ¤èµ{¦¡³£·|¦³¡]½Æ¼Æ¡^®Úªº«H©À´N·U¥[¼W±j¡CEuler¡Bd'Alembert¡BLagrange µ¥¤H³£¹Á¸ÕµÛ¥hÃÒ©ú¡Cª½¨ì Gauss¡]1777¡ã1855¦~¡^ªº¥X²{¡A¤~¨ÏÃÒ©ú¦³ªø¨¬ªº¶i¨B¡C1799¦~¡AGauss ¦b Helmstädt ¤j¾Çªº³Õ¤h½×¤å¤¤¡A´N±N«e¤HªºÃÒ©úªº¯Ê¥¢¤@¤@«ü¥X¡AµM«á´£¥X¦Û¤vªº¨£¸Ñ¡C°£¤F³Õ¤h½×¤å¥~¡A¦b1815¦~¤Î1816¦~¡A¥L¤S¤À§O´£¥X¥t¥~¨âºØªºÃÒªk¡C¨ì¤F±ß¦~¡]1849¦~¡^Gauss ¤S¦^¨ì³Õ¤h½×¤åªº¤èªk¡A´£¥X§ó§¹¾ãªº»¡©ú¡C

¦b³Õ¤h½×¤å¤¤¡AGauss ªº·Qªk¬O³o¼Ëªº¡G°²©w f(z) ¬O­Ó½Æ¼Æ z=x+yi ªº¦h¶µ¦¡¡Af(z) ¬O­Ó½Æ¼Æ¡A¥i¥H¼g¦¨¬°

f(z)=p(x,y)+q(x,y)i

p(x,y)¡Bq(x,y) ³£¬O¹ê¼Æ¡A³£¬O x,y ªº¨ç¼Æ¡C¨D f(x)=0 ªº®Ú´Nµ¥©ó¨D p(x,y)=0, q(x,y)=0 ªº¦@¸Ñ¡Cp(x,y)=0 ¤Î q(x,y)=0 ¦b xy ¥­­±¤W¦U¥Nªí¤@±ø¦±½u¡A©Ò¥H Gauss ªº¥D­n½×ÂI´N¬O¦b»¡©ú³o¨â±ø¦±½u¤@©w·|¦³¥æÂIªº¡F³o­Ó¥æÂI©Ò¹ïÀ³ªº z ­È¡A¦ÛµM´N¬O f(z)=0 ªº®Ú¤F¡C

Gauss ªº²Ä¤G­ÓÃÒ©ú¥Îªº¬O¨â­Ó¦h¶µ¦¡ªºµ²¦¡ (resultant)¡C­Y $g(x)=a_m x^m+a_{m-1} x^{m-1}+\cdots+a_0$¡B $h(x)=b_n x^n+b_{n-1} x^{n-1}+\cdots+b_0$ ¬°¨â­Ó¦h¶µ¦¡¡A«h¦æ¦C¦¡

\begin{displaymath}
R(g,h)=
\begin{array}{\vert cccccc\vert c}
a_m & a_{m-1}&\cd...
...& & & & &\\
& & b_n & b_{n-1}& \cdots & b_0 & \\
\end{array}\end{displaymath}

´N¬O³o¨â­Ó¦h¶µ¦¡ªºµ²¦¡¡Cg(x)=0 »P h(x)=0 ¦³¦@¸Ñªº±ø¥ó¬O R(g,h)=0¡A¥H g(x)=x(x-a)=x2-ax, h(x)=(x-1)(x-b)=x2-(1+b)x+b ¬°¨Ò¡A

\begin{displaymath}
R(g,h)=
\begin{array}{\vert cccc\vert}
1 & -a & 0 & 0 \\
...
... &-(b+1)&b& 0 \\
0 & 1 &-(b+1)& b\\
\end{array}=(a-1)b(a-b)
\end{displaymath}

