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version 1.2, 2001/07/26 09:19:34 version 1.3, 2001/07/28 03:31:09
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 %$OpenXM: OpenXM/doc/sci-semi2001/factor-resume.tex,v 1.1 2001/07/26 07:55:04 noro Exp $  %$OpenXM: OpenXM/doc/sci-semi2001/factor-resume.tex,v 1.2 2001/07/26 09:19:34 noro Exp $
 \documentclass[12pt]{jarticle}  \documentclass[12pt]{jarticle}
 %\oddsidemargin -0.25in  %\oddsidemargin -0.25in
 %\evensidemargin -0.25in  %\evensidemargin -0.25in
Line 113  $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D
Line 113  $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D
 \end{enumerate}  \end{enumerate}
   
 $B$5$F(B, $BNc$($P(B $f(x) = x^2+11508x+28386587$ $B$r0x?tJ,2r$9$k>l9g(B, $B$I$NJ}K!(B  $B$5$F(B, $BNc$($P(B $f(x) = x^2+11508x+28386587$ $B$r0x?tJ,2r$9$k>l9g(B, $B$I$NJ}K!(B
 $B$,$h$$$@$m$&$+(B. $B@52r$O(B $f(x)=(x+3581)(x+7927)$ $B$@$,(B,  $B$,$h$$$@$m$&$+(B. $f(x)$ $B$O(B $f(x)=(x+3581)(x+7927)$ $B$HJ,2r$5$l$k$,(B,
 $28386587=3581\cdot 7927$ $B$H$$$&AG0x?tJ,2r$,!V4cNO!W$GJ,$+$k?M$OB?J,>/(B  $28386587=3581\cdot 7927$ $B$H$$$&AG0x?tJ,2r$,!V4cNO!W$GJ,$+$k?M$OB?J,>/(B
 $B$J$$$H;W$&(B. $B<B:](B, $BB?9`<0$N0x?tJ,2r$KHf$Y$F(B, $B@0?t$NAG0x?tJ,2r$N$[$&$,$O(B  $B$J$$$H;W$&(B. $B<B:](B, $BB?9`<0$N0x?tJ,2r$KHf$Y$F(B, $B@0?t$NAG0x?tJ,2r$N$[$&$,$O(B
 $B$k$+$K:$Fq$JLdBj$G$"$k(B. $B$^$?(B, $BAG0x?tJ,2r$,4JC1$G$b(B, $BAG0x?t$,B?$9$.$k$H(B,  $B$k$+$K:$Fq$JLdBj$G$"$k(B. $B$^$?(B, $BAG0x?tJ,2r$,4JC1$G$b(B, $BAG0x?t$,B?$9$.$k$H(B,
Line 151  $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H
Line 151  $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H
 $B$3$l$+$i=R$Y$kJ}K!$O(B, $B0x;R$N7A(B ($B<!?t(B)$B$r2>Dj$7$F(B, $B$=$N78?t$r6a;w$K(B  $B$3$l$+$i=R$Y$kJ}K!$O(B, $B0x;R$N7A(B ($B<!?t(B)$B$r2>Dj$7$F(B, $B$=$N78?t$r6a;w$K(B
 $B$h$j5a$a$F$$$/J}K!$G$"$k(B. $B$3$N>l9g$K;X?K$H$J$k86M}$O(B  $B$h$j5a$a$F$$$/J}K!$G$"$k(B. $B$3$N>l9g$K;X?K$H$J$k86M}$O(B
 $B!V@0?t(B $m$ $B$,(B 0 $\Leftrightarrow$ $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k!W(B  $B!V@0?t(B $m$ $B$,(B 0 $\Leftrightarrow$ $m$ $B$O$I$s$J@0?t$G$b3d$j@Z$l$k!W(B
 $B$"$k$$$O(B  
 $B!V@0?t(B $m$ $B$,(B 0 $\Leftrightarrow$ $m$ $B$O==J,Bg$-$$@0?t$G3d$j@Z$l$k!W(B  
 $B$H$$$&$b$N$G$"$k(B. $B$3$l$rMQ$$$F(B, $B$?$H$($P(B  $B$H$$$&$b$N$G$"$k(B. $B$3$l$rMQ$$$F(B, $B$?$H$($P(B
   
 \begin{enumerate}  \begin{enumerate}
Line 164  $h_1$ $B$r8+$D$1$k(B. 
Line 162  $h_1$ $B$r8+$D$1$k(B. 
   
