| version 1.9, 2003/12/18 10:26:20 |
version 1.15, 2020/09/01 09:25:32 |
|
|
| @comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.8 2003/04/19 15:44:59 noro Exp $ |
@comment $OpenXM: OpenXM/src/asir-doc/parts/builtin/num.texi,v 1.14 2020/08/27 11:05:27 takayama Exp $ |
| \BJP |
\BJP |
| @node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B |
@node $B?t$N1i;;(B,,, $BAH$_9~$_H!?t(B |
| @section $B?t$N1i;;(B |
@section $B?t$N1i;;(B |
|
|
| * fac:: |
* fac:: |
| * igcd igcdcntl:: |
* igcd igcdcntl:: |
| * ilcm:: |
* ilcm:: |
| |
* isqrt:: |
| * inv:: |
* inv:: |
| * prime lprime:: |
* prime lprime:: |
| * random:: |
* random:: |
|
|
| * conj real imag:: |
* conj real imag:: |
| * eval deval:: |
* eval deval:: |
| * pari:: |
* pari:: |
| * setprec:: |
* setbprec setprec:: |
| * setmod:: |
* setmod:: |
| * lrandom:: |
* lrandom:: |
| * ntoint32 int32ton:: |
* ntoint32 int32ton:: |
| |
* setround:: |
| |
* inttorat:: |
| @end menu |
@end menu |
| |
|
| \JP @node idiv irem,,, $B?t$N1i;;(B |
\JP @node idiv irem,,, $B?t$N1i;;(B |
| Line 747 These functions works also for polynomials with comple |
|
| Line 750 These functions works also for polynomials with comple |
|
| @code{deval} $B$OG\@:EYIbF0>.?t$r7k2L$H$7$F(B |
@code{deval} $B$OG\@:EYIbF0>.?t$r7k2L$H$7$F(B |
| @code{eval} $B$N>l9g(B, $BM-M}?t$O$=$N$^$^;D$k(B. |
@code{eval} $B$N>l9g(B, $BM-M}?t$O$=$N$^$^;D$k(B. |
| @item |
@item |
| @code{eval} $B$K$*$$$F$O(B, $B7W;;$O(B @b{PARI} (@ref{pari}) $B$,9T$&(B. |
@code{eval} $B$K$*$$$F$O(B, $B7W;;$O(B @b{MPFR} $B%i%$%V%i%j$,9T$&(B. |
| @code{deval} $B$K$*$$$F$O(B, $B7W;;$O(B C $B?t3X%i%$%V%i%j$N4X?t$rMQ$$$F9T$&(B. |
@code{deval} $B$K$*$$$F$O(B, $B7W;;$O(B C $B?t3X%i%$%V%i%j$N4X?t$rMQ$$$F9T$&(B. |
| @item |
@item |
| @code{deval} $B$OJ#AG?t$O07$($J$$(B. |
@code{deval} $B$OJ#AG?t$O07$($J$$(B. |
| Line 755 These functions works also for polynomials with comple |
|
| Line 758 These functions works also for polynomials with comple |
|
| @code{eval} $B$K$*$$$F$O(B, |
@code{eval} $B$K$*$$$F$O(B, |
| @var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B. |
@var{prec} $B$r;XDj$7$?>l9g(B, $B7W;;$O(B, 10 $B?J(B @var{prec} $B7eDxEY$G9T$o$l$k(B. |
| @var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B. |
@var{prec} $B$N;XDj$,$J$$>l9g(B, $B8=:_@_Dj$5$l$F$$$k@:EY$G9T$o$l$k(B. |
| (@xref{setprec}.) |
(@xref{setbprec setprec}.) |
| @item |
@item |
| @table @t |
@table @t |
| @item $B07$($kH!?t$O(B, $B<!$NDL$j(B. |
@item $B07$($kH!?t$O(B, $B<!$NDL$j(B. |
|
|
| double float. Rational numbers remain unchanged in results from @code{eval}. |
double float. Rational numbers remain unchanged in results from @code{eval}. |
| @item |
@item |
| In @code{eval} the computation is done |
In @code{eval} the computation is done |
| by @b{PARI}. (@xref{pari}.) In @code{deval} the computation is |
by @b{MPFR} library. In @code{deval} the computation is |
| done by the C math library. |
done by the C math library. |
| @item |
@item |
| @code{deval} cannot handle complex numbers. |
@code{deval} cannot handle complex numbers. |
| Line 798 done by the C math library. |
|
| Line 801 done by the C math library. |
|
| When @var{prec} is specified, computation will be performed with a |
When @var{prec} is specified, computation will be performed with a |
| precision of about @var{prec}-digits. |
precision of about @var{prec}-digits. |
| If @var{prec} is not specified, computation is performed with the |
If @var{prec} is not specified, computation is performed with the |
| precision set currently. (@xref{setprec}.) |
precision set currently. (@xref{setbprec setprec}.) |
| @item |
@item |
| Currently available numerical functions are listed below. |
Currently available numerical functions are listed below. |
| Note they are only a small part of whole @b{PARI} functions. |
|
| |
|
| @table @t |
@table @t |
