% proflycee-tools-arithm.tex
% Copyright 2023-2025 Cédric Pierquet
% Released under the LaTeX Project Public License v1.3c or later, see http://www.latex-project.org/lppl.txt

%===trad fr/en wip

\RequirePackage{ifthen}
\RequirePackage{etoolbox}
\RequirePackage{modulus}

%%-----Nombres premiers (code venant de la documentation de xint :-))
\NewDocumentCommand\pflboolestpremier{ m }{%
  \ifnumodd{#1}%
    {%
      \ifnumless{#1}{8}%
        {%
          \ifnumequal{#1}{1}{0}{1}%
        }%le cas 3,5,7 qui sont premiers
        {%
          \if\xintiloop[3+2]%
            \ifnum#1<\numexpr\xintiloopindex*\xintiloopindex\relax%
              \expandafter\xintbreakiloopanddo\expandafter1\expandafter.%
            \fi%
              \ifnum#1=\numexpr(#1/\xintiloopindex)*\xintiloopindex\relax%
            \else
              \repeat 00\expandafter0%
          \else%
            \expandafter1%
          \fi%
        }%
    }%fin de la partie impaire
    {%
      \ifnumequal{#1}{2}{1}{0}%
    }%partie paire
}%

\NewCommandCopy\pflisprimebool\pflboolestpremier

\NewDocumentCommand\pflestpremier{ O{est premier} O{n'est pas premier} m }{%
  \num{#3} \xintifboolexpr{\pflboolestpremier{#3} == 1}{#1}{#2}\relax%
}%

\NewCommandCopy\pflisprime\pflestpremier

%%------Décomposition, l3
\ExplSyntaxOn

% Variables globales
\int_new:N \l_pfl_n_int
\int_new:N \l_pfl_factor_int
\int_new:N \l_pfl_exp_int
\tl_new:N  \l_pfl_result_tl
\bool_new:N \l_pfl_star_bool
\bool_new:N \l_pfl_more_bool %booléen auxiliaire pour savoir s'il faut un séparateur

\NewDocumentCommand \pfldecompnb { s m }
{
  \IfBooleanTF{#1}
  { \bool_set_true:N  \l_pfl_star_bool }
  { \bool_set_false:N \l_pfl_star_bool }
  \pfl_decompnb:n { #2 }
}

\NewCommandCopy\pflnumberfact\pfldecompnb

\cs_new_protected:Npn \pfl_decompnb:n #1
{
  \tl_set:Ne \l_tmpa_tl { \xinteval{#1} }
  %\tl_set:Nn \l_tmpa_tl { #1 }
  %compat xint := remplacer toutes les occurrences de 'exemple' par 'test'
  %\regex_replace_all:nnN {\*\*} {^} \l_tmpa_tl
  \int_set:Nn \l_pfl_n_int { \tl_use:N \l_tmpa_tl }
  \int_set:Nn \l_pfl_factor_int { 2 }
  \int_zero:N  \l_pfl_exp_int
  \tl_clear:N  \l_pfl_result_tl
  % Boucle principale
  \bool_while_do:nn { \int_compare_p:n { \l_pfl_n_int > 1 } }
  {
    \int_compare:nNnTF
    { \int_mod:nn { \l_pfl_n_int } { \l_pfl_factor_int } } = { 0 }
    {
      % divisible : on divise et on incrémente l'exposant
      \int_set:Nn \l_pfl_n_int
      { \int_div_truncate:nn { \l_pfl_n_int } { \l_pfl_factor_int } }
      \int_incr:N \l_pfl_exp_int
    }
    {
      % non divisible : on prépare le booléen "y aura-t-il encore un facteur après ?"
      \bool_set:Nn \l_pfl_more_bool { \int_compare_p:n { \l_pfl_n_int > 1 } }
      \pfl_append_factor:n { \l_pfl_factor_int }
      \int_incr:N \l_pfl_factor_int
      \int_zero:N  \l_pfl_exp_int
    }
  }
  % dernier facteur : plus de reste => plus de séparateur
  \bool_set_false:N \l_pfl_more_bool
  \pfl_append_factor:n { \l_pfl_factor_int }
  \ensuremath{\tl_use:N \l_pfl_result_tl}
}

% Ajoute le facteur #1 ; se base sur \l_pfl_exp_int et \l_pfl_more_bool
\cs_new_protected:Npn \pfl_append_factor:n #1
{
  \int_compare:nNnT { \l_pfl_exp_int } > { 0 }
  {
    \bool_if:NTF \l_pfl_star_bool
    {
      % afficher l'exposant même s'il vaut 1
      \tl_put_right:Nx \l_pfl_result_tl
      {
        \num{\int_to_arabic:n {#1}}^{\int_to_arabic:n{\l_pfl_exp_int}}
        \bool_if:NT \l_pfl_more_bool { \times }
      }
    }
    {
      % pas d'étoile : omettre ^1
      \int_compare:nNnTF { \l_pfl_exp_int } = { 1 }
      {
        \tl_put_right:Nx \l_pfl_result_tl
        {
          \num{\int_to_arabic:n {#1}}
          \bool_if:NT \l_pfl_more_bool { \times }
        }
      }
      {
        \tl_put_right:Nx \l_pfl_result_tl
        {
          \num{\int_to_arabic:n {#1}}^{\int_to_arabic:n{\l_pfl_exp_int}}
          \bool_if:NT \l_pfl_more_bool { \times }
        }
      }
    }
  }
}

\ExplSyntaxOff

%%------ConversionsBases
%dec->bin avec blocs de 4 chiffres
\setKVdefault[CONVDECBIN]{%
  AffBase=true,show base=true
}
\NewDocumentCommand\ConversionDecBin{ s O{} m }{%
  \useKVdefault[CONVDECBIN]
  \setKV[CONVDECBIN]{#2}% on paramètres les nouvelles clés et on les simplifie
  %en/fr
  \setKVbooltruedefaultmulti[CONVDECBIN]{show base}{AffBase}%
  %next
  \def\resbrut{\xintDecToBin{#3}}%
  \pfllenstr{\resbrut}[\nbchiffres]%
  \def\nbgrp{\fpeval{4*ceil(\nbchiffres/4,0)}}%
  \IfBooleanTF{#1}%
    {\num{#3}\ifboolKV[CONVDECBIN]{AffBase}{_{10}}{}=\num[digit-group-size=4]{\resbrut}\ifboolKV[CONVDECBIN]{AffBase}{_{2}}{}}%
    {\num{#3}\ifboolKV[CONVDECBIN]{AffBase}{_{10}}{}=\num[digit-group-size=4,minimum-integer-digits=\nbgrp]{\resbrut}\ifboolKV[CONVDECBIN]{AffBase}{_{2}}{}}%
}
\NewCommandCopy\pflconvdecbin\ConversionDecBin

\setKVdefault[CONVBINHEX]{%
  %Epaisseur=0.75pt,%
  AffBase=true,show base=true,%
  Details=true,details=true
}
%bourrage de 0 avant
\ExplSyntaxOn
\NewExpandableDocumentCommand{\PLstrzeros}{m}
{
  \int_compare:nT { #1 > 0 }
  {
    0 \prg_replicate:nn { #1 - 1 } { 0 }
  }
}
\ExplSyntaxOff


