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version 1.2, 2001/07/26 09:19:34

version 1.5, 2001/07/28 07:07:20

Line 1

%$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 1.4 2001/07/28 06:37:39 noro Exp $

\documentclass[12pt]{jarticle}

%\oddsidemargin -0.25in

%\evensidemargin -0.25in

Line 51 computer $B$H$$$&8@MU$O(B, $BJ8;zDL$j$K2r<a$9$l$P!V

\end{itemize}

$B$3$N$h$&$K(B CPU $B$X$NL?Na$N0l$D0l$D$OC1=c$J$b$N$P$+$j$G$"$k(B.

$B07$($k?t$NBg$-$5$O(B, $B$9$J$o$A%l%8%9%?$NBg$-$5$H8@$C$F$h$$(B. $B$?$H$($P(B,

$B07$($k?t$NBg$-$5$O(B, $B%l%8%9%?$NBg$-$5$G7h$^$k$H8@$C$F$h$$(B. $B$?$H$($P(B,

32 $B%S%C%H%l%8%9%?$H$$$&$N$O(B 0 $B$^$?$O(B 1 $B$rI=$95-21AuCV$,(B 32 $B8D$"$k(B

$B%l%8%9%?$G$"$k$,(B, $B$3$N%l%8%9%?$O(B 0 $B$+$i(B $2^{32}-1$ $B$^$G$N@0?t$7$+(B

$BJ];}$G$-$J$$$3$H$K$J$k(B.

Line 62 CPU $B$NDs6!$9$k5!G=$@$1$G$O(B, $B?t3X$K;H$&$K$OITB

$B$NCM$r(B $2^{32}$ $B$G3d$C$?M>$j$G$"$k(B. $B$"$k$$$O(B,

$BEEBn$N$h$&$K(B $1.234567 \times 10^{20}$ $B$H$$$&CM$rJV$5$l$k$N$b:$$k(B.

$B8m:9$,F~$C$F$7$^$&$H(B, $B?t3XE*$K$O0UL#$N$J$$7k2L$H$J$j$+$M$J$$(B.

$B$3$N$3$H$+$i(B, $B:GDc8B(B, $BBg$-$J@0?t$r07$($J$$$H:$$k$H$$$&$3$H$,$o$+$k(B.

$B$9$J$o$A(B, $B?t3X$K;H$&$K$O(B, $B07$($k@0?t$NBg$-$5$K@)8B$,$"$C$F$O$J$i$J$$(B.

$B$3$N$?$a$K$OG$0U$NBg$-$5$N@0?t$r07$&$?$a$N%W%m%0%i%`$r=q$1$P$h$$(B.

$B$9$J$o$A(B, $B%a%b%j>e$K(B, $BNc$($P(B 32$B%S%C%H@0?t$rJB$Y$F(B, $B%3%s%T%e!<%?$K(B

$B!VI.;;!W$r$5$;$l$P$h$$(B. $B$3$N>l9g(B, $B?M4V$H0[$J$k$N$O(B, $B?M4V$N>l9g(B,

Line 71 CPU $B$NDs6!$9$k5!G=$@$1$G$O(B, $B?t3X$K;H$&$K$OITB

$BNc$H$7$F(B, $B@0?t$NB-$7;;$O<!$N$h$&$K$J$k(B.

\begin{tabular}{ccccc}\\

& $2^{64}$ & $2^{32}$ & $1$ \\

& 5 & 4001257187 & 1914644777 & (= $3^{42}$) \\

+ & & 2830677074 & 689956897 & (= $3^{40}$) \\ \hline

& 6 & 2536966965 & 2604601674 & (= $10\times 3^{40}$)\\

Line 113 $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D

Line 114 $a^2-4b = t^2$ ($t$ : $B@0?t(B) $B$H$+$1$k$+$I$&$+D

\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

$B$,$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

$B$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,

Line 130 $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H

Line 131 $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H

$BCf4VCM$NDjM}(B

$B!V(B$f(a) < 0$, $f(b) > 0$ $B$J$i(B $a$, $b$ $B$N4V$K(B $f(c)=0$ $B$J$k(B $c$ $B$,$"$k(B.$B!W(B

$B$r;H$C$F:,$rC5$9J}K!$G$"$k!VFsJ,K!!W$"$k$$$O@\@~$rMQ$$$k%K%e!<%H%sK!(B

$B$,$"$k(B ($B$$$:$l$b9b9;?t3X(B C $B$K$"$k(B). $B$3$l$i$H(B, $B@0?t:,$NM-L5$rD4$Y$l$P(B

$B$,$"$k(B ($B$$$:$l$b9b9;?t3X(B C $B$K$"$k(B). $BLdBj$O@0?t:,$NM-L5$J$N$G(B,

$B$h$$(B, $B$H$$$&$3$H$+$i(B, $BM-8B2s$G7W;;$,$G$-$k$3$H$,J,$+$k(B. $B$7$+$7(B, $BB?9`<0(B

