简化平方根

Question 1

问题中的算法非常低效：当然，除非原始整数是完全平方数。

本答案（按时间顺序）：

一种模仿最简单分解算法的宏方法，
一种使用 OP 中的算法的可扩展方法。更新非常尴尬的是，作者没有理解 OP 的算法，在找到了I诸如I^2divided 之类的简化方法后N，他进行了递归操作，却N<-N/I^2没有理解算法可以停止在那里。（作为一个苍白的借口，他首先实现了“自下而上”的方式，这需要递归，与“自上而下”（效率较低）的方式相反）。因此，答案已更新，向所有慷慨、值得信赖的早期投票者致歉。

我再次更新（抱歉），因为我现在阅读了更多的xintexpr.sty文档，并且为了提高效率，我从切换i=sqrt(N)..1到i=-sqrt(N)++（没有--可用的，因此使用减号的技巧）。前者预先生成整个floor(\sqrt{N})数字列表（sqrt意味着中的截断平方根\xintiiexpr），后者是一个迭代器，它不会预先生成任何东西。此外，前一种语法只能生成大约5000值（sqrt(N)..[-1]..1没有这样的限制，但仍然会预先生成所有内容）。
与方法 1 类似的更快（高中分解类型）算法的可扩展实现。

老实说，2.、3. 甚至 1. 可能最好完全使用来编写，\numexpr因为它们假装处理大于的数字有点2^31牵强，需要花费时间对 10 位素数进行数万次除法才能得出它无平方的结论...实现 2. 具有内在限制，2^62因为平方根必须是 TeX 数字（由于内部的一些构造）。

在 2. 和 3. 中，我们稍微超出了合理范围，使用递归序列的语法的可能性\xintexpr。符号有点麻烦。此外，xintexpr.sty 1.2g它还改变了相关的语法。

最后（2017）我还添加了无包 numexpr 仅可扩展的方法。

第一种方法（我们改变算法）

这并不是说这是一个简单的问题。谷歌搜索后发现，数学家们目前显然认为找到整数的平方自由基可能和完全分解因数一样困难：https://math.stackexchange.com/questions/171568/finding-the-radical-of-an-integer和https://math.stackexchange.com/questions/14667/square-free-integers-factorization。

这是一种模仿最简单形式的分解算法的方法（使用宏）。

包xintexpr仅用于允许输入诸如 1e7或甚至表达式。它还加载xinttools语法中使用的内容。

除此之外，所有操作都是通过提供的宏完成的xint。由于我们在示例中几乎只处理数字，因此<2^31我们可以采用一种变体，其中所有操作都使用唯一的来完成\numexpr，这自然会更快。

代码使用\xintiiDivision同时计算商和余数。这就是为什么\xintAssign使用将它们存储到两个宏\A和中\B。代码检查是否\B消失以检测可被整除Q=P^2。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}

\makeatletter
\def\Rsqrt@ {%
    \let\Nrad\N
    \def\Nroot {1}% 
% we will always have original N = \Nrad times \Nroot^2
% first we check powers of 2
    \def\P{2}%
    \def\Q{4}% \Q is always square of \P
    \xintloop
% try to divide \Nrad by 4. If possible, multiply \Nroot by 2
        \xintAssign\xintiiDivision{\Nrad}{\Q}\to \A\B
        \xintiiifZero{\B}
           {\let\Nrad\A
            \edef\Nroot{\xintiiMul{\Nroot}{\P}}% 
            \iftrue}
           {\iffalse}%
    \repeat
% try to divide \Nrad by 9=3^2, then by 25=5^2, etc...
% unfortunately we divide by all odd integers, but only odd prime
% integers would be really needed
    \def\P{3}%
    \xintloop
      \edef\Q{\xintiiSqr{\P}}%
      \xintiiifGt{\Q}{\Nrad}
         {\iffalse}%
         {\xintloop
             \xintAssign\xintiiDivision{\Nrad}{\Q}\to \A\B
             \xintiiifZero{\B}
                {\let\Nrad\A
                 \edef\Nroot{\xintiiMul{\P}{\Nroot}}% 
                 \iftrue}
                {\iffalse}%
          \repeat 
          \edef\P{\xintiiAdd{2}{\P}}%
          \iftrue
         }%
    \repeat
% at this stage \N = \Nrad times \Nroot^2
% and \Nrad is square-free.
    \xintiiifOne{\Nroot}{}{\Nroot}%
    \xintiiifOne{\Nrad} {}{\sqrt{\Nrad}}%
}%

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
}

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing
% \begin{multicols}{4}
% \xintFor* #1 in {\xintSeq {10000}{10100}}\do
%        {$\sqrt{#1}=\Rsqrt{#1}$\par}
% \end{multicols}

% $\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}, \Rsqrt{1e21}$

\pdfsetrandomseed 123456789

\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 1000)^2
             *\pdfuniformdeviate 1000\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<50
\repeat
\end{multicols}
    
\end{document}

第二种方法（与OP中的算法相同，可扩展的实现）

使用原始算法。这里我们定义\ExtractRadical可扩展的返回值为A,B。不可扩展的包装器从上述更快的方法中N=A^2 B重新循环以生成，区分负数或的情况。\RsqrtA\sqrt{B}NN=0, 1

我添加了代码注释来解释实现。早期版本非常蹩脚（参见答案顶部）并且需要额外xintexpr 1.2g内容，但现在不再如此。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}

% Aim: given N find largest I such as I^2 divides N.
% Then compute J=N/I^2 and print I\sqrt{J}.