©Ò¥H·í a=1 ©Î b=0 ©Î a=b ®É¡AR(g,h)=0¡A¦Ó¦¹®É g(x)=0¡Ah(x)=0 ¦³¦@¸ÑÅãµM¬O¹ïªº¡C

¦pªG f(z) ªº¦¸¼Æ n ¬°©_¼Æ¡A«h f(z)=0 ¦Ü¤Ö¦³¤@¹ê®Ú¡A³o¬O¦]¬° z ¬O«Ü¤jªº¥¿¹ê¼Æ®É¡Af(z) »P¨ä­º¶µ«Y¼Æ¦P¸¹¡A¦Ó·í z ¬O«Ü¤jªº­t¹ê¼Æ®É¡Af(z) »P¨ä­º¶µ«Y¼Æ²§¸¹¡A©Ò¥H¥Ñ¤¤¶¡­È©w²z¡A¥iª¾ f(z)=0 ¥²¦³¤@®Ú¡]¨£¹Ï¤@¡^¡A



¹Ï¤@¡Gf(a)f(b)< 0¡A©Ò¥Hf(x)=0¦³¸Ñ¡C

·í n ¬°°¸¼Æ¡A¥B¥u§t 2 ªº¤@¦¸¦]¼Æ®É¡AGauss ¦Ò¼{

\begin{eqnarray*}
g(z,w)&=& f(z+w)+f(z-w),\\
h(z,w)&=& f(z+w)-\frac{F(z-w)}{w}
\end{eqnarray*}


¥¦­Ì¹ï z ¦Ó¨¥¦U¬° n ¦¸¤Î n-1 ¦¸¡C¥¦­Ìªºµ²¦¡ R(g,h) ¬° w ªº n(n-1) ¦¸¦h¶µ¦¡¡A¦ý¬O¦]¬°¦b g »P h ¤¤¡Aw ªº©_¦¸¶µ¨Ã¤£¥X²{¡A©Ò¥H R(g,h)=0 ¥i¥H¬Ý¦¨ t=w2 ªº $\frac{n(n-1)}{2}$ ¦¸¤èµ{¦¡¡F¥¦¬O©_¦¸ªº¡A©Ò¥H¦³­Ó¸Ñ t=t0¡C ¦]¦¹ $g(z,\sqrt{t_0})=0$ »P $h(z,\sqrt{t_0})=0$ ¦³¦@¸Ñ¡A¦¹¦@¸Ñ¥²¬° $f(z+\sqrt{t_0})=0$ ªº¸Ñ¡A¥Ñ¦¹±oª¾ f(z)=0 ¦³¸Ñ¡C ¦pªG n §t¦³ 2 ªº¤G¦¸¥H¤Wªº¦]¼Æ¡A«h¤@¦A¨Ï¥Îµ²¦¡¡A¤]¥i¥H±o¨ì¦P¼Ëªºµ²½×¡C

Gauss ªº²Ä¤T­ÓÃÒ©ú¡A¥Îªº¬O½ÆÅܨç¼Æªº¿n¤À²z½×¡A¬Û·í²`¤J¡A¦b¦¹²¤¹L¤£ªí¡C

Gauss ¹ï¥N¼Æ¾Ç°ò¥»©w²z©Òµ¹ªºÃÒ©ú¬O¹º®É¥Nªº¡C±q§Æþ¥H¨Ó¡A¼Æ¾Çªº¹êÅ饲¥ý§ä¨ì¡A¤~¯à½Í¥¦ªº©Ê½è¡F¤]´N¬O»¡ÃÒ©úªº¹Lµ{¥²¶·±q¤wª¾ªº¶q¥Xµo¡A¤@¨B¤@¨B§@¥X¤¤¶¡¹Lµ{ªº¸É§U¶q¡A¥H¹F¨ì³Ì«á©Ò­nªº¶q¡C³o¬O©Ò¿×«Øºc«¬ªºÃÒ©ú¡C Gauss ªº¤èªk«o¤£«ö¨B´N¯Z¡A¥L¥u¥Î¾Ç²zªÖ©w©Ò­n¶qªº¦s¦b©Ê¡A¦Ó©Ò­nªº¶q¬O§_«ö¨B´N¯Z¥i±o¡A«o¤£¬OÃÒ©ú©ÒÃö¤ßªº¡A³o¬O¤@ºØ¦s¦b«¬ªºÃÒ©ú¡C

±q Gauss ¶}©l¡A¦s¦b«¬ªºÃÒ©ú³vº¥½«¬°­·®ð¡AÅܦ¨²{¥N¼Æ¾Çªº¯S¦â¤§¤@¡C

 
¹ï¥~·j´MÃöÁä¦r¡G
¡D¥N¼Æ¾Ç°ò¥»©w²z
¡DCardano
¡DEuler
¡Dd'Alembert
¡DLagrange
¡DWallis
¡DGauss
¡D«Øºc«¬ªºÃÒ©ú
¡D¦s¦b«¬ªºÃÒ©ú
 