$B$3$l$i$NJ}K!$K$h$jHf3SE*MF0W$K:,$,C5$;$k(B. $B$7$+$7(B, $BB?9`<0(B

$B$N<!?t$,(B 4 $B<!0J>e$N>l9g(B, $B0x;R$N<!?t$,$5$^$6$^$G$"$k$?$a(B, $B:,$rC5$9J}K!(B

$B$rE,MQ$9$k$N$O:$Fq$G$"$m$&(B. $B$=$3$G(B, $B%3%s%T%e!<%?$K9g$C$?J}K!$rC5$9$3$H(B

$B$K$9$k(B. $B%R%s%H$H$7$F$O(B,

\begin{itemize}

\item $B!V6a;w!W$r$&$^$/;H$&(B

\item $B%3%s%T%e!<%?$O7+$jJV$7$,F@0U(B

\item $B%3%s%T%e!<%?$O7+$jJV$7(B, $B;n9T:x8m$,F@0U(B ($B$J$K$r$d$i$;$F$bJ86g$r8@$o$J$$(B)

\end{itemize}

$B$H$$$&(B 2 $BE@$G$"$k(B. $BA0<T$O(B, $BCf4VCM$NDjM}$,<B?t$K$*$1$k6a;w$H(B

$B7k$SIU$$$?$h$&$K(B, $BB>$N%?%$%W$N6a;w$,;H$($J$$$+(B, $B$H$$$&$3$H$G$"$k(B.

$B8e<T$O(B, $B$=$N$h$&$J6a;w$r7+$jJV$7$FL\E*$NJ,2r$K6aIU$$$F$$$3$&(B, $B$H$$$&(B

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

$B$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

$B$"$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

\begin{enumerate}

Line 160 $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H

Line 158 $(a^2-4b)/4 = 4717584$ $B$,(B $2172^2$ $B$G$"$k$3$H

$h_1$ $B$r8+$D$1$k(B.

\item $f(x)-g_k(x)h_k(x)$ $B$N78?t$,(B $p^k$ $B$G3d$j@Z$l$k$h$&$K(B $g_k$, $h_k$ $B$r(B

$B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$)

$B=g<!:n$C$F$$$/(B ($k=2,3,\ldots$) --- $B$@$s$@$s!V@:EY!W$,>e$,$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.

\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.

$B$^$:(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

$$f(x) = a_0(x)+p \cdot a_1(x)+p^2\cdot a_2(x)+\cdots$$

Line 250 $tr \equiv 2 \bmod 3$}\\

Line 248 $tr \equiv 2 \bmod 3$}\\

\noindent

$B$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

$$f \equiv b_0c_0 \bmod 3$$ $B$,@.$jN)$D(B.

$f \equiv b_0c_0 \bmod 3$ $B$,@.$jN)$D(B.

$B$b$H$N<0$KLa$k$H(B,

$$gh \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$h$j(B

$$f-gh \equiv a_0-b_0c_0+3(a_1-(c_0b_1+b_0c_1)) \bmod 3^2$$

Line 282 $2r+t \equiv 0 \bmod 3$}\\

Line 280 $2r+t \equiv 0 \bmod 3$}\\

\noindent

$B$3$l$G(B $$f \equiv (b_0+3b_1)(c_0+3c_1) \bmod 3^2$$ $B$H$J$k(B $b_1$, $c_1$

$B$,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,

($i=2,3,\ldots$) $B$H$*$$$F(B $(q,r,s,t)$ $B$NO"N)0l<!9gF1<0$r=g<!(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, $BA0$HF1MM$K(B $(q,r,s,t)$ $B$NO"N)0l<!(B

$B9gF1<0$,F@$i$l$k(B.

$B0J2<F1MM$K(B $b_i = qx+r, c_i = sx+t$

($i=3,4,\ldots$) $B$H$*$$$F(B $(q,r,s,t)$ $B$NO"N)0l<!9gF1<0$r=g<!(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\

}) \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]

\label{gh}

\begin{center}

Line 310 $k$ & $g_k$ & $h_k$ \\ \hline

Line 317 $k$ & $g_k$ & $h_k$ \\ \hline

\caption{($g_k$, $h_k$)}

\end{center}

\end{table}

$BI=(B 1 $B$O(B, $B$3$NA`:n$rB3$1$?$H$-$N(B, $B3F%9%F%C%W$K$*$1$k(B $g_k$ $h_k$ $B$r<($9(B.

$BI=(B 1 $B$O(B, $B$3$NA`:n$rB3$1$?$H$-$N(B, $B3F%9%F%C%W$K$*$1$k(B $g_k$, $h_k$ $B$r<($9(B.