% Algorithm: compute first truncated square root Imax of N.
% With I = Imax try to divide N by I^2, if it does not work
% repeat with I replaced by I-1 and so on. 
% As soon as it works the seeked-for I has been found.

% **** Notice: embarrassingly the author of this answer initially continued
% **** the algorithm recursively with N<-N/I^2, which was very stupid, but
% **** explainable from the fact that he had implemented first another (much
% **** faster) algorithm which divided not from the top down, but from the
% **** bottom up.

% The code is far simpler now. And it does not require xintexpr 1.2g anymore,
% earlier versions of xintexpr.sty work, too.

% The iteration over i used Imax..1 syntax which requires Imax 
% to be <2^31. Else we could use Imax..[-1]..1, but we don't
% really consider realistic to iterate over 2^31 or more values !

% After an update we use (-Imax)++ syntax; this also requires Imax<2^31.

\def\ExtractRadical #1{%
  \xinttheiiexpr 
  subs(
  % we return I, #1//I^2 where I is biggest integer such as I^2 divides #1.
     (I, #1//I^2), 
  % The I is computed via the "seq" here. Theoretically this "seq" 
  % evaluates as many values as the last list indicates.
  % But here we omit all i's such that i^2 does not divide #1
  % and as soon as we have found one, we stop here and now by 
  % "break". We work topdown, at the worst I=1.
% The i=A..B syntax pre-generates all values, which is wasteful
% and limited to about at most 5000 values.
%       I=seq((#1/:i^2)?{omit}{break(i)}, i=sqrt(#1)..1)
% On the contrary the N++ syntax does not pre-generate anything.
       I=seq((#1/:i^2)?{omit}{break(-i)}, i=-sqrt(#1)++)
% There is currently no "n--" only "n++", thus we tricked with a minus sign.
      )
  \relax
}

\makeatletter

\def\Rsqrt@ {\expandafter\Rsqrt@@\romannumeral-`0\ExtractRadical\N,}

% The #2#3 trick is to get rid of a space after the comma
% because \ExtractRadical does \xinttheiiexpr which in case
% of comma separated values on output always inserts such a space.
% Naturally as the typesetting is in math mode the space is
% not a real problem (it is not a problem either in \xintiiifOne 
% as here its argument is already expanded anyhow).
\def\Rsqrt@@ #1,#2#3,{\xintiiifOne{#1}{}{#1}\xintiiifOne{#2#3}{}{\sqrt{#2#3}}}

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
 }

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing

\begin{multicols}{4}
\xintFor* #1 in {\xintSeq {10000}{10099}}\do
       {$\sqrt{#1}=\Rsqrt{#1}$\par}
\end{multicols}

% $\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}$%, 

% this one does not work because 10^10.5 > 2^31 causes an arithmetic
% overflow in the "sqrt(J)..1" part. 
% It would not overflow with "sqrt(J)..[-1]..1"
% but then we can wait long time ... 
% from 31622776601 downto
%      10000000000 that's a lot of iterations !
%$\Rsqrt{1e21}$
% The update uses n++ syntax, but this also requires abs(n) to be <2^31
% hence the same remark applies: a "Number too big" error is generated.
% Better actually than to wait the completion of 21622776601 iterations ;-)


% \stop

\pdfsetrandomseed 123456789

% we try with smaller numbers... 1000 replaced by 100...
\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 100)^2
             *\pdfuniformdeviate 100\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<51
\repeat
\end{multicols}

\end{document}

第三种方法：同样是更快的算法，但具有可扩展性。

按照原样进行编码会更合理\numexpr。详细信息请参阅代码注释。示例现在有 51 个随机示例，有趣的是，缺失的一个示例（来自第一种方法）原来是一个随机方块（pdftex 随机数生成器的随机种子设置为123456789）。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}[2016/03/19]%
% needs xintexpr 1.2g due to 
%  - changed meaning of iter
%  - shift by 1 in [L][n] syntax


% syntax \ExtractRadical {N or <integer expression>} expands to "A, B" with
% N=A^2 B, B square-free
% Algorithm:
% main variable a triple (P, I, J) where
%  - always N = I^2 J
%  - J's prime factors < P have multiplicity one.
% START: (2, 1, N)
% ITER:  (P, I, J)
         % Q=P^2
         % is Q > J ?
         %   - yes: STOP(I, J)
         %   - no:
         %         does Q divide J ?
         %         - yes: I<-I*P, J<-J/Q and repeat until Q does not divide J
         %         - no; continue with (P+2, I, J). Except if P=2 then we go
         %                      on with P=3.
% Also works with N=0 (produces 1, 0) and with N=1 (produces 1, 1)
%

\newcommand*\ExtractRadical [1]{%
  \xinttheiiexpr 
  iter (2, 1, #1; % starting triple P=2, I=1, J=N
  subs(subs(subs(subs(
  % apart from Q=P^2, these substitutions are mainly because [@][n] syntax
  % is cumbersome; and inefficient as it allows n to be itself a complicated
  % expression, hence does some rather unneeded parsing here of n= 0, 1, 2.
  % We really need some better syntax like iter(P=2, I=1, J=#1;...) and then
  % work with P, I, J standing for the last values.
  % Or at least something like subs(..., (Q, P, I, J)=(...)).
  % (not yet with xintexpr 1.2g). 
  (Q>J)?
      {break(I, J)}
      {(J/:Q)?
          {(n)?{P+2}{3}, I, J}
      % must use parentheses here: ([@][1]). Else ]/: will confuse parser.
      % I could have used again subs, but well. 
          {iter(P*I,J//Q;(([@][1])/:Q)?{break((n)?{P+2}{3},@)}
                                       {(P*[@][0],([@][1])//Q)},e=1++)
          }
      }
  , Q=P^2), P=[@][0]), I=[@][1]), J=[@][2]), n=0++)
    \relax
}             
                  