| @code{sin}, @code{cos}, @code{tan}, |
@code{sin}, @code{cos}, @code{tan}, |
| Line 846 Napier's number (@t{exp}(1)) |
|
| Line 848 Napier's number (@t{exp}(1)) |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{ctrl}, @fref{setprec}, @fref{pari}. |
@fref{ctrl}, @fref{setbprec setprec}. |
| @end table |
@end table |
| |
|
| \JP @node pari,,, $B?t$N1i;;(B |
\JP @node pari,,, $B?t$N1i;;(B |
| Line 889 Napier's number (@t{exp}(1)) |
|
| Line 891 Napier's number (@t{exp}(1)) |
|
| $BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B |
$BH!?tCM$NI>2A$r9bB.$K9T$&$3$H$,$G$-$k(B. @b{PARI} $B$OB>$N%W%m%0%i%`$+$i(B |
| $B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B |
$B%5%V%k!<%A%s%i%$%V%i%j$H$7$FMQ$$$k$3$H$,$G$-(B, $B$^$?(B, @samp{gp} $B$H$$$&(B |
| @b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B |
@b{PARI}$B%i%$%V%i%j$N%$%s%?%U%'!<%9$K$h$j(B UNIX $B$N%"%W%j%1!<%7%g%s$H$7$F(B |
| $BMxMQ$9$k$3$H$b$G$-$k(B. $B8=:_$N%P!<%8%g%s$O(B @b{2.0.17beta} $B$G$$$/$D$+$N(B ftp |
$BMxMQ$9$k$3$H$b$G$-$k(B. |
| site ($B$?$H$($P(B @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}) |
|
| $B$+$i(B anonymous ftp $B$G$-$k(B. |
|
| @item |
@item |
| $B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B. |
$B:G8e$N0z?t(B @var{prec} $B$G7W;;@:EY$r;XDj$G$-$k(B. |
| @var{prec} $B$r>JN,$7$?>l9g(B @code{setprec()} $B$G;XDj$7$?@:EY$H$J$k(B. |
@var{prec} $B$r>JN,$7$?>l9g(B @code{setprec()} $B$G;XDj$7$?@:EY$H$J$k(B. |
| @item |
@item |
| $B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$NH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B |
$B8=;~E@$G<B9T$G$-$k(B @b{PARI} $B$N<g$JH!?t$O<!$NDL$j$G$"$k(B. $B$$$:$l$b(B |
| 1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B. |
1 $B0z?t$G(B @b{Asir} $B$,BP1~$G$-$k7?$N0z?t$r$H$kH!?t$G$"$k(B. |
| |
$B<B9T$G$-$kH!?t$NA4%j%9%H$O(B asir-contrib $B$N(B ox_pari $B%^%K%e%"%k$r;2>H(B. |
| $B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B. |
$B$J$*3F!9$N5!G=$K$D$$$F$O(B @b{PARI} $B$N%^%K%e%"%k$r;2>H$N$3$H(B. |
| \E |
\E |
| \BEG |
\BEG |
| Line 913 function evaluations as well as arithmetic operations |
|
| Line 914 function evaluations as well as arithmetic operations |
|
| speed. It can also be used from other external programs as a library. |
speed. It can also be used from other external programs as a library. |
| It provides a language interface named @samp{gp} to its library, which |
It provides a language interface named @samp{gp} to its library, which |
| enables a user to use @b{PARI} as a calculator which runs on UNIX. |
enables a user to use @b{PARI} as a calculator which runs on UNIX. |
| The current version is @b{2.0.17beta}. It can be obtained by several ftp |
|
| sites. (For example, @code{ftp://megrez.ceremab.u-bordeaux.fr/pub/pari}.) |
|
| @item |
@item |
| The last argument (optional) @var{int} specifies the precision in digits |
The last argument (optional) @var{int} specifies the precision in digits |
| for bigfloat operation. |
for bigfloat operation. |
| If the precision is not explicitly specified, operation will be performed |
If the precision is not explicitly specified, operation will be performed |
| with the precision set by @code{setprec()}. |
with the precision set by @code{setprec()}. |
| @item |
@item |
| Currently available functions of @b{PARI} system are as follows. |
Some of currently available functions of @b{PARI} system are as follows. |
| Note these are only a part of functions in @b{PARI} system. |
Note these are only a part of functions in @b{PARI} system. |
| |
A complete list of functions which can be called from asir |
| |
is in the ox_pari manual of the asir-contrib. |
| For details of individual functions, refer to the @b{PARI} manual. |
For details of individual functions, refer to the @b{PARI} manual. |
| (Some of them can be seen in the following example.) |