 \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.  \item $g_1$, $h_1$ $B$,@52r$KBP1~$7$F$$$l$P(B, $BE,Ev$J(B $k$ $B$N$H$3$m$G$[$s$H$K3d$j@Z$l$k$@$m$&(B.
 \end{enumerate}  \end{enumerate}
 $B$H$$$&%?%$%W$N%"%k%4%j%:%`$r9=@.$9$k$N$G$"$k(B. $B8@$$$+$($k$H<!$N$h$&$K$J$k(B.  $B$H$$$&%?%$%W$N%"%k%4%j%:%`$r9=@.$9$k(B. $B8@$$$+$($k$H<!$N$h$&$K$J$k(B.
 $B0J2<(B, $B4JC1$N$?$a(B $f(x)$ $B$*$h$S$=$N0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.  $B0J2<(B, $B4JC1$N$?$a(B $f(x)$ $B$*$h$S$=$N0x;R$N78?t$OA4$F@5$G$"$k$H$9$k(B.
 $B$^$:(B, $f(x)$ $B$N3F78?t$r(B $p$-$B?J?t$GI=$7(B, $B3F(B $p^k$ $B$4$H$K$^$H$a$F(B  $B$^$:(B, $f(x)$ $B$N3F78?t$r(B $p$-$B?J?t$GI=$7(B, $B3F(B $p^k$ $B$4$H$K$^$H$a$F(B
 $$f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$$  $$f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$$
Line 250  $tr \equiv 2 \bmod 3$}\\
Line 248  $tr \equiv 2 \bmod 3$}\\
 \noindent  \noindent
 $B$r0UL#$9$k$,(B, $B$3$l$i$O%Z%"$H$7$F$OF1$8$b$N$G$"$k(B.  $B$r0UL#$9$k$,(B, $B$3$l$i$O%Z%"$H$7$F$OF1$8$b$N$G$"$k(B.
 $b_0=x^2+1$, $c_0=x^2+x+2$ $B$H$9$k$H3N$+$K(B  $b_0=x^2+1$, $c_0=x^2+x+2$ $B$H$9$k$H3N$+$K(B
 $$f \equiv b_0c_0 \bmod 3$$ $B$,@.$jN)$D(B.  $f \equiv b_0c_0 \bmod 3$ $B$,@.$jN)$D(B.
 $B$b$H$N<0$KLa$k$H(B,  $B$b$H$N<0$KLa$k$H(B,
 $$gh \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$h$j(B  $$gh \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$h$j(B
 $$f-gh \equiv a_0-b_0c_0+3(a_1-(c_0b_1+b_0c_1)) \bmod 3^2$$  $$f-gh \equiv a_0-b_0c_0+3(a_1-(c_0b_1+b_0c_1)) \bmod 3^2$$
Line 282  $2r+t \equiv 0 \bmod 3$}\\
Line 280  $2r+t \equiv 0 \bmod 3$}\\
   
 \noindent  \noindent
 $B$3$l$G(B $$f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$H$J$k(B $b_1$, $c_1$  $B$3$l$G(B $$f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$H$J$k(B $b_1$, $c_1$
 $B$,5a$^$C$?$3$H$K$J$k(B. $B0J2<F1MM$K(B $$b_i = qx+r, c_i = sx+t$$  $B$,5a$^$C$?$3$H$K$J$k(B. $B<!$O(B $a_2$, $b_2$, $c_2$ $B$^$G$H$C$F(B $\bmod 3^3$ $B$G8+$H(B,
   $$f \equiv a_0+3a_1+3^2a_2 \equiv (b_0+3b_1+3^2b_2)(c_0+3c_1+3^2c_2) \bmod 3^3$$
   $B$h$j(B
   $$((a_0+3a_1)-(b_0+3b_1)(c_0+3c_1))+3^2(a_2-(c_0b_2+b_0c_2)) \equiv 0 \bmod 3^3$$
   ($3^3$ $B$G3d$j@Z$l$k9`$O<N$F$?(B.) $B@hF,ItJ,$O(B $3^2$ $B$G3d$j@Z$l$k$N$G(B,
   $BN>JU$r(B $3^2$ $B$G3d$k$H(B
   $$((a_0+3a_1)-(b_0+3b_1)(c_0+3c_1))/3^2+(a_2-(c_0b_2+b_0c_2)) \equiv 0 \bmod 3$$
   $b_2 = qx+r, c_2 = sx+t$ $B$H$*$/$H(B, $k=1$ $B$HF1MM$N(B $(q,r,s,t)$ $B$NO"N)0l<!(B
   $B9gF1<0$,F@$i$l$k(B.
   $B0J2<F1MM$K(B $b_i = qx+r, c_i = sx+t$
 ($i=2,3,\ldots$) $B$H$*$$$F(B $(q,r,s,t)$ $B$NO"N)0l<!9gF1<0$r=g<!(B  ($i=2,3,\ldots$) $B$H$*$$$F(B $(q,r,s,t)$ $B$NO"N)0l<!9gF1<0$r=g<!(B
 $B2r$$$F$$$1$P(B  $B2r$$$F$$$1$P(B
 $$f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1\  $$f \equiv (b_0+\ldots+3^{k-1}b_{k-1})(c_0+\ldots+3^{k-1}c_{k-1\
 }) \bmod 3^k$$  }) \bmod 3^k$$
 $B$9$J$o$A(B$$f \equiv g_kh_k \bmod 3^k$$ $B$H$J$k(B $g_k$, $h_k$ $B$,7h$^$k(B.  $B$9$J$o$A(B$f \equiv g_kh_k \bmod 3^k$ $B$H$J$k(B $g_k$, $h_k$ $B$,7h$^$k(B.
 \begin{table}[hbtp]  \begin{table}[hbtp]
 \label{gh}  \label{gh}
 \begin{center}  \begin{center}
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