%la conversion complète
\newcommand\ConversionBinHex[2][]{%
  \useKVdefault[CONVBINHEX]%
  \setKV[CONVBINHEX]{#1}% on paramètres les nouvelles clés et on les simplifie
  %en/fr
  \setKVbooltruedefaultmulti[CONVBINHEX]{show base}{AffBase}%
  \setKVbooltruedefaultmulti[CONVBINHEX]{details}{Details}%
  %next
  \def\chbrut{#2}%
  \pfllenstr{\chbrut}[\nbchiffres]%nb de chiffres du binaire
  \xdef\nbgrp{\fpeval{4*ceil(\nbchiffres/4,0)}} %nb de chiffres avec blocs de 4
  \xdef\nbblocs{\fpeval{\nbgrp/4}} %nb de blocs
  %on rajoute des zeros si besoin := OK
  \xdef\resinter{\chbrut}%
  \num[digit-group-size=4]{\chbrut}\ifboolKV[CONVBINHEX]{AffBase}{_{2}}{}=%
  \ifboolKV[CONVBINHEX]{Details}{%
  \ifnum\nbchiffres<\nbgrp%
    \xdef\nbz{\inteval{\nbgrp-\nbchiffres}}%
    \xdef\resinter{\PLstrzeros{\nbz}\chbrut}%
    \num[digit-group-size=4,minimum-integer-digits=\nbgrp]{\resinter}=%
  \fi%
  %découpage par blocs et conversion en hexa := OK
  \newcount\cpt%
  \cpt0%
  \loop\ifnum \cpt<\nbblocs%
    \def\iinit{\fpeval{4*\cpt+1}}%
    \def\ifinal{\fpeval{4*(\cpt+1)}}%
    \StrMid{\resinter}{\iinit}{\ifinal}[\blocinter]%
    {\underbracket{\blocinter}_{\xintBinToHex{\blocinter}}\,}%
    \advance\cpt by 1%
  \repeat%
  \!=%
  }%
  {}%
  \xintBinToHex{\chbrut}\ifboolKV[CONVBINHEX]{AffBase}{_{16}}{}%
}
\NewCommandCopy\pflconvbinhex\ConversionBinHex

%hexa-bin par bloc de 4
\setKVdefault[CONVHEXBIN]{%
  %Epaisseur=0.75pt,
  AffBase=true,show base=true,%
  Details=true,details=true
}

\newcommand\ConvHexBinBloc[1]{%
  \def\binbrut{\xintHexToBin{#1}}%
  \pfllenstr{\binbrut}[\nbchiffresbinbrut]%nb
  \PLstrzeros{\xinteval{4-\nbchiffresbinbrut}}{\binbrut}%
}

\newcommand\ConversionHexBin[2][]{%
  \useKVdefault[CONVHEXBIN]%
  \setKV[CONVHEXBIN]{#1}% on paramètres les nouvelles clés et on les simplifie
  %en/fr
  \setKVbooltruedefaultmulti[CONVHEXBIN]{show base}{AffBase}%
  \setKVbooltruedefaultmulti[CONVHEXBIN]{details}{Details}%
  %next
  %\def\thicktraitshexbin{\useKV[CONVHEXBIN]{Epaisseur}}%
  \ifboolKV[CONVHEXBIN]{AffBase}{#2_{16}=}{#2=}%
  \pfllenstr{#2}[\nbchiffreshex]%nb
  \ifboolKV[CONVHEXBIN]{Details}%
  {%
    \foreach \i in {1,...,\nbchiffreshex}{%
      \StrChar{#2}{\i}[\tmpcharhex]%
      {\underbracket{\ConvHexBinBloc{\tmpcharhex}}_{\tmpcharhex}\,}
    }%
    \ifboolKV[CONVHEXBIN]{AffBase}{{}_{2}}{}%
  }%
  {%
    \foreach \i in {1,...,\nbchiffreshex}{%
      \StrChar{#2}{\i}[\tmpcharhex]%
      \ConvHexBinBloc{\tmpcharhex}\,%
    }%
    \!\ifboolKV[CONVHEXBIN]{AffBase}{{}_{2}}{}%
  }%
}
\NewCommandCopy\pflconvhexbin\ConversionHexBin


%hexa/bin->dec avec écriture polynomiale
\defKV[CONVTODEC]{%
  BaseDep=\def\basedepart{#1},base from=\def\basedepart{#1},
}
\setKVdefault[CONVTODEC]{%
  BaseDep=2,base from=2,%
  AffBase=true,show base=true,%
  Details=true,details=true,%
  Zeros=true,zeros=true
}

\ExplSyntaxOn
\newcommand\convertbasetobasedix[2]{%
  \int_from_base:nn {#1}{#2}
}
\ExplSyntaxOff

\newcommand\ConversionVersDec[2][]{%
  \useKVdefault[CONVTODEC]
  \setKV[CONVTODEC]{#1}% on paramètres les nouvelles clés et on les simplifie
  %en/fr
  \setKVbooltruedefaultmulti[CONVTODEC]{show base}{AffBase}%
  \setKVbooltruedefaultmulti[CONVTODEC]{details}{Details}%
  \setKVbooltruedefaultmulti[CONVTODEC]{zeros}{Zeros}%
  %next
  \def\nbdepart{#2}%
  \pfllenstr{\nbdepart}[\nbchiffres]%
  \StrChar{\nbdepart}{1}[\chiffre]%
  %si on est en base 16
  \xintifboolexpr{\basedepart == 16}%
  {%
    \nbdepart\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
    \ifboolKV[CONVTODEC]{Details}{%
      \xintHexToDec{\chiffre}\times\basedepart^{\inteval{\nbchiffres-1}}%
      \newcount\cpt%
      \cpt2%
      \loop\ifnum \cpt<\inteval{\nbchiffres+1}%
        \def\puiss{\inteval{\nbchiffres-\cpt}}%
        \StrChar{\nbdepart}{\cpt}[\chiffre]%
        \ifboolKV[CONVTODEC]{Zeros}%
          {%
            +\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
          }%
          {%
            \ifnum\xintHexToDec{\chiffre} > 0%
              +\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
            \fi%
          }%
        \advance\cpt by 1%
      \repeat%
      =%
      }%
      {}%
    \num{\xintHexToDec{\nbdepart}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
  }%
  {}%
  \xintifboolexpr{\basedepart == 2}%
  {%
    \num[digit-group-size=4]{\nbdepart}\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
    \ifboolKV[CONVTODEC]{Details}{%
      \chiffre\times\basedepart^{\inteval{\nbchiffres-1}}%
      \newcount\cpt%
      \cpt2%
      \loop\ifnum \cpt<\inteval{\nbchiffres+1}%
        \def\puiss{\inteval{\nbchiffres-\cpt}}%
        \StrChar{\nbdepart}{\cpt}[\chiffre]%
        \ifboolKV[CONVTODEC]{Zeros}%
        {%
          +\chiffre\times\basedepart^{\puiss}%
        }%
        {%
          \ifnum\chiffre > 0%
            +\chiffre\times\basedepart^{\puiss}%
          \fi%
        }%
      \advance\cpt by 1%
      \repeat%
      =%
      }%
      {}%
    \num{\xintBinToDec{\nbdepart}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
  }%
  {}%
}
\NewCommandCopy\pflconvversdec\ConversionVersDec
\NewCommandCopy\pflconvtodec\ConversionVersDec