(Some of them can be seen in the following example.) |
| \E |
\E |
| Line 1061 For details of individual functions, refer to the @b{P |
|
| Line 1062 For details of individual functions, refer to the @b{P |
|
| |
|
| \BJP |
\BJP |
| @item |
@item |
| @b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k$,(B, $B:#8e(B |
@b{Asir} $B$GMQ$$$F$$$k$N$O(B @b{PARI} $B$N$[$s$N0lIt$N5!G=$G$"$k(B. |
| $B$h$jB?$/$N5!G=$,MxMQ$G$-$k$h$&2~NI$9$kM=Dj$G$"$k(B. |
|
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| @b{Asir} currently uses only a very small subset of @b{PARI}. |
@b{Asir} currently uses only a very small subset of @b{PARI}. |
| We will improve @b{Asir} so that it can provide more functions of |
|
| @b{PARI}. |
|
| \E |
\E |
| @end itemize |
@end itemize |
| |
|
| Line 1087 We will improve @b{Asir} so that it can provide more f |
|
| Line 1085 We will improve @b{Asir} so that it can provide more f |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| \EG @item References |
\EG @item References |
| @fref{setprec}. |
@fref{setbprec setprec}. |
| @end table |
@end table |
| |
|
| \JP @node setprec,,, $B?t$N1i;;(B |
\JP @node setbprec setprec,,, $B?t$N1i;;(B |
| \EG @node setprec,,, Numbers |
\EG @node setbprec setprec,,, Numbers |
| @subsection @code{setprec} |
@subsection @code{setbprec}, @code{setprec} |
| |
@findex setbprec |
| @findex setprec |
@findex setprec |
| @cindex PARI |
|
| |
|
| @table @t |
@table @t |
| @item setprec([@var{n}]) |
@item setbprec([@var{n}]) |
| \JP :: @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. |
@itemx setprec([@var{n}]) |
| \EG :: Sets the precision for @b{bigfloat} operations to @var{n} digits. |
\JP :: @b{setbprec}, @b{setprec} $B$O(B @b{bigfloat} $B$N@:EY$r$=$l$>$l(B 2 $B?J(B, 10$B?J(B @var{n} $B7e$K@_Dj$9$k(B. |
| |
\EG :: @b{setbprec}, @b{setprec} set the precision for @b{bigfloat} operations to @var{n} bits, @var{n} digits respectively. |
| @end table |
@end table |
| |
|
| @table @var |
@table @var |
| Line 1117 We will improve @b{Asir} so that it can provide more f |
|
| Line 1116 We will improve @b{Asir} so that it can provide more f |
|
| $B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. |
$B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N7e?t$r(B @var{n} $B7e$K@_Dj$9$k(B. |
| $B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B. |
$B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B. |
| @item |
@item |
| @b{bigfloat} $B$N7W;;$O(B @b{PARI} (@ref{pari}) $B$K$h$C$F9T$o$l$k(B. |
@b{bigfloat} $B$N7W;;$O(B @b{MPFR} $B%i%$%V%i%j$K$h$C$F9T$o$l$k(B. |
| @item |
@item |
| @b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B. |
@b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B. |
| @b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B. |
@b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B. |
| Line 1127 We will improve @b{Asir} so that it can provide more f |
|
| Line 1126 We will improve @b{Asir} so that it can provide more f |
|
| \E |
\E |
| \BEG |
\BEG |
| @item |
@item |
| When an argument is given, it |
When an argument @var{n} is given, these functions |
| sets the precision for @b{bigfloat} operations to @var{n} digits. |
set the precision for @b{bigfloat} operations to @var{n} bits or @var{n} digits. |
| The return value is always the previous precision in digits regardless of |
The return value is always the previous precision regardless of |
| the existence of an argument. |
the existence of an argument. |
| |
|
| @item |
@item |
| @b{Bigfloat} operations are done by @b{PARI}. (@xref{pari}.) |
@b{Bigfloat} operations are done by @b{MPFR} library. |
| @item |
@item |
| This is effective for computations in @b{bigfloat}. |
This is effective for computations in @b{bigfloat}. |
| Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.' |
Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.' |