\makeatletter
\def\Rsqrt@ {\expandafter\Rsqrt@@\romannumeral-`0\ExtractRadical\N,}

\def\Rsqrt@@ #1,#2,{\xintiiifOne{#1}{}{#1}\xintiiifOne{#2}{}{\sqrt{#2}}}

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
 }

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing

% \xintFor* #1 in {\xintSeq {0}{50}}\do 
% {\ExtractRadical {#1}\par}

% \ExtractRadical {128}

% \ExtractRadical {1024}
% \stop

% $\Rsqrt{5000}$

% \stop

% \begin{multicols}{4}
% \xintFor* #1 in {\xintSeq {10000}{10099}}\do
%        {$\sqrt{#1}=\Rsqrt{#1}$\par}
% \end{multicols}

$\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}, \Rsqrt {1e21}$%, 

\pdfsetrandomseed 123456789

\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 1000)^2
             *\pdfuniformdeviate 1000\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<51
\repeat
\end{multicols}

\end{document}

更新（2017 年）。

如果需要，这里有一个非封装的可扩展宏。它扩展到I,J原始位置N并且I**2 times J无J平方。仅使用\numexpr。仅限于<2**31整数。从低到大工作，与基本因式分解方法相同。停止标准应该改进，下面的评论也适用于此处。

\makeatletter

\newcommand\ExtractRadical[1]{%
    \romannumeral0%
    \expandafter
    \ExtractRadical@two@i\expandafter1\expandafter,\the\numexpr#1.%
}%
\def\ExtractRadical@two@i #1,#2.{%
    \ifnum4>#2 \expandafter\ExtractRadical@two@done\fi
    \expandafter\ExtractRadical@two@ii\the\numexpr#2/4;#1,#2.%
}%
\edef\ExtractRadical@two@done #1;#2,#3.%
    {\space#2,#3}% (not sole #2 for readability)
\def\ExtractRadical@two@ii #1;#2,#3.{%
    \ifnum\numexpr#1*4=#3 
      \expandafter\@firstoftwo
    \else
      \expandafter\@secondoftwo
    \fi
   {\expandafter\ExtractRadical@two@i\the\numexpr2*#2,#1.}%
   {\ExtractRadical@i 3;#2,#3.}%
}%
\def\ExtractRadical@i #1;{%
    \expandafter\ExtractRadical@ii\the\numexpr#1*#1.#1;%
}%
\def\ExtractRadical@ii #1.#2;#3,#4.{%
    \ifnum#1>#4 \expandafter\ExtractRadical@done\fi
    \expandafter\ExtractRadical@iii\the\numexpr#4/#1.#1;#4.#2;#3.%
}%
\def\ExtractRadical@iii #1.#2;#3.{%
    \ifnum\numexpr#1*#2=#3
      \expandafter\@firstoftwo
    \else
      \expandafter\@secondoftwo
    \fi
    \ExtractRadical@update
    \ExtractRadical@next
    #1.#2;#3.%
}%
\def\ExtractRadical@update #1.#2;#3.#4;#5.{%
    \expandafter\ExtractRadical@ii
    \the\numexpr#2\expandafter.%
    \the\numexpr#4\expandafter;%
    \the\numexpr#4*#5,#1.%
}%
\def\ExtractRadical@next #1.#2;#3.#4;#5.{%
    \expandafter\ExtractRadical@i\the\numexpr2+#4;#5,#3.%
}%
\edef\ExtractRadical@done #1;#2.#3;#4.{\space#4,#2}%

\makeatother

Answer

问题中的算法非常低效：当然，除非原始整数是完全平方数。

本答案（按时间顺序）：

一种模仿最简单分解算法的宏方法，
一种使用 OP 中的算法的可扩展方法。更新非常尴尬的是，作者没有理解 OP 的算法，在找到了I诸如I^2divided 之类的简化方法后N，他进行了递归操作，却N<-N/I^2没有理解算法可以停止在那里。（作为一个苍白的借口，他首先实现了“自下而上”的方式，这需要递归，与“自上而下”（效率较低）的方式相反）。因此，答案已更新，向所有慷慨、值得信赖的早期投票者致歉。

我再次更新（抱歉），因为我现在阅读了更多的xintexpr.sty文档，并且为了提高效率，我从切换i=sqrt(N)..1到i=-sqrt(N)++（没有--可用的，因此使用减号的技巧）。前者预先生成整个floor(\sqrt{N})数字列表（sqrt意味着中的截断平方根\xintiiexpr），后者是一个迭代器，它不会预先生成任何东西。此外，前一种语法只能生成大约5000值（sqrt(N)..[-1]..1没有这样的限制，但仍然会预先生成所有内容）。
与方法 1 类似的更快（高中分解类型）算法的可扩展实现。

老实说，2.、3. 甚至 1. 可能最好完全使用来编写，\numexpr因为它们假装处理大于的数字有点2^31牵强，需要花费时间对 10 位素数进行数万次除法才能得出它无平方的结论...实现 2. 具有内在限制，2^62因为平方根必须是 TeX 数字（由于内部的一些构造）。

在 2. 和 3. 中，我们稍微超出了合理范围，使用递归序列的语法的可能性\xintexpr。符号有点麻烦。此外，xintexpr.sty 1.2g它还改变了相关的语法。