\newcommand\ConversionBaseDix[3][]{%1=options,%2=nb,%3=basedep ??
  \useKVdefault[CONVTODEC]
  \setKV[CONVTODEC]{#1}% on paramètres les nouvelles clés et on les simplifie
  %en/fr
  \setKVbooltruedefaultmulti[CONVTODEC]{show base}{AffBase}%
  \setKVbooltruedefaultmulti[CONVTODEC]{details}{Details}%
  \setKVbooltruedefaultmulti[CONVTODEC]{zeros}{Zeros}%
  %next
  \def\NBdepart{#2}%
  \def\basedepart{#3}%
  \pfllenstr{\NBdepart}[\nbchiffres]%
  \StrChar{\NBdepart}{1}[\chiffre]%
  \NBdepart\ifboolKV[CONVTODEC]{AffBase}{_{\basedepart}}{} =%
  \ifboolKV[CONVTODEC]{Details}{%
    \xintHexToDec{\chiffre}\times\basedepart^{\inteval{\nbchiffres-1}}%
    \newcount\cpt%
    \cpt2%
    \loop\ifnum \cpt<\inteval{\nbchiffres+1}%
      \def\puiss{\inteval{\nbchiffres-\cpt}}%
      \StrChar{\NBdepart}{\cpt}[\chiffre]%
      \ifboolKV[CONVTODEC]{Zeros}%
        {%
          +\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
        }%
        {%
          \ifnum\xintHexToDec{\chiffre} > 0%
            +\xintHexToDec{\chiffre}\times\basedepart^{\puiss}%
          \fi%
        }%
      \advance\cpt by 1%
    \repeat%
    =%
    }%
    {}%
  \num{\convertbasetobasedix{#2}{#3}}\ifboolKV[CONVTODEC]{AffBase}{_{10}}{}%
}
\NewCommandCopy\pflconvdix\ConversionBaseDix
\NewCommandCopy\pflconvten\ConversionBaseDix

%%------CONVFROMDEC
\newcommand\PLnoeud[2]{\tikz[remember picture,baseline=(#1.base)]\node[shape=rectangle,inner sep=0pt](#1){#2};}
\ExplSyntaxOn
\newcommand\convertbasedixtobase[2]{%
  \int_to_Base:nn {#1}{#2}
}
\ExplSyntaxOff

\defKV[convfromten]{%
  Couleur=\def\PLConvCouleur{#1},color=\def\PLConvCouleur{#1},%
  DecalH=\def\PLConvDecalH{#1},hsep=\def\PLConvDecalH{#1},%
  DecalV=\def\PLConvDecalV{#1},vsep=\def\PLConvDecalV{#1},%
  Noeud=\def\PLConvNoeud{#1},node=\def\PLConvNoeud{#1}
}

\setKVdefault[convfromten]{%
  Couleur=red,color=red,%
  DecalH=2pt,hsep=2pt,%
  DecalV=3pt,vsep=3pt,%
  Rect=true,rect=true,%
  Noeud=EEE,node=EEE,%
  CouleurRes=false,res color=false
}

\newcommand\ConversionDepuisBaseDix[3][]{%
  \useKVdefault[convfromten]%
  \setKV[convfromten]{#1}%
  %en/fr
  \setKVbooltruedefaultmulti[convfromten]{rect}{Rect}%
  \setKVboolfalsedefaultmulti[convfromten]{res color}{CouleurRes}%
  %next
  \xdef\ValRes{\xintDecToHex{#2}}%
  \xdef\ValA{#2}\xdef\ValB{#3}%
  \xdef\ValTMP{#2}%
  \xdef\ValMU{\inteval{#3-1}}%
  \ensuremath{%
    \left\lbrace\begin{array}{@{\,}r@{\;=\;}l@{\;+\;}r}
      %1ere division
      \xdef\ValQ{\fpeval{trunc(\ValTMP/(#3),0)}}\xdef\ValR{\fpeval{\ValTMP-(#3)*\ValQ}}
      \num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\PLnoeud{\PLConvNoeud1}{\num{\ValR}}%
      \xdef\ValTMP{\ValQ}%
      \whiledo {\ValTMP > \ValMU}%
      {%
        \xdef\ValQ{\fpeval{trunc(\ValTMP/(#3),0)}}\xdef\ValR{\fpeval{\ValTMP-(#3)*\ValQ}}%
        \\ \num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\num{\ValR}
        \xdef\ValTMP{\ValQ}%
      }
      %dernière
      \xdef\ValQ{\fpeval{trunc(\ValTMP/(#3),0)}}\xdef\ValR{\fpeval{\ValTMP-(#3)*\ValQ}}%
      \\ \num{\ValTMP}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}\PLnoeud{\PLConvNoeud2}{\num{\ValR}}%
    \end{array} \right| \Rightarrow \num{#2}_{10}=\ifboolKV[convfromten]{CouleurRes}{\mathcolor{\PLConvCouleur}{\convertbasedixtobase{#2}{#3}_{#3}}}{\convertbasedixtobase{#2}{#3}_{#3}}}%
  \ifboolKV[convfromten]{Rect}%
  {%
    \IfSubStr{\PLConvDecalH}{/}%
      {\StrCut{\PLConvDecalH}{/}{\PLConvDecalHg}{\PLConvDecalHd}}%
      {\def\PLConvDecalHg{\PLConvDecalH}\def\PLConvDecalHd{\PLConvDecalH}}%
    \begin{tikzpicture}[remember picture]
      \draw[overlay,rounded corners=4pt,\PLConvCouleur,thick] ($(\PLConvNoeud1.north west)+(-\PLConvDecalHg,\PLConvDecalV)$) rectangle ($(\PLConvNoeud2.south east)+(\PLConvDecalHd,-\PLConvDecalV)$) ;
      \draw[overlay,rounded corners=4pt,\PLConvCouleur,thick,->,>=latex] ($(\PLConvNoeud2.east)+(\PLConvDecalHd,0)$)--++(0,{0.75\baselineskip}) ;
    \end{tikzpicture}%
  }{}%
}
\NewCommandCopy\pflconvdepuisdix\ConversionDepuisBaseDix
\NewCommandCopy\pflconvfromten\ConversionDepuisBaseDix

%%------PRESPGCD
\DeclareMathOperator{\PLpgcd}{PGCD}
\defKV[prespgcd]{%
  Couleur=\def\PLPGCDCouleur{#1},color=\def\PLPGCDCouleur{#1},%
  DecalRect=\def\PLPGCDDecal{#1},rect sep=\def\PLPGCDDecal{#1},%
  Noeud=\def\PLPGCDNoeud{#1},node=\def\PLPGCDNoeud{#1}
}

\setKVdefault[prespgcd]{%
  Couleur=red,color=red,%
  DecalRect=2pt,rect sep=2pt,%
  Rectangle=true,rect=true,%
  Noeud=FFF,node=FFF,%
  CouleurResultat=false,res color=false,%
  AfficheConclusion=true,disp ccl=true,%
  AfficheDelimiteurs=true,disp delim=true
}