| Line 1146 Therefore, it is safe to specify a larger value. |
|
| Line 1145 Therefore, it is safe to specify a larger value. |
|
| |
|
| @example |
@example |
| [1] setprec(); |
[1] setprec(); |
| 9 |
15 |
| [2] setprec(100); |
[2] setprec(100); |
| 9 |
15 |
| [3] setprec(100); |
[3] setprec(100); |
| 96 |
99 |
| |
[4] setbprec(); |
| |
332 |
| @end example |
@end example |
| |
|
| @table @t |
@table @t |
| \JP @item $B;2>H(B |
\JP @item $B;2>H(B |
| @fref{ctrl}, @fref{eval deval}, @fref{pari}. |
@fref{ctrl}, @fref{eval deval}. |
| @end table |
@end table |
| |
|
| |
\JP @node setround,,, $B?t$N1i;;(B |
| |
\EG @node setround,,, Numbers |
| |
@subsection @code{setround} |
| |
@findex setround |
| |
|
| |
@table @t |
| |
@item setround([@var{mode}]) |
| |
\JP :: @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B. |
| |
\EG :: Sets the rounding mode @var{mode}. |
| |
@end table |
| |
|
| |
@table @var |
| |
@item return |
| |
\JP $B@0?t(B |
| |
\EG integer |
| |
@item mode |
| |
\JP $B@0?t(B |
| |
\EG integer |
| |
@end table |
| |
|
| |
@itemize @bullet |
| |
\BJP |
| |
@item |
| |
$B0z?t$,$"$k>l9g(B, @b{bigfloat} $B$N4]$a%b!<%I$r(B @var{mode} $B$K@_Dj$9$k(B. |
| |
$B0z?t$N$"$k$J$7$K$+$+$o$i$:(B, $B0JA0$K@_Dj$5$l$F$$$?CM$rJV$9(B. |
| |
$B4]$a%b!<%I$N0UL#$O<!$N$H$*$j(B. |
| |
@table @code |
| |
@item 0 |
| |
Round to nearest |
| |
@item 1 |
| |
Round toward 0 |
| |
@item 2 |
| |
Round toward +infinity |
| |
@item 3 |
| |
Round toward -infinity |
| |
@end table |
| |
@item |
| |
@b{bigfloat} $B$G$N7W;;$KBP$7M-8z$G$"$k(B. |
| |
@b{bigfloat} $B$N(B flag $B$r(B on $B$K$9$kJ}K!$O(B, @code{ctrl} $B$r;2>H(B. |
| |
\E |
| |
\BEG |
| |
@item |
| |
When an argument @var{mode} is given, these functions |
| |
set the rounding mode for @b{bigfloat} operations to @var{mode}. |
| |
The return value is always the previous rounding mode regardless of |
| |
the existence of an argument. |
| |
The meanings of rounding modes are as follows |
| |
@table @code |
| |
@item 0 |
| |
Round to nearest |
| |
@item 1 |
| |
Round toward 0 |
| |
@item 2 |
| |
Round toward +infinity |
| |
@item 3 |
| |
Round toward -infinity |
| |
@end table |
| |
|
| |
@item |
| |
This is effective for computations in @b{bigfloat}. |
| |
Refer to @code{ctrl()} for turning on the `@b{bigfloat} flag.' |
| |
\E |
| |
@end itemize |
| |
|
| |
@example |
| |
[1] setprec(); |
| |
15 |
| |
[2] setprec(100); |
| |
15 |
| |
[3] setprec(100); |
| |
99 |
| |
[4] setbprec(); |
| |
332 |
| |
@end example |
| |
|
| |
@table @t |
| |
\JP @item $B;2>H(B |
| |
@fref{ctrl}, @fref{eval deval}. |
| |
@end table |
| |
|
| |
|
| \JP @node setmod,,, $B?t$N1i;;(B |
\JP @node setmod,,, $B?t$N1i;;(B |
| \EG @node setmod,,, Numbers |
\EG @node setmod,,, Numbers |
| @subsection @code{setmod} |
@subsection @code{setmod} |
| Line 1265 integer. These functions are used in such a case. |
|
| Line 1347 integer. These functions are used in such a case. |
|
| \EG @item References |
\EG @item References |
| \JP @fref{$BJ,;67W;;(B}, @fref{$B?t$N7?(B}. |
\JP @fref{$BJ,;67W;;(B}, @fref{$B?t$N7?(B}. |
| \EG @fref{Distributed computation}, @fref{Types of numbers}. |
\EG @fref{Distributed computation}, @fref{Types of numbers}. |
| |
@end table |
| |
|
| |
\JP @node inttorat,,, $B?t$N1i;;(B |
| |
\EG @node inttorat,,, Numbers |
| |
@subsection @code{inttorat} |
| |
@findex inttorat |
| |
|
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@table @t |
| |
@item inttorat(@var{a},@var{m},@var{b}) |
| |
\JP :: $B@0?t(B-$BM-M}?tJQ49$r9T$&(B. |
| |
\EG :: Perform the rational reconstruction. |
| |
@end table |
| |
|
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@table @var |
| |
@item return |
| |
\JP $B%j%9%H$^$?$O(B 0 |
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\EG list or 0 |
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@item a |