最后（2017）我还添加了无包 numexpr 仅可扩展的方法。

第一种方法（我们改变算法）

这并不是说这是一个简单的问题。谷歌搜索后发现，数学家们目前显然认为找到整数的平方自由基可能和完全分解因数一样困难：https://math.stackexchange.com/questions/171568/finding-the-radical-of-an-integer和https://math.stackexchange.com/questions/14667/square-free-integers-factorization。

这是一种模仿最简单形式的分解算法的方法（使用宏）。

包xintexpr仅用于允许输入诸如 1e7或甚至表达式。它还加载xinttools语法中使用的内容。

除此之外，所有操作都是通过提供的宏完成的xint。由于我们在示例中几乎只处理数字，因此<2^31我们可以采用一种变体，其中所有操作都使用唯一的来完成\numexpr，这自然会更快。

代码使用\xintiiDivision同时计算商和余数。这就是为什么\xintAssign使用将它们存储到两个宏\A和中\B。代码检查是否\B消失以检测可被整除Q=P^2。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}

\makeatletter
\def\Rsqrt@ {%
    \let\Nrad\N
    \def\Nroot {1}% 
% we will always have original N = \Nrad times \Nroot^2
% first we check powers of 2
    \def\P{2}%
    \def\Q{4}% \Q is always square of \P
    \xintloop
% try to divide \Nrad by 4. If possible, multiply \Nroot by 2
        \xintAssign\xintiiDivision{\Nrad}{\Q}\to \A\B
        \xintiiifZero{\B}
           {\let\Nrad\A
            \edef\Nroot{\xintiiMul{\Nroot}{\P}}% 
            \iftrue}
           {\iffalse}%
    \repeat
% try to divide \Nrad by 9=3^2, then by 25=5^2, etc...
% unfortunately we divide by all odd integers, but only odd prime
% integers would be really needed
    \def\P{3}%
    \xintloop
      \edef\Q{\xintiiSqr{\P}}%
      \xintiiifGt{\Q}{\Nrad}
         {\iffalse}%
         {\xintloop
             \xintAssign\xintiiDivision{\Nrad}{\Q}\to \A\B
             \xintiiifZero{\B}
                {\let\Nrad\A
                 \edef\Nroot{\xintiiMul{\P}{\Nroot}}% 
                 \iftrue}
                {\iffalse}%
          \repeat 
          \edef\P{\xintiiAdd{2}{\P}}%
          \iftrue
         }%
    \repeat
% at this stage \N = \Nrad times \Nroot^2
% and \Nrad is square-free.
    \xintiiifOne{\Nroot}{}{\Nroot}%
    \xintiiifOne{\Nrad} {}{\sqrt{\Nrad}}%
}%

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
}

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing
% \begin{multicols}{4}
% \xintFor* #1 in {\xintSeq {10000}{10100}}\do
%        {$\sqrt{#1}=\Rsqrt{#1}$\par}
% \end{multicols}

% $\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}, \Rsqrt{1e21}$

\pdfsetrandomseed 123456789

\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 1000)^2
             *\pdfuniformdeviate 1000\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<50
\repeat
\end{multicols}
    
\end{document}

第二种方法（与OP中的算法相同，可扩展的实现）

使用原始算法。这里我们定义\ExtractRadical可扩展的返回值为A,B。不可扩展的包装器从上述更快的方法中N=A^2 B重新循环以生成，区分负数或的情况。\RsqrtA\sqrt{B}NN=0, 1

我添加了代码注释来解释实现。早期版本非常蹩脚（参见答案顶部）并且需要额外xintexpr 1.2g内容，但现在不再如此。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}

% Aim: given N find largest I such as I^2 divides N.
% Then compute J=N/I^2 and print I\sqrt{J}.

% Algorithm: compute first truncated square root Imax of N.
% With I = Imax try to divide N by I^2, if it does not work
% repeat with I replaced by I-1 and so on. 
% As soon as it works the seeked-for I has been found.

% **** Notice: embarrassingly the author of this answer initially continued
% **** the algorithm recursively with N<-N/I^2, which was very stupid, but
% **** explainable from the fact that he had implemented first another (much
% **** faster) algorithm which divided not from the top down, but from the
% **** bottom up.

% The code is far simpler now. And it does not require xintexpr 1.2g anymore,
% earlier versions of xintexpr.sty work, too.

% The iteration over i used Imax..1 syntax which requires Imax 
% to be <2^31. Else we could use Imax..[-1]..1, but we don't
% really consider realistic to iterate over 2^31 or more values !

% After an update we use (-Imax)++ syntax; this also requires Imax<2^31.

\def\ExtractRadical #1{%
  \xinttheiiexpr 
  subs(
  % we return I, #1//I^2 where I is biggest integer such as I^2 divides #1.
     (I, #1//I^2), 
  % The I is computed via the "seq" here. Theoretically this "seq" 
  % evaluates as many values as the last list indicates.
  % But here we omit all i's such that i^2 does not divide #1
  % and as soon as we have found one, we stop here and now by 
  % "break". We work topdown, at the worst I=1.
% The i=A..B syntax pre-generates all values, which is wasteful
% and limited to about at most 5000 values.
%       I=seq((#1/:i^2)?{omit}{break(i)}, i=sqrt(#1)..1)
% On the contrary the N++ syntax does not pre-generate anything.
       I=seq((#1/:i^2)?{omit}{break(-i)}, i=-sqrt(#1)++)
% There is currently no "n--" only "n++", thus we tricked with a minus sign.
      )
  \relax
}