\RequirePackage{xintgcd}

\newcommand\PresentationPGCD[3][]{%
  \useKVdefault[prespgcd]%
  \setKV[prespgcd]{#1}%
  %en/fr
  \setKVbooltruedefaultmulti[prespgcd]{rect}{Rect}%
  \setKVboolfalsedefaultmulti[prespgcd]{res color}{CouleurRes}%
  \setKVbooltruedefaultmulti[prespgcd]{disp ccl}{AfficheConclusion}%
  \setKVbooltruedefaultmulti[prespgcd]{disp delim}{AfficheDelimiteurs}%
  %next
  \xdef\respgcd{\xinteval{gcd(#2,#3)}}
  \xdef\ValA{#2}\xdef\ValB{#3}%on stocke les valeurs du départ
  \ensuremath{%
    \ifboolKV[prespgcd]{AfficheDelimiteurs}%
    {\left\lbrace}%
    {}%
    \begin{array}{@{\,}r@{\;=\;}l@{\;+\;}r}
      %1ère division
      \xdef\ValQ{\fpeval{trunc((\ValA)/(\ValB),0)}}\xdef\ValR{\fpeval{\ValA-(\ValB)*(\ValQ)}}%
      \num{\ValA}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}%
      \xintifboolexpr{\ValR == \respgcd}%
      {\PLnoeud{\PLPGCDNoeud1}{\num{\ValR}}}%noeud si c'est le pgcd
      {\num{\ValR}}%
      \xdef\ValA{\ValB}\xdef\ValB{\ValR}%nouvelles valeurs
      \whiledo {\ValR > 0}%
      {%
        \xdef\ValQ{\fpeval{trunc((\ValA)/(\ValB),0)}}\xdef\ValR{\fpeval{\ValA-(\ValB)*(\ValQ)}}%
        \\%
        \num{\ValA}\uppercase{&}\num{\ValB}\times\num{\ValQ}\uppercase{&}%
        \xintifboolexpr{\ValR == \respgcd}%
        {\PLnoeud{\PLPGCDNoeud1}{\num{\ValR}}}%noeud si c'est le pgcd
        {\num{\ValR}}%
        \xdef\ValA{\ValB}\xdef\ValB{\ValR}%nouvelles valeurs
      }%
    \end{array}%
    \ifboolKV[prespgcd]{AfficheDelimiteurs}%
    {\right|}%
    {}%
    \ifboolKV[prespgcd]{AfficheConclusion}%
    {%
      \Rightarrow \PLpgcd\left(\num{#2};\num{#3}\right)=\ifboolKV[prespgcd]{CouleurResultat}{\mathcolor{\PLPGCDCouleur}{\num{\respgcd}}}{\num{\respgcd}}%
    }%
    {}%
  }%
  \ifboolKV[prespgcd]{Rectangle}%
  {%
    \begin{tikzpicture}[remember picture]
      \draw[overlay,rounded corners=4pt,\PLPGCDCouleur,thick] ($(\PLPGCDNoeud1.north west)+(-\PLPGCDDecal,\PLPGCDDecal)$) rectangle ($(\PLPGCDNoeud1.south east)+(\PLPGCDDecal,-\PLPGCDDecal)$) ;
    \end{tikzpicture}%
  }{}%
}
\NewCommandCopy\pflprespgcd\PresentationPGCD

%%===égalité de Bezout
\NewDocumentCommand\AffCoeffBezout{ m }{%
  \xintifboolexpr{#1 < 0}%
    {\left( \num{#1} \right)}%
    {\num{#1}}%
}
\NewDocumentCommand\EgaliteBezout{ O{black} m m }{%
  \xintAssign{\xintBezout{#2}{#3}}\to\TmpU\TmpV\TmpD%
  \ensuremath{\num{#2} \times \mathcolor{#1}{\AffCoeffBezout{\TmpU}} + \AffCoeffBezout{#3} \times \mathcolor{#1}{\AffCoeffBezout{\TmpV}} = \num{\TmpD}}%
}
\NewCommandCopy\pflbezout\EgaliteBezout

%%===Équations diophantiennes
\makeatletter
\if@loadcancel
  \RequirePackage[thicklines]{cancel}%comme PfC
\fi
\makeatother

\NewDocumentCommand\AffCoeffDioph{ m }{%
  \xintifboolexpr{#1 < 0}%
    {\left( \num{#1} \right)}%
    {\num{#1}}%
}
\NewDocumentCommand\AffCoeffDiophSign{ m }{%
  \xintifboolexpr{#1 < 0}%
    {\num{#1}}%
    {+\num{#1}}%
}

\defKV[eqdioph]{%
  Lettre=\def\LettreSolEDioph{#1},letter=\def\LettreSolEDioph{#1},%
  Couleur=\def\CouleurSolEDioph{#1},color=\def\CouleurSolEDioph{#1},%
  Inconnues=\def\InconnuesSolEDioph{#1},unknowns=\def\InconnuesSolEDioph{#1},%
  Entier=\def\KKK{#1},integer=\def\KKK{#1}
}

\setKVdefault[eqdioph]{%
  Lettre=E,letter=E,%
  Couleur=black,color=black,%
  Inconnues=x/y,unknowns=x/y,%
  Entier=k,integer=k,%
  Cadres=false,border=false,%
  PresPGCD=true,pres gcd=true
}

\makeatletter

\NewDocumentCommand\EquationDiophantienne{ O{} m }{%
    \useKVdefault[eqdioph]%
    \setKV[eqdioph]{#1}%
  %en/fr
  \setKVboolfalsedefaultmulti[eqdioph]{border}{Cadres}%
  \setKVbooltruedefaultmulti[eqdioph]{pres gcd}{PresPGCD}%
  %next
  %====multilng
  \if@pfllngfr
    \def\labeldapres{\text{ d'après }}
  \fi
  \if@pfllngen
    \def\labeldapres{\text{ from }}
  \fi
  \if@pfllngde
    \def\labeldapres{\text{ nach }}
  \fi
  \if@pfllnges
    \def\labeldapres{\text{ según }}
  \fi
    \setlength{\parindent}{0pt}%
    % Extractions des paramètres
    \StrBefore[1]{\InconnuesSolEDioph}{/}[\XXX]%
    \StrBehind[1]{\InconnuesSolEDioph}{/}[\YYY]%
    \StrBefore{#2}{\XXX}[\AA]%
    \StrBetween{#2}{\XXX}{\YYY}[\BB]%
    \StrBehind{#2}{=}[\CC]%
    \IfStrEq{\AA}{}%
        {\def\AA{1}}{}%
    \IfStrEq{\AA}{-}%
        {\def\AA{-1}}{}%
    \pfllenstr{\BB}[\lgtB]%
    \xintifboolexpr{ \lgtB > 1 }% +b ou -b
        {%
            \StrDel{\BB}{+}[\BB]%
        }%
        {%
            \IfStrEq{\BB}{-}%
                {\def\BB{-1}}{}%
            \IfStrEq{\BB}{+}%
                {\def\BB{1}}{}%
        }%
    % Calcul du PGCD
    \xdef\PGCDD{\xinteval{gcd(\AA,\BB)}}%

