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version 1.3, 2001/07/28 03:31:09

version 1.4, 2001/07/28 06:37:39

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%$OpenXM: OpenXM/doc/sci-semi2001/factor-resume.tex,v 1.2 2001/07/26 09:19:34 noro Exp $

%$OpenXM: OpenXM/doc/sci-semi2001/factor-resume.tex,v 1.3 2001/07/28 03:31:09 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.

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$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 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

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$B$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

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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}

Line 287 $$((a_0+3a_1)-(b_0+3b_1)(c_0+3c_1))+3^2(a_2-(c_0b_2+b_

($3^3$ $B$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

$b_2 = qx+r, c_2 = sx+t$ $B$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 $b_i = qx+r, c_i = sx+t$

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

($i=3,4,\ldots$) $B$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$$

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 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 510 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

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$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

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$B$J$,$i(B, $B<B$O$3$C$=$j@$$NCf$NLr$KN)$C$F$$$k$H$$$&(B, $B?t3X$N2{$N?<$5$rI=$7(B

$B$F$$$k$H$$$&5$$,$9$k(B.

\begin{thebibliography}{99}

Line 528 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}

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