\makeatletter

\def\Rsqrt@ {\expandafter\Rsqrt@@\romannumeral-`0\ExtractRadical\N,}

% The #2#3 trick is to get rid of a space after the comma
% because \ExtractRadical does \xinttheiiexpr which in case
% of comma separated values on output always inserts such a space.
% Naturally as the typesetting is in math mode the space is
% not a real problem (it is not a problem either in \xintiiifOne 
% as here its argument is already expanded anyhow).
\def\Rsqrt@@ #1,#2#3,{\xintiiifOne{#1}{}{#1}\xintiiifOne{#2#3}{}{\sqrt{#2#3}}}

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
 }

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing

\begin{multicols}{4}
\xintFor* #1 in {\xintSeq {10000}{10099}}\do
       {$\sqrt{#1}=\Rsqrt{#1}$\par}
\end{multicols}

% $\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}$%, 

% this one does not work because 10^10.5 > 2^31 causes an arithmetic
% overflow in the "sqrt(J)..1" part. 
% It would not overflow with "sqrt(J)..[-1]..1"
% but then we can wait long time ... 
% from 31622776601 downto
%      10000000000 that's a lot of iterations !
%$\Rsqrt{1e21}$
% The update uses n++ syntax, but this also requires abs(n) to be <2^31
% hence the same remark applies: a "Number too big" error is generated.
% Better actually than to wait the completion of 21622776601 iterations ;-)


% \stop

\pdfsetrandomseed 123456789

% we try with smaller numbers... 1000 replaced by 100...
\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 100)^2
             *\pdfuniformdeviate 100\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<51
\repeat
\end{multicols}

\end{document}

第三种方法：同样是更快的算法，但具有可扩展性。

按照原样进行编码会更合理\numexpr。详细信息请参阅代码注释。示例现在有 51 个随机示例，有趣的是，缺失的一个示例（来自第一种方法）原来是一个随机方块（pdftex 随机数生成器的随机种子设置为123456789）。

\documentclass[a4paper]{article}

\usepackage{geometry}

\usepackage{xintexpr}[2016/03/19]%
% needs xintexpr 1.2g due to 
%  - changed meaning of iter
%  - shift by 1 in [L][n] syntax


% syntax \ExtractRadical {N or <integer expression>} expands to "A, B" with
% N=A^2 B, B square-free
% Algorithm:
% main variable a triple (P, I, J) where
%  - always N = I^2 J
%  - J's prime factors < P have multiplicity one.
% START: (2, 1, N)
% ITER:  (P, I, J)
         % Q=P^2
         % is Q > J ?
         %   - yes: STOP(I, J)
         %   - no:
         %         does Q divide J ?
         %         - yes: I<-I*P, J<-J/Q and repeat until Q does not divide J
         %         - no; continue with (P+2, I, J). Except if P=2 then we go
         %                      on with P=3.
% Also works with N=0 (produces 1, 0) and with N=1 (produces 1, 1)
%

\newcommand*\ExtractRadical [1]{%
  \xinttheiiexpr 
  iter (2, 1, #1; % starting triple P=2, I=1, J=N
  subs(subs(subs(subs(
  % apart from Q=P^2, these substitutions are mainly because [@][n] syntax
  % is cumbersome; and inefficient as it allows n to be itself a complicated
  % expression, hence does some rather unneeded parsing here of n= 0, 1, 2.
  % We really need some better syntax like iter(P=2, I=1, J=#1;...) and then
  % work with P, I, J standing for the last values.
  % Or at least something like subs(..., (Q, P, I, J)=(...)).
  % (not yet with xintexpr 1.2g). 
  (Q>J)?
      {break(I, J)}
      {(J/:Q)?
          {(n)?{P+2}{3}, I, J}
      % must use parentheses here: ([@][1]). Else ]/: will confuse parser.
      % I could have used again subs, but well. 
          {iter(P*I,J//Q;(([@][1])/:Q)?{break((n)?{P+2}{3},@)}
                                       {(P*[@][0],([@][1])//Q)},e=1++)
          }
      }
  , Q=P^2), P=[@][0]), I=[@][1]), J=[@][2]), n=0++)
    \relax
}             
                  
\makeatletter
\def\Rsqrt@ {\expandafter\Rsqrt@@\romannumeral-`0\ExtractRadical\N,}

\def\Rsqrt@@ #1,#2,{\xintiiifOne{#1}{}{#1}\xintiiifOne{#2}{}{\sqrt{#2}}}

\newcommand* \Rsqrt[1]{%
   \begingroup
     \edef\N{\xinttheiexpr #1\relax}%
     \xintiiifSgn \N
        {\pm\edef\N{\xintiiAbs{\N}}\xintiiifOne\N{}{\Rsqrt@}i}
        {0}
        {\xintiiifOne \N{1}{\Rsqrt@}}
   \endgroup
 }

\makeatother
\usepackage{multicol}

\begin{document}

\parindent0pt\def\columnseprule{.4pt}%

% testing

% \xintFor* #1 in {\xintSeq {0}{50}}\do 
% {\ExtractRadical {#1}\par}

% \ExtractRadical {128}

% \ExtractRadical {1024}
% \stop

% $\Rsqrt{5000}$

% \stop

% \begin{multicols}{4}
% \xintFor* #1 in {\xintSeq {10000}{10099}}\do
%        {$\sqrt{#1}=\Rsqrt{#1}$\par}
% \end{multicols}