    % Texte multilingue : Énoncé de l'équation
    \if@pfllngfr
        On cherche à résoudre l'équation diophantienne :\[ \num{\AA}\XXX + \AffCoeffDioph{\BB}\YYY=\num{\CC} \xintifboolexpr{ \PGCDD == 1 'or' \xintiiRem{\CC}{\PGCDD} != 0 }{\qquad (\LettreSolEDioph)}{} \]%
        \ifboolKV[eqdioph]{PresPGCD}%
            {D'après l'algorithme d'Euclide : \PresentationPGCD[Rectangle=false]{\xinteval{abs(\AA)}}{\xinteval{abs(\BB)}}.}%
            {Le PGCD de \num{\AA} et de \num{\BB} vaut \num{\PGCDD}.}%
    \fi
    \if@pfllngen
        We seek to solve the Diophantine equation: \[ \num{\AA}\XXX + \AffCoeffDioph{\BB}\YYY=\num{\CC} \xintifboolexpr{ \PGCDD == 1 'or' \xintiiRem{\CC}{\PGCDD} != 0 }{\qquad (\LettreSolEDioph)}{} \]%
        \ifboolKV[eqdioph]{PresPGCD}%
            {Using the Euclidean algorithm: \PresentationPGCD[Rectangle=false]{\xinteval{abs(\AA)}}{\xinteval{abs(\BB)}}.}%
            {The GCD of \num{\AA} and \num{\BB} is \num{\PGCDD}.}%
    \fi
    \if@pfllngde
        Man sucht die Lösungen der diophantischen Gleichung: \[ \num{\AA}\XXX + \AffCoeffDioph{\BB}\YYY=\num{\CC} \xintifboolexpr{ \PGCDD == 1 'or' \xintiiRem{\CC}{\PGCDD} != 0 }{\qquad (\LettreSolEDioph)}{} \]%
        \ifboolKV[eqdioph]{PresPGCD}%
            {Nach dem euklidischen Algorithmus: \PresentationPGCD[Rectangle=false]{\xinteval{abs(\AA)}}{\xinteval{abs(\BB)}}.}%
            {Der GGT von \num{\AA} und \num{\BB} ist \num{\PGCDD}.}%
    \fi
    \if@pfllnges
        Se busca resolver la ecuación diofántica: \[ \num{\AA}\XXX + \AffCoeffDioph{\BB}\YYY=\num{\CC} \xintifboolexpr{ \PGCDD == 1 'or' \xintiiRem{\CC}{\PGCDD} != 0 }{\qquad (\LettreSolEDioph)}{} \]%
        \ifboolKV[eqdioph]{PresPGCD}%
            {Según el algoritmo de Euclides: \PresentationPGCD[Rectangle=false]{\xinteval{abs(\AA)}}{\xinteval{abs(\BB)}}.}%
            {El MCD de \num{\AA} y \num{\BB} es \num{\PGCDD}.}%
    \fi