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@itemx m |
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@itemx b |
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\JP $B@0?t(B |
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\EG integer |
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@end table |
| |
|
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@itemize @bullet |
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\BJP |
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@item |
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$B@0?t(B @var{a} $B$KBP$7(B, @var{xa=y} mod @var{m} $B$rK~$?$9@5@0?t(B @var{x}, $B@0?t(B @var{y} |
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(@var{x}, @var{|y|} < @var{b}, @var{GCD(x,y)=1}) $B$r5a$a$k(B. |
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@item |
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$B$3$N$h$&$J(B @var{x}, @var{y} $B$,B8:_$9$k$J$i(B @var{[y,x]} $B$rJV$7(B, $BB8:_$7$J$$>l9g$K$O(B 0 $B$rJV$9(B. |
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@item |
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@var{b} $B$r(B @var{floor(sqrt(m/2))} $B$H<h$l$P(B, @var{x}, @var{y} $B$OB8:_$9$l$P0l0U$G$"$k(B. |
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@var{floor(sqrt(m/2))} $B$O(B @code{isqrt(floor(m/2))} $B$G7W;;$G$-$k(B. |
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@end itemize |
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\E |
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\BEG |
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@item |
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For an integer @var{a}, find a positive integer @var{x} and an intger @var{y} satisfying |
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@var{xa=y} mod @var{m}, @var{x}, @var{|y|} < @var{b} and @var{GCD(x,y)=1}. |
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@item |
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If such @var{x}, @var{y} exist then a list @var{[y,x]} is returned. Otherwise 0 is returned. |
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@item |
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If @var{b} is set to @var{floor(sqrt(M/2))}, then @var{x} and @var{y} are unique if they |
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exist. @var{floor(sqrt(M/2))} can be computed by @code{floor} and @code{isqrt}. |
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@end itemize |
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\E |
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|
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@example |
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[2121] M=lprime(0)*lprime(1); |
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9996359931312779 |
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[2122] B=isqrt(floor(M/2)); |
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70697807 |
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[2123] A=234234829304; |
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234234829304 |
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[2124] inttorat(A,M,B); |
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[-20335178,86975031] |
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@end example |
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|
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@table @t |
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\JP @item $B;2>H(B |
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\EG @item References |
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@fref{ceil floor rint dceil dfloor drint}, @fref{isqrt}. |
| @end table |
@end table |
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