$\Rsqrt{-10}, \Rsqrt{-1}, \Rsqrt{-16}$

$\Rsqrt {1e6}, \Rsqrt {1e7}, \Rsqrt {1e21}$%, 

\pdfsetrandomseed 123456789

\begin{multicols}{3}
\xintiloop [1+1]
    \edef\Z {\xinttheiiexpr 
             (\pdfuniformdeviate 1000)^2
             *\pdfuniformdeviate 1000\relax }%
    $\sqrt{\Z}=\Rsqrt{\Z}$\par
\ifnum\xintiloopindex<51
\repeat
\end{multicols}

\end{document}

更新（2017 年）。

如果需要，这里有一个非封装的可扩展宏。它扩展到I,J原始位置N并且I**2 times J无J平方。仅使用\numexpr。仅限于<2**31整数。从低到大工作，与基本因式分解方法相同。停止标准应该改进，下面的评论也适用于此处。

\makeatletter

\newcommand\ExtractRadical[1]{%
    \romannumeral0%
    \expandafter
    \ExtractRadical@two@i\expandafter1\expandafter,\the\numexpr#1.%
}%
\def\ExtractRadical@two@i #1,#2.{%
    \ifnum4>#2 \expandafter\ExtractRadical@two@done\fi
    \expandafter\ExtractRadical@two@ii\the\numexpr#2/4;#1,#2.%
}%
\edef\ExtractRadical@two@done #1;#2,#3.%
    {\space#2,#3}% (not sole #2 for readability)
\def\ExtractRadical@two@ii #1;#2,#3.{%
    \ifnum\numexpr#1*4=#3 
      \expandafter\@firstoftwo
    \else
      \expandafter\@secondoftwo
    \fi
   {\expandafter\ExtractRadical@two@i\the\numexpr2*#2,#1.}%
   {\ExtractRadical@i 3;#2,#3.}%
}%
\def\ExtractRadical@i #1;{%
    \expandafter\ExtractRadical@ii\the\numexpr#1*#1.#1;%
}%
\def\ExtractRadical@ii #1.#2;#3,#4.{%
    \ifnum#1>#4 \expandafter\ExtractRadical@done\fi
    \expandafter\ExtractRadical@iii\the\numexpr#4/#1.#1;#4.#2;#3.%
}%
\def\ExtractRadical@iii #1.#2;#3.{%
    \ifnum\numexpr#1*#2=#3
      \expandafter\@firstoftwo
    \else
      \expandafter\@secondoftwo
    \fi
    \ExtractRadical@update
    \ExtractRadical@next
    #1.#2;#3.%
}%
\def\ExtractRadical@update #1.#2;#3.#4;#5.{%
    \expandafter\ExtractRadical@ii
    \the\numexpr#2\expandafter.%
    \the\numexpr#4\expandafter;%
    \the\numexpr#4*#5,#1.%
}%
\def\ExtractRadical@next #1.#2;#3.#4;#5.{%
    \expandafter\ExtractRadical@i\the\numexpr2+#4;#5,#3.%
}%
\edef\ExtractRadical@done #1;#2.#3;#4.{\space#4,#2}%

\makeatother

Question 2

这是一个基于 LuaLaTeX 的解决方案。代码设置了一个名为的 LaTeX 宏\rsqrt，它调用一个名为的 Lua 函数rsqrt。后者实现了您提出的简化算法——并进行了以下改进：

对于n=0或n=1，代码仅返回n（没有平方根符号）；并且
\sqrt{n/i²}如果该项等于，则应小心地省略该项1，即，如果n是“平方数”（4、9、16 等）或较小平方数的乘积。例如，如果n=36，则\rsqrt{36}表明，6因为36 = 6^2 = 2^2*3^2。

不执行输入完整性检查，即用户负责提供一个参数，该参数\rsqrt要么是非负整数，要么计算结果为非负整数。因此，可以写入\rsqrt{1e6}和\rsqrt{3.6e7}：宏将分别返回1000和6000。

请注意，宏\rsqrt必须在数学模式下使用，因为它可能输出\sqrt指令。

% !TEX TS-program = lualatex  
%% Note: Code updated 2019/10/26 to work with LaTeX 2019-10-01 

%% Create an external file to contain the Lua code
\begin{filecontents*}[overwrite]{rsqrt.lua}

function rsqrt ( n )
  -- n : a non-negative whole number (or something
  --     that evaluates to a non-neg. whole number)
  if n == 0 or n == 1 then   -- Nothing to do
    return ( n )
  else
    i = math.floor ( math.sqrt ( n ) )
    while i > 1 do
      if ( n % i^2 == 0 ) then     -- n is divisible by i^2
        k = math.floor ( n / i^2 ) -- 'math.floor' makes k an explicit integer
        if k == 1 then  -- n is a "square" number (or a product of square numbers)
          return ( i )
        else
          return ( i .. "\\sqrt{" .. k .. "}" )
        end
      end
      i = i-1
    end
    -- No simplification possible:
    return ( "\\sqrt{" .. n .. "}" )
  end
end
-- Define a vector (in form of a Lua table) of whole numbers
nvec = {0,1,2,3,4,5,7,12,16,18,27,32}

-- Lua function to print 3-column array:
function PrintArray()
  for i=1,#nvec do
    u = nvec[i]
    tex.sprint ( math.floor(u) .. 
                "& \\sqrt{" .. math.floor(u) .. 
                "}&" .. rsqrt(u) .. "\\\\" )
  end
end