    \par\smallskip

    % Suite de la macro : Solutions ou non
    \xintifboolexpr{ \xintiiRem{\CC}{\PGCDD} == 0 }% solutions obligatoires
        {%
            \xintifboolexpr{ \PGCDD == 1}%
                {%
                    \if@pfllngfr
                        Les entiers \num{\xinteval{abs(\AA)}} et \num{\xinteval{abs(\BB)}} sont premiers entre eux, donc l'équation $(\LettreSolEDioph)$ admet une infinité de solutions.\par
                        \xdef\AAA{\AA}\xdef\BBB{\BB}\xdef\CCC{\CC}%
                    \fi
                    \if@pfllngen
                        The integers \num{\xinteval{abs(\AA)}} and \num{\xinteval{abs(\BB)}} are coprime, so the equation $(\LettreSolEDioph)$ has infinitely many solutions.\par
                        \xdef\AAA{\AA}\xdef\BBB{\BB}\xdef\CCC{\CC}%
                    \fi
                    \if@pfllngde
                        Die Zahlen \num{\xinteval{abs(\AA)}} und \num{\xinteval{abs(\BB)}} sind teilerfremd, also hat die Gleichung $(\LettreSolEDioph)$ unendlich viele Lösungen.\par
                        \xdef\AAA{\AA}\xdef\BBB{\BB}\xdef\CCC{\CC}%
                    \fi
                    \if@pfllnges
                        Los números \num{\xinteval{abs(\AA)}} y \num{\xinteval{abs(\BB)}} son primos entre sí, por lo que la ecuación $(\LettreSolEDioph)$ tiene infinitas soluciones.\par
                        \xdef\AAA{\AA}\xdef\BBB{\BB}\xdef\CCC{\CC}%
                    \fi
                }%
                {%
                    \if@pfllngfr
                        Le PGCD de \num{\AA} et \num{\BB} divise \num{\CC}, donc on peut simplifier l'équation diophantienne par \num{\PGCDD}.
                    \fi
                    \if@pfllngen
                        The GCD of \num{\AA} and \num{\BB} divides \num{\CC}, so we can simplify the Diophantine equation by \num{\PGCDD}.
                    \fi
                    \if@pfllngde
                        Der GGT von \num{\AA} und \num{\BB} teilt \num{\CC}, also können wir die diophantische Gleichung durch \num{\PGCDD} vereinfachen.
                    \fi
                    \if@pfllnges
                        El MCD de \num{\AA} y \num{\BB} divide a \num{\CC}, por lo que podemos simplificar la ecuación diofántica por \num{\PGCDD}.
                    \fi
                    \xdef\AAA{\xintiieval{\AA/\PGCDD}}\xdef\BBB{\xintiieval{\BB/\PGCDD}}\xdef\CCC{\xintiieval{\CC/\PGCDD}}%
                    %
                    \[ \num{\AA}\XXX+\AffCoeffDioph{\BB}\YYY=\num{\CC} \underset{\div\num{\PGCDD}}{\Longleftrightarrow} \num{\AAA}\XXX+\AffCoeffDioph{\BBB}\YYY=\num{\CCC} \qquad (\LettreSolEDioph) \]%
                    \if@pfllngfr
                        Les entiers \num{\AAA} et \num{\BBB} sont premiers entre eux, donc l'équation $(\LettreSolEDioph)$ admet une infinité de solutions.\par
                    \fi
                    \if@pfllngen
                        The integers \num{\AAA} and \num{\BBB} are coprime, so the equation $(\LettreSolEDioph)$ has infinitely many solutions.\par
                    \fi
                    \if@pfllngde
                        Die Zahlen \num{\AAA} und \num{\BBB} sind teilerfremd, also hat die Gleichung $(\LettreSolEDioph)$ unendlich viele Lösungen.\par
                    \fi
                    \if@pfllnges
                        Los números \num{\AAA} y \num{\BBB} son primos entre sí, por lo que la ecuación $(\LettreSolEDioph)$ tiene infinitas soluciones.\par
                    \fi
                }%
            \xintAssign{\xintBezout{\AAA}{\BBB}}\to\TmpU\TmpV\TmpD
            %
            \if@pfllngfr
                On détermine une solution particulière de $(E)$ :
            \fi
            \if@pfllngen
                We determine a particular solution of $(E)$:
            \fi
            \if@pfllngde
                Wir bestimmen eine spezielle Lösung von $(E)$:
            \fi
            \if@pfllnges
                Determinamos una solución particular de $(E)$:
            \fi
            \[ \num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffBezout{\TmpU}} + \AffCoeffBezout{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffBezout{\TmpV}} = \num{\TmpD}
            \xintifboolexpr{ \CCC != 1}%
                {%
                    \underset{\times\AffCoeffDioph{\CCC}}{\implies}
                    \num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpU*\CCC}}} + \AffCoeffBezout{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpV*\CCC}}} = \num{\CCC}
                }%
                {}%
                \qquad ({\LettreSolEDioph}_0)
             \]%
            %
            \if@pfllngfr
                Par soustraction :
            \fi
            \if@pfllngen
                By subtraction:
            \fi
            \if@pfllngde
                Durch Subtraktion:
            \fi
            \if@pfllnges
                Restando:
            \fi
            %
            \[%
            {\renewcommand\arraystretch{1.25}%
                \begin{array}{ @{\,} c @{\,} c @{\;\times\;} c @{\;+\;} c @{\;\times\;} c @{\;=\;} c }
                    & \num{\AAA} & \XXX & \AffCoeffDioph{\BBB} & \YYY & \num{\CCC} \\
                    -~~~~~ & \num{\AAA} & \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpU*\CCC}}} & \AffCoeffDioph{\BBB} & \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\TmpV*\CCC}}} & \num{\CCC} \\ \hline
                    & \num{\AAA} & \left( \XXX \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpU*\CCC}}} \right)& \AffCoeffDioph{\BBB} & \left( \YYY \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpV*\CCC}}} \right) & 0\\
                \end{array}}
            \]%
            \def\TmpPartieA{\XXX \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpU*\CCC}}}}%
            \def\TmpPartieB{\YYY \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\TmpV*\CCC}}}}%
            %
            \if@pfllngfr
                On en déduit que $\num{\AAA} \times \underbrace{\left( \TmpPartieA \right)}_{\text{entier}} = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$, et donc que $\num{\AAA} \mid \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$.\par\smallskip
                Or \num{\xinteval{abs(\AAA)}} et \num{\xinteval{abs(\BBB)}} sont premiers entre eux, donc d'après le théorème de Gauss, on a $\num{\AAA} \mid \TmpPartieB$.\par
                Il existe donc un entier $\KKK$ tel que $\TmpPartieB = \num{\AAA} \times \KKK$, ce qui donne
                $\ifboolKV[eqdioph]{Cadres}
                    {\boxed{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}}
                    {\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}
                $.\par
            \fi
            \if@pfllngen
                We deduce that $\num{\AAA} \times \underbrace{\left( \TmpPartieA \right)}_{\text{integer}} = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$, and therefore that $\num{\AAA} \mid \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$.\par\smallskip
                Since \num{\xinteval{abs(\AAA)}} and \num{\xinteval{abs(\BBB)}} are coprime, by Gauss's theorem, we have $\num{\AAA} \mid \TmpPartieB$.\par
                There exists an integer $\KKK$ such that $\TmpPartieB = \num{\AAA} \times \KKK$, which gives
                $\ifboolKV[eqdioph]{Cadres}
                    {\boxed{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}}
                    {\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}
                $.\par
            \fi
            \if@pfllngde
                Daraus folgt, dass $\num{\AAA} \times \underbrace{\left( \TmpPartieA \right)}_{\text{ganze Zahl}} = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$, und somit $\num{\AAA} \mid \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$.\par\smallskip
                Da \num{\xinteval{abs(\AAA)}} und \num{\xinteval{abs(\BBB)}} teilerfremd sind, folgt nach dem Satz von Gauss, dass $\num{\AAA} \mid \TmpPartieB$.\par
                Es existiert eine ganze Zahl $\KKK$, sodass $\TmpPartieB = \num{\AAA} \times \KKK$, was ergibt
                $\ifboolKV[eqdioph]{Cadres}
                    {\boxed{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}}
                    {\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}
                $.\par
            \fi
            \if@pfllnges
                Se deduce que $\num{\AAA} \times \underbrace{\left( \TmpPartieA \right)}_{\text{entero}} = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$, y por lo tanto que $\num{\AAA} \mid \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right)$.\par\smallskip
                Como \num{\xinteval{abs(\AAA)}} y \num{\xinteval{abs(\BBB)}} son primos entre sí, según el teorema de Gauss, tenemos $\num{\AAA} \mid \TmpPartieB$.\par
                Existe un entero $\KKK$ tal que $\TmpPartieB = \num{\AAA} \times \KKK$, lo que da
                $\ifboolKV[eqdioph]{Cadres}
                    {\boxed{\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}}
                    {\YYY = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}
                $.\par
            \fi
            % Suite des calculs
            \begin{align*}
                \num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \left( \TmpPartieB \right) & \implies \num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \big( \underbrace{\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK}_{\mathclap{\YYY}}  \mathcolor{\CouleurSolEDioph}{\AffCoeffDiophSign{\xinteval{-\CCC*\TmpV}}} \big) \\
                & \implies \num{\AAA} \times \left( \TmpPartieA \right) = \num{\xinteval{-\BBB}} \times \left( \num{\AAA}\KKK \right) \\
                & \implies \TmpPartieA = \num{\xinteval{-\BBB}}\KKK \\
                & \implies \ifboolKV[eqdioph]{Cadres}
                    {\boxed{\XXX = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}}
                    {\XXX = \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}
            \end{align*}
            %
            \if@pfllngfr
                Ainsi, si $\XXX$ et $\YYY$ sont solutions de $(\LettreSolEDioph)$, alors il existe un entier $\KKK$ tel que ${\XXX=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}$ et ${\YYY=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}$.\par\medskip
                Réciproquement, soit $\KKK$ un entier quelconque :
            \fi
            \if@pfllngen
                Therefore, if $\XXX$ and $\YYY$ are solutions of $(\LettreSolEDioph)$, then there exists an integer $\KKK$ such that ${\XXX=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}$ and ${\YYY=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}$.\par\medskip
                Conversely, let $\KKK$ be any integer:
            \fi
            \if@pfllngde
                Daher sind, wenn $\XXX$ und $\YYY$ Lösungen von $(\LettreSolEDioph)$ sind, dann existiert eine ganze Zahl $\KKK$, sodass ${\XXX=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}$ und ${\YYY=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}$.\par\medskip
                Umgekehrt sei $\KKK$ eine beliebige ganze Zahl:
            \fi
            \if@pfllnges
                Por lo tanto, si $\XXX$ y $\YYY$ son soluciones de $(\LettreSolEDioph)$, entonces existe un entero $\KKK$ tal que ${\XXX=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK}$ y ${\YYY=\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK}$.\par\medskip
                Recíprocamente, sea $\KKK$ un entero cualquiera:
            \fi
            %
            \begin{align*}
                \num{\AAA} \times \left( \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \right) + \AffCoeffDioph{\BBB} \times  \left( \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right) & = \num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpU}}} + \cancel{\AffCoeffDioph{\AAA} \times \AffCoeffDioph{\xinteval{-\BBB}} \KKK} + \AffCoeffDioph{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpV}}} +  \cancel{\AffCoeffDioph{\BBB} \times \AffCoeffDioph{\AAA} \KKK} \\
                & = \underbrace{\num{\AAA} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpU}}} +  \AffCoeffDioph{\BBB} \times \mathcolor{\CouleurSolEDioph}{\AffCoeffDioph{\xinteval{\CCC*\TmpV}}}}_{=\,\num{\CCC} \labeldapres ({\LettreSolEDioph}_0)} \\
                & = \num{\CCC}
            \end{align*}
            %
            \if@pfllngfr
                On en déduit que $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ est solution de $(\LettreSolEDioph)$.\par\medskip
                On en déduit que $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ est solution de $(\LettreSolEDioph)$.\par\medskip
                En conclusion, les solutions de $(E)$ sont donc les couples $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK \right)$, avec $\KKK$ un entier relatif.
            \fi
            \if@pfllngen
                We deduce that $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ is a solution of $(\LettreSolEDioph)$.\par\medskip
                In conclusion, the solutions of $(E)$ are the pairs $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK \right)$, with $\KKK$ any integer.
            \fi
            \if@pfllngde
                Daraus folgt, dass $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ eine Lösung von $(\LettreSolEDioph)$ ist.\par\medskip
                Zusammenfassend sind die Lösungen von $(E)$ die Paare $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK \right)$, wobei $\KKK$ eine ganze Zahl ist.
            \fi
            \if@pfllnges
                Se deduce que $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}}  \AffCoeffDiophSign{\AAA}\KKK \right)$ es solución de $(\LettreSolEDioph)$.\par\medskip
                En conclusión, las soluciones de $(E)$ son los pares $\left(\mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpU}}} \AffCoeffDiophSign{\xinteval{-\BBB}}\KKK \mathpunct{}; \mathcolor{\CouleurSolEDioph}{\num{\xinteval{\CCC*\TmpV}}} \AffCoeffDiophSign{\AAA}\KKK \right)$, con $\KKK$ un entero.
            \fi
        }%
        {%
            \if@pfllngfr
                Le PGCD de \num{\AA} et \num{\BB} ne divise pas \num{\CC}, donc l'équation $(\LettreSolEDioph)$ n'admet aucune solution.
            \fi
            \if@pfllngen
                The GCD of \num{\AA} and \num{\BB} does not divide \num{\CC}, so the equation $(\LettreSolEDioph)$ has no solution.
            \fi
            \if@pfllngde
                Der GGT von \num{\AA} und \num{\BB} teilt nicht \num{\CC}, also hat die Gleichung $(\LettreSolEDioph)$ keine Lösung.
            \fi
            \if@pfllnges
                El MCD de \num{\AA} y \num{\BB} no divide a \num{\CC}, por lo que la ecuación $(\LettreSolEDioph)$ no tiene solución.
            \fi
        }%
}