ªþ¿ý¡G¥t¤@ºØÃÒªk¡]°ª¤¤¥Í¤]¯àÀ´ªº¡^

·í½Æ¼Æ z ¦b¥­­±¤WÅܰʮɡA|f(z)| ¤]¸òµÛÅÜ°Ê¡C¦Ò¼{ªÅ¶¡¤¤ªº¤@­Ó¦±­±¡A¥¦¦b xy ¥­­±¤W¤@ÂI (x,y) ¤Wªº°ª«×¬° |f(x+yi)|¡C¦]¬° |f(x+yi)| ÀHµÛ z=x+yi °µ³sÄòÅܤơA©Ò¥H³o­Ó¦±­±ªº°ª«×¤£·|¦³¬ðµMÂ_Â÷ªº²{¶H¡C

°²³] $f(z)=a_n z^n+a_{n-1} z^{n-1}+\cdots+a_0$¡A·í z ÂIÂ÷­ìÂI«Ü»·®É¡A¤]´N¬O·í |z| «Ü¤j®É¡A¦]¬°

\begin{eqnarray*}
\vert f(z)\vert& \geq & \vert a_n\vert\vert z\vert^n-(\vert a_...
...vert z\vert}+\cdots
+\frac{\vert a_0\vert}{\vert z\vert^n})]\\
\end{eqnarray*}


©Ò¥H |f(z)| ¥i¥HÅܱo«Ü¤j¡C ¥Ñ¦¹¥iª¾¡A³o­Ó¦±­±¦b¸û»·ªº³¡¤À¡A°ª«×·|Åܱo«Ü¤j¡A©Ò¥H±q´X¦ó¹Ï§ÎªºÆ[ÂI¨Ó¬Ý¡A¾ã­Ó¦±­±·|¦Ü¤Ö§t¦³¤@­Ó³Ì§CÂI f(z0)¡C§Ú­Ìªº¥Øªº´N¬O­n»¡ |f(z0)|=0¡A¥ç§Y f(z0)=0¡C



¹Ï¤G

¦pªG¤£µM¡A«h |f(z0)>0|¡Cµe­Ó¹Ï¨Ó»¡¡]¨£¹Ï¤G¡^¡A³oªí¥Ü f(z) ¨M¤£·|¸¨¤J¥H­ìÂI¬°¤¤¤ß¡A|f(z0)| ªø¬°¥b®|ªº¶ê¤º¡C§Ú­Ì¦Ò¼{ z0 ªþªñªºÂI z0+w¡AÆ[¹î f(z0+w) ªºÅܤƱ¡§Î¡G

\begin{eqnarray*}
f(z_0+w)&=& a_n (z_0+w)^n+a_{n-1}(z_0+w)^{n-1}+\cdots+a_0 \\
...
...inus0.1pt{\fontfamily{cwM2}\fontseries{m}\selectfont \char 225}}
\end{eqnarray*}


¥O $w_0 = r_0(\cos{\theta_0}+i \cos{\theta_0})$, $w = r(\cos \theta+i \sin \theta)$¡A«h

\begin{displaymath}
w w_0 = r_0 r[\cos(\theta_0+\theta)+i \sin(\theta_0+\theta)]
\end{displaymath}

§Ú­Ì¥i¥H¿ï¾A·íªº £c ¨¤¤Î«Ü¤pªº r ­È¡A¨Ï±o f(z0)+w0 w ¸¨¤J¶ê°é¤º¡A¦Ó¥B·í r ­È°÷¤p®É¡A§t w2 ¦]¼Æªº¦U¶µ¡A¹ï f(z0+w) ªº¦ì¸m¤£·|¤Þ°_¦³·N¸qªºÅܤơA¤]´N¬O»¡ f(z0+w) ·|ÀHµÛ f(z0)+w0 w ¤]¦b¶ê°é¤º¡C³o´N»P¥ý«e¶ê°é¤ºµL­Èªº°²©w¬Û¥Ù¬Þ¤F¡C¤]´N¬O»¡ |f(z0)|>0 ªº°²©w¤£¯à¦¨¥ß¡C

   

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