\end{filecontents*}

\documentclass{article}

%% Load Lua code from external file:
\directlua{dofile("rsqrt.lua")}

%% TeX-side code: "wrapper" macro that invokes the Lua function:
\newcommand\rsqrt[1]{\directlua{tex.sprint(rsqrt(#1))}}

\begin{document}
\[
\renewcommand\arraystretch{1.25}
\begin{array}{@{} rcc @{}}
  \verb+n+ & \verb+\sqrt+ & \verb+\rsqrt+ \\ % print header row
  \directlua{PrintArray()}    % create and print body of 'array'
\end{array}
\]
\end{document}

Answer

这是一个基于 LuaLaTeX 的解决方案。代码设置了一个名为的 LaTeX 宏\rsqrt，它调用一个名为的 Lua 函数rsqrt。后者实现了您提出的简化算法——并进行了以下改进：

对于n=0或n=1，代码仅返回n（没有平方根符号）；并且
\sqrt{n/i²}如果该项等于，则应小心地省略该项1，即，如果n是“平方数”（4、9、16 等）或较小平方数的乘积。例如，如果n=36，则\rsqrt{36}表明，6因为36 = 6^2 = 2^2*3^2。

不执行输入完整性检查，即用户负责提供一个参数，该参数\rsqrt要么是非负整数，要么计算结果为非负整数。因此，可以写入\rsqrt{1e6}和\rsqrt{3.6e7}：宏将分别返回1000和6000。

请注意，宏\rsqrt必须在数学模式下使用，因为它可能输出\sqrt指令。

% !TEX TS-program = lualatex  
%% Note: Code updated 2019/10/26 to work with LaTeX 2019-10-01 

%% Create an external file to contain the Lua code
\begin{filecontents*}[overwrite]{rsqrt.lua}

function rsqrt ( n )
  -- n : a non-negative whole number (or something
  --     that evaluates to a non-neg. whole number)
  if n == 0 or n == 1 then   -- Nothing to do
    return ( n )
  else
    i = math.floor ( math.sqrt ( n ) )
    while i > 1 do
      if ( n % i^2 == 0 ) then     -- n is divisible by i^2
        k = math.floor ( n / i^2 ) -- 'math.floor' makes k an explicit integer
        if k == 1 then  -- n is a "square" number (or a product of square numbers)
          return ( i )
        else
          return ( i .. "\\sqrt{" .. k .. "}" )
        end
      end
      i = i-1
    end
    -- No simplification possible:
    return ( "\\sqrt{" .. n .. "}" )
  end
end
-- Define a vector (in form of a Lua table) of whole numbers
nvec = {0,1,2,3,4,5,7,12,16,18,27,32}

-- Lua function to print 3-column array:
function PrintArray()
  for i=1,#nvec do
    u = nvec[i]
    tex.sprint ( math.floor(u) .. 
                "& \\sqrt{" .. math.floor(u) .. 
                "}&" .. rsqrt(u) .. "\\\\" )
  end
end

\end{filecontents*}

\documentclass{article}

%% Load Lua code from external file:
\directlua{dofile("rsqrt.lua")}

%% TeX-side code: "wrapper" macro that invokes the Lua function:
\newcommand\rsqrt[1]{\directlua{tex.sprint(rsqrt(#1))}}

\begin{document}
\[
\renewcommand\arraystretch{1.25}
\begin{array}{@{} rcc @{}}
  \verb+n+ & \verb+\sqrt+ & \verb+\rsqrt+ \\ % print header row
  \directlua{PrintArray()}    % create and print body of 'array'
\end{array}
\]
\end{document}

Question 3

在expl3：

\documentclass{article}

\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\rsqrt}{m}
 {
  \manual_rsqrt:n { #1 }
 }

\int_new:N \l_manual_rsqrt_int

\cs_new_protected:Nn \manual_rsqrt:n
 {
  \int_set:Nn \l_manual_rsqrt_int { \fp_to_decimal:n { trunc(sqrt(#1),0) } }
  \bool_until_do:nn
   {
    \int_compare_p:n { \int_mod:nn { #1 } { \l_manual_rsqrt_int * \l_manual_rsqrt_int } == 0 }
   }
   {
    \int_decr:N \l_manual_rsqrt_int
   }
  \int_compare:nTF { \l_manual_rsqrt_int == 1 }
   {
    \sqrt{#1}
   }
   {
    \int_to_arabic:n { \l_manual_rsqrt_int }
    \int_compare:nF { #1 == \l_manual_rsqrt_int*\l_manual_rsqrt_int }
     {
      \sqrt{ \int_to_arabic:n { #1/(\l_manual_rsqrt_int*\l_manual_rsqrt_int) } }
     }
   }
 }
\ExplSyntaxOff

\begin{document}

$\rsqrt{4}$

$\rsqrt{8}$

$\rsqrt{18}$

$\rsqrt{12}$

$\rsqrt{7}$

\end{document}

如果要处理参数 0 和 1 以及负参数，可以将主定义更改为

\NewDocumentCommand{\rsqrt}{m}
 {
  \int_compare:nTF { #1 < 0 }
   {
    \int_compare:nTF { #1 = -1 } { i } { \manual_rsqrt:n { -#1 } i }
   }
   {
    \int_compare:nTF { #1 < 2 } { #1 } { \manual_rsqrt:n { #1 } }
   }
 }

现在输入

$\rsqrt{0}$ and $\rsqrt{1}$

$\rsqrt{-1}$ and $\rsqrt{-4}$ and $\rsqrt{-32}$

将输出

Answer

在expl3：

\documentclass{article}

\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\rsqrt}{m}
 {
  \manual_rsqrt:n { #1 }
 }