\makeatother

\NewCommandCopy\pflequadioph\EquationDiophantienne

%liste diviseurs
\setKVdefault[listdiv]{%
  AffNom=true,show name=true
}
\NewDocumentCommand\ListeDiviseurs{ s O{} m }{%
  \useKVdefault[listdiv]%
  \setKV[listdiv]{#2}%
  %en/fr
  \setKVbooltruedefaultmulti[listdiv]{show name}{AffNom}%
  %next
  \xdef\tmplistdiv{1}%
  \xdef\argcal{\xinteval{#3}}%
  \xintFor* ##1 in {\xintSeq{2}{\argcal}}\do{%
    \xintifboolexpr{ \xintiiRem{\argcal}{##1} == 0 }%
    {%
      \xdef\tmplistdiv{\tmplistdiv /\num{##1}}%
    }%
    {}%
  }%
  \ensuremath{\ifboolKV[listdiv]{AffNom}{\IfBooleanTF{#1}{\mathscr{D}}{\mathcal{D}}_{\num{\argcal}}=}{}\EcritureEnsemble[\strut]{\tmplistdiv}}%
}
\NewCommandCopy\pfllistediv\ListeDiviseurs
\NewCommandCopy\pfllistdiv\ListeDiviseurs

%arbre diviseurs
\defKV[arbrediviseurs]{%
  EspaceNiveau=\def\TmpEspNiv{#1},level sep=\def\TmpEspNiv{#1},%
  EspaceFeuille=\def\TMpEspFeuille{#1},child sep=\def\TMpEspFeuille{#1},%
  CouleurDetails=\def\TmpCoulDetails{#1},details color=\def\TmpCoulDetails{#1},%
  Echelle=\def\TmpEchelle{#1},scale=\def\TmpEchelle{#1}
}

\setKVdefault[arbrediviseurs]{%
  EspaceNiveau=2.25,level sep=2.25,%
  EspaceFeuille=0.66,child sep=0.66,%
  Details=true,details=true,%
  CouleurDetails=red,details color=red,%
  Echelle=1,scale=1,%
  Fleches=true,arrows=true
}