$BI=$G8+$k$H(B, $k = 12$ $B$+$i(B $k = 13$ $B$GJQ2=$,$J$$$3$H$,J,$+$k(B. $B<B:]$K(B

$f-g_{13}h_{13}$ $B$r7W;;$7$F$_$k$H(B 0 $B$G$"$k$3$H$,J,$+$k(B.

$B$9$J$o$A(B

Line 344 $p$ $B$,AG?t$N$H$-(B, $GF(p) = \{0,1,\cdots,p-1\}$$

Line 351 $p$ $B$,AG?t$N$H$-(B, $GF(p) = \{0,1,\cdots,p-1\}$$

$B$3$l$O8@$$BX$($k$H(B

$B!V(B$a {\not \equiv} 0 \bmod p$ $B$J$i(B $ab \equiv 1 \bmod p$ $B$J$k(B $b$ $B$,$"$k!W(B

$B$H$$$&$3$H$G$"$k(B.

$B!V(B$a$, $p$ $B$,8_$$$KAG$N$H$-@0?t(B $b$, $q$ $B$,B8:_$7$F(B $ab+qp=1$$B!W(B

$B!V(B$a$, $p$ $B$,8_$$$KAG$N$H$-@0?t(B $b$, $q$ $B$,B8:_$7$F(B $ab+pq=1$$B!W(B

$B$H=q$1$P(B, $B8+$?$3$H$,$"$k$+$b$7$l$J$$(B. $B$3$l$O(B, $B%f!<%/%j%C%I$N8_=|K!$N(B

$BI{;:J*$H$7$FF@$i$l$k7k2L$G$"$k(B.

\end{enumerate}

$B$9$J$o$A(B, $GF(p)$ $B$OBN(B($B%?%$(B)$B$r$J$9(B.

$B85$N8D?t$,M-8B8D(B ($p$ $B8D(B)$B$J$N$GM-8BBN$H$h$V(B. $B$5$F(B, $a_0 \equiv f \bmod p$

$B$r(B $BBN(B $GF(p)$ $B$K78?t$r$b$D0lJQ?tB?9`<0$H8+$k$H(B, $a_0 \equiv b_0c_0 \bmod p$

Line 503 p^k$ $B$G$N0x?tJ,2r$K;}$A>e$2$F(B, $B:G=*E*$K@0?t>e

Line 509 p^k$ $B$G$N0x?tJ,2r$K;}$A>e$2$F(B, $B:G=*E*$K@0?t>e

$B$$$i$l$F$$$k$3$H$b>R2p$7$?(B. $B%3%s%Q%/%H%G%#%9%/$N?.Mj@-$b(B, $BM-8BBN$rMQ$$(B

$B$?Id9f$G$"$k(B Reed-Solomon $BId9f$G;Y$($i$l$F$$$k$3$H$r9M$($l$P(B, $BM-8BBN$,(B

IT $B<R2q$rN"$G;Y$($F$$$k$H$$$C$F$b2a8@$G$O$J$$$@$m$&(B. $B?t3X$r@$$NCf$NLr(B

$B$KN)$F$h$&(B, $B$J$I$H9M$($F$$$k?t3X<T$O$[$H$s$I$$$J$$$@$m$&$7(B, $B$^$?(B, $B$=$&(B

$B$KN)$F$h$&(B, $B$J$I$H9M$($F$$$k?t3X<T$O$"$^$j$$$=$&$K$J$$$7(B, $B$^$?(B, $B$=$&(B

$B$$$&$7$,$i$_$+$i<+M3$G$"$k$H$3$m$,?t3X$NH/E8$N8;@t$H$$$($k$N$+$b$7$l$J(B

$B$$(B. $B$7$+$7(B, $B$=$N$h$&$K$7$FF@$i$l$?7k2L$,(B, $B8e$GA[A|$b$D$+$J$$$H$3$m$G1~(B

$BMQ$5$l$k>l9g$b$"$k(B. $B$^$?(B, $B0lHL$K4w$_7y$o$l$k860x$H$J$k!VFq$7$5!W$,(B, $B0E(B

Line 521 Seminumerical Algorithms, Third ed. Addison-Wesley (19

Line 527 Seminumerical Algorithms, Third ed. Addison-Wesley (19

$B?t3X65<<(B (2001).

\bibitem{ASIR}

$BLnO$(B $BB>(B, $B7W;;5!Be?t%7%9%F%`(B Risa/Asir.

$BLnO$(B $BB>(B, $B7W;;5!Be?t%7%9%F%`(B Risa/Asir (1994-2001).

{\tt ftp://archives.cs.ehime-u.ac.jp/pub/asir2000/} (1994-2001).

{\tt ftp://archives.cs.ehime-u.ac.jp/pub/asir2000/}

\end{thebibliography}

$BK\9F$G=R$Y$?$3$H$O(B, $B<g$K%3%s%T%e!<%?$,CB@8$7$?8e$K9M0F$5$l$?J}K!$G(B, $BJ8(B

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