\int_new:N \l_manual_rsqrt_int

\cs_new_protected:Nn \manual_rsqrt:n
 {
  \int_set:Nn \l_manual_rsqrt_int { \fp_to_decimal:n { trunc(sqrt(#1),0) } }
  \bool_until_do:nn
   {
    \int_compare_p:n { \int_mod:nn { #1 } { \l_manual_rsqrt_int * \l_manual_rsqrt_int } == 0 }
   }
   {
    \int_decr:N \l_manual_rsqrt_int
   }
  \int_compare:nTF { \l_manual_rsqrt_int == 1 }
   {
    \sqrt{#1}
   }
   {
    \int_to_arabic:n { \l_manual_rsqrt_int }
    \int_compare:nF { #1 == \l_manual_rsqrt_int*\l_manual_rsqrt_int }
     {
      \sqrt{ \int_to_arabic:n { #1/(\l_manual_rsqrt_int*\l_manual_rsqrt_int) } }
     }
   }
 }
\ExplSyntaxOff

\begin{document}

$\rsqrt{4}$

$\rsqrt{8}$

$\rsqrt{18}$

$\rsqrt{12}$

$\rsqrt{7}$

\end{document}

如果要处理参数 0 和 1 以及负参数，可以将主定义更改为

\NewDocumentCommand{\rsqrt}{m}
 {
  \int_compare:nTF { #1 < 0 }
   {
    \int_compare:nTF { #1 = -1 } { i } { \manual_rsqrt:n { -#1 } i }
   }
   {
    \int_compare:nTF { #1 < 2 } { #1 } { \manual_rsqrt:n { #1 } }
   }
 }

现在输入

$\rsqrt{0}$ and $\rsqrt{1}$

$\rsqrt{-1}$ and $\rsqrt{-4}$ and $\rsqrt{-32}$

将输出

Question 4

如果您不介意跳出 LaTeX，这里有一个使用该pythontex包的解决方案。我将其命名sroot为 simple root。显然，您可以随意命名。此版本需要

pdflatex *filename*.tex，，文档的执行顺序pythontex filename*.tex。 pdflatex *filename*.tex

\documentclass{article}
\usepackage{pythontex}
\begin{document}
\newcommand{\sroot}[1]{\ensuremath{\py{simpleroot(#1)}}}
\begin{pycode}
from math import *
def simpleroot(n):
    if n==0:
        return(str(0))
    j=int(sqrt(n))
    flag_continue=True
    while flag_continue:
        b=n*1./(j*j)
        if b==int(b):
            mystring=str(j)+'\\sqrt{'+str(int(b)) +'}'
            flag_continue=False
        else:
            j-=1

        if int(b)==1:
            mystring=str(j)
        if int(b)==n and b>1:
            mystring='\\sqrt{'+str(int(b)) +'}'

    return(mystring)
\end{pycode}

This is a test.

The $\sqrt{1}$ is \sroot{1}.

The $\sqrt{4}$ is \sroot{4}.

The $\sqrt{7}$ is \sroot{7}.

The $\sqrt{8}$ is \sroot{8}.

The $\sqrt{18}$ is \sroot{18}.

The $\sqrt{23}$ is \sroot{23}.

The $\sqrt{27}$ is \sroot{27}.

The $\sqrt{32}$ is \sroot{32}.

The $\sqrt{64}$ is \sroot{64}.

The $\sqrt{80}$ is \sroot{80}.

The $\sqrt{1000}$ is \sroot{1000}.

The $\sqrt{3000033}$ is \sroot{3000033}.

Goodbye.
\end{document}

Answer

如果您不介意跳出 LaTeX，这里有一个使用该pythontex包的解决方案。我将其命名sroot为 simple root。显然，您可以随意命名。此版本需要

pdflatex *filename*.tex，，文档的执行顺序pythontex filename*.tex。 pdflatex *filename*.tex

\documentclass{article}
\usepackage{pythontex}
\begin{document}
\newcommand{\sroot}[1]{\ensuremath{\py{simpleroot(#1)}}}
\begin{pycode}
from math import *
def simpleroot(n):
    if n==0:
        return(str(0))
    j=int(sqrt(n))
    flag_continue=True
    while flag_continue:
        b=n*1./(j*j)
        if b==int(b):
            mystring=str(j)+'\\sqrt{'+str(int(b)) +'}'
            flag_continue=False
        else:
            j-=1

        if int(b)==1:
            mystring=str(j)
        if int(b)==n and b>1:
            mystring='\\sqrt{'+str(int(b)) +'}'

    return(mystring)
\end{pycode}

This is a test.

The $\sqrt{1}$ is \sroot{1}.

The $\sqrt{4}$ is \sroot{4}.

The $\sqrt{7}$ is \sroot{7}.

The $\sqrt{8}$ is \sroot{8}.

The $\sqrt{18}$ is \sroot{18}.

The $\sqrt{23}$ is \sroot{23}.

The $\sqrt{27}$ is \sroot{27}.

The $\sqrt{32}$ is \sroot{32}.

The $\sqrt{64}$ is \sroot{64}.

The $\sqrt{80}$ is \sroot{80}.

The $\sqrt{1000}$ is \sroot{1000}.

The $\sqrt{3000033}$ is \sroot{3000033}.

Goodbye.
\end{document}

简化平方根

答案1

第一种方法（我们改变算法）

第二种方法（与OP中的算法相同，可扩展的实现）

第三种方法：同样是更快的算法，但具有可扩展性。

更新（2017 年）。

答案2

答案3

答案4

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