\NewDocumentCommand\ArbreDiviseurs{ O{} D<>{} m }{%
  \useKVdefault[arbrediviseurs]%
  \setKV[arbrediviseurs]{#1}%
  %en/fr
  \setKVbooltruedefaultmulti[arbrediviseurs]{details}{Details}%
  \setKVbooltruedefaultmulti[arbrediviseurs]{arrows}{Fleches}%
  %next
  % test avec CPoulain ^^
  \xdef\tmpcalc{\xinteval{#3}}%
  \xdef\tmparg{}%
  \newcount\anp\newcount\bnp\newcount\cnp%
  \newcount\pileb\newcount\exposant%
  \exposant=0\relax%
  \anp=\tmpcalc\relax%
  \bnp=2\relax%
  \pileb=2\relax%
  \whiledo{\the\anp > 1}{%
    \modulo{\the\anp}{\the\bnp}%
    \ifnum\remainder=0\relax%
    \cnp=\numexpr\anp/\bnp\relax%
    \exposant=\numexpr\exposant+1\relax%
    \anp=\cnp\relax%
    \else%
    \ifnum\exposant>0\relax%
    \xdef\tmparg{\tmparg\the\pileb,\the\exposant*}%
    %\expandafter\UpdatetoksCPier\Foo\nil%
    \fi%
    \bnp=\numexpr\bnp+1\relax%
    \pileb=\bnp\relax%
    \exposant=0\relax%
    \fi%
  }%
  \xdef\tmparg{\tmparg\the\bnp,{\the\exposant}}%
  %\expandafter\UpdatetoksCPier\the\bnp,{\the\exposant}\nil%
  % \xdef\argnum{\xinteval{#2}}%
  % \def\decompotmp{\DecompoPremierExposant{#2}}%
  \setsepchar{*/,}%
  \xdef\Foo{\tmparg}%
  \readlist*\readcaractdiv{\Foo}%
  % fin test CPoulain ^^
  \xdef\arbredivnbdivprem{\readcaractdivlen}%nombre diviseurs permiers
  \xdef\arbredivnbdiv{1}%
  \xintFor* ##1 in {\xintSeq{1}{\arbredivnbdivprem}}\do{\xdef\arbredivnbdiv{\xinteval{\arbredivnbdiv*(\readcaractdiv[##1,2]+1)}}}%nombre diviseurs
  \def\HauteurTotale{\xinteval{\arbredivnbdiv-1}}%
  \ifboolKV[arbrediviseurs]{Fleches}%
  {\tikzstyle{flechearbrediv} = [line width=\fpeval{\TmpEchelle*0.6}pt,->,>=latex]}%
  {\tikzstyle{flechearbrediv} = [line width=\fpeval{\TmpEchelle*0.6}pt]}%
  \begin{tikzpicture}[scale=\TmpEchelle,every node/.style={scale=\TmpEchelle},#2]
    % grille d'aide
    % \draw[xstep=\TmpEspNiv,ystep=\TMpEspFeuille,thin,lightgray] (0,{(-\arbredivnbdiv+1)*\TMpEspFeuille}) grid ({\arbredivnbdivprem*\TmpEspNiv},0) ;
    % dernier niveau OK
    \xdef\nbsommets{\arbredivnbdiv}%
    \xintFor* ##1 in {\xintSeq{1}{\nbsommets}}\do%
    {%
      \itemtomacro\readcaractdiv[\arbredivnbdivprem,2]\tmppuiss%
      \xdef\tmpdiv{\xinteval{\tmppuiss+1}}%
      \node[outer sep=0pt] (N-\arbredivnbdivprem-##1) at ({\arbredivnbdivprem*\TmpEspNiv},{-(##1-1)*\TMpEspFeuille}) {$\num{\readcaractdiv[\arbredivnbdivprem,1]}^{\xinteval{\xintiiRem{\xinteval{##1-1}}{\tmpdiv}}}$} ;%
    }
    % niveaux suivants, en descendant
    \xintFor* ##2 in {\xintSeq{\arbredivnbdivprem-1}{1}}\do%
    {%
      \xdef\nbsommets{\xintieval{\nbsommets/(\readcaractdiv[##2+1,2]+1)}}%
      \xdef\espacementsommets{\xinteval{\arbredivnbdiv/(\nbsommets)}}%
      \xdef\OffsetNiveau{\xintieval{(\HauteurTotale-(\nbsommets-1)*\espacementsommets)}}%
      \xintFor* ##3 in {\xintSeq{1}{\nbsommets}}\do%
      {%
        \itemtomacro\readcaractdiv[##2,2]\tmppuiss%
        \xdef\tmpdiv{\xinteval{\tmppuiss+1}}%
        \node[outer sep=0pt] (N-##2-##3) at ({##2*\TmpEspNiv},{(-(##3-1)*\espacementsommets-0.5*\OffsetNiveau)*\TMpEspFeuille}) {$\num{\readcaractdiv[##2,1]}^{\xinteval{\xintiiRem{\xinteval{##3-1}}{\tmpdiv}}}$} ;
      }%
    }%
    % racine
    \coordinate (Racine) at ({0},{-0.5*(\arbredivnbdiv-1)*(\TMpEspFeuille)}) ;
    % \draw (Racine) node {$\Omega$} ;
    % les flèches, qui partent de la racine
    \xintFor* ##4 in {\xintSeq{1}{\xinteval{\readcaractdiv[1,2]+1}}}\do%
    {%
      \draw[flechearbrediv] (Racine) -- (N-1-##4) ;
    }%
    % les flèches successives
    \foreach \nivdep in {1,...,\xinteval{\arbredivnbdivprem-1}}%
    {%
      \xdef\nbsommetsniv{1}%
      \foreach \i in {1,...,\nivdep}{\xdef\nbsommetsniv{\xinteval{\nbsommetsniv*(\readcaractdiv[\i,2]+1)}}}%calcul du nombre de sommets de départ
      % boucle sur sommets de départ
      \foreach \numsom in {1,...,\nbsommetsniv}%
      {%
        \xdef\nivplusun{\xinteval{\nivdep+1}}%
        \foreach \nbsousbranches in {1,...,\xinteval{\readcaractdiv[\nivplusun,2]+1}}%
        {%
          \xdef\sumsomarriv{\xinteval{ (\readcaractdiv[\nivplusun,2]+1)*(\numsom-1) + \nbsousbranches }}%
          \draw[flechearbrediv] (N-\nivdep-\numsom) -- (N-\nivplusun-\sumsomarriv) ;
        }
      }%
    }%
    \ifboolKV[arbrediviseurs]{Details}%
    {%
      % essai de génération des calculs := ouaiissssss
      \foreach \i in {1,...,\arbredivnbdiv}{%
        \xdef\resdiv{1}%
        % on extrait la premiere puissance := on obtient la répartition ?
        \itemtomacro\readcaractdiv[-1,2]\tmppuiss\xdef\tmpdiv{\xinteval{(\tmppuiss+1)}}%
        % création de la liste des puissances !!
        \xdef\calculdiviseur{\num{\readcaractdiv[-1,1]}^{\xinteval{\xintiiRem{\xinteval{\i-1}}{\tmpdiv}}}}%
        \xdef\resdiv{\xinteval{\resdiv*(\readcaractdiv[-1,1])^(\xinteval{\xintiiRem{\xinteval{\i-1}}{\tmpdiv}})}}%
        % on complète avec les autres parties ??
        \foreach \nbprem in {2,...,\arbredivnbdivprem}{%
          % il reste à stocker les produits des puissances, en descendant... erf.....
          \xdef\tmppuisscumul{1}%
          \foreach \k in {1,...,\xinteval{\nbprem-1}}{%
            \itemtomacro\readcaractdiv[-\k,2]\tmppuissrepet%
            \xdef\tmppuisscumul{\xinteval{\tmppuisscumul*(\tmppuissrepet+1)}}%
          }%
          \itemtomacro\readcaractdiv[-\nbprem,2]\tmppuiss%
          \xdef\tmpdivniv{\xinteval{(\tmppuiss+1)}}%
          % \xdef\tmpdiv{\xinteval{(\tmppuissrepet+1)}}%
          \xdef\calculdiviseur{\num{\readcaractdiv[-\nbprem,1]}^{\xinteval{\xintiiRem{\xintieval{trunc((\i-1)/(\tmppuisscumul),0)}}{\tmpdivniv}}}\times\calculdiviseur}%
          \xdef\resdiv{\xinteval{\resdiv*(\readcaractdiv[-\nbprem,1])^(\xinteval{\xintiiRem{\xintieval{trunc((\i-1)/(\tmppuisscumul),0)}}{\tmpdivniv}})}}%
        }%
        \draw ([xshift=1cm]N-\arbredivnbdivprem-\i) node[right,\TmpCoulDetails] {$\calculdiviseur=\num{\resdiv}$} ;
      }%
    }{}%
  \end{tikzpicture}%
}
\NewCommandCopy\pflarbrediv\ArbreDiviseurs
\NewCommandCopy\pfltreediv\ArbreDiviseurs

%somme des chiffres
\ifthenelse{\isundefined{\SommeChiffres}}%
  {%
    \NewDocumentCommand\SommeChiffres{ m }{%
      \pfllenstr{#1}[\nbchiffres]%
      \xdef\tmpres{0}%
      \foreach \i in {1,...,\nbchiffres}{\StrChar{#1}{\i}[\tmpchf]\xdef\tmpres{\xinteval{\tmpres+\tmpchf}}}%
      \ensuremath{\num{\tmpres}}%
    }
  }%
  {%
    \RenewDocumentCommand\SommeChiffres{ m }{%
      \pfllenstr{#1}[\nbchiffres]%
      \xdef\tmpres{0}%
      \foreach \i in {1,...,\nbchiffres}{\StrChar{#1}{\i}[\tmpchf]\xdef\tmpres{\xinteval{\tmpres+\tmpchf}}}%
      \ensuremath{\num{\tmpres}}%
    }
  }%
\NewCommandCopy\pflsomchiff\SommeChiffres
\NewCommandCopy\pflsumdigits\SommeChiffres

%====CHIFFREMENTS !!
\xdef\aLPHaBeTMajusc{ABCDEFGHIJKLMNOPQRSTUVWXYZ}
\xdef\aLPHaBeTminusc{abcdefghijklmnopqrstuvwxyz}

\NewDocumentCommand\InverseModulo{ s m m }{%
  \xdef\PGCDD{\xinteval{gcd(#2,#3)}}%
  \xdef\resinvmod{#2{} n'est pas inversible modulo #3.}%
  \xintFor* ##1 in {\xintSeq{1}{#3}}\do{\xintifboolexpr{\xintiiRem{\xinteval{#2*##1}}{#3} == 1}{\xdef\resinvmod{##1}}{}}%
  \IfBooleanTF{#1}%
    {%
      On a $\text{PGCD}(#2;#3)=\num{\PGCDD}$.~%
      \xintifboolexpr{\PGCDD != 1}%
        {Le PGCD étant différent de 1, on en déduit que #2 n'est pas inversible modulo #3.\relax}%
        {Le PGCD étant égal à 1, on en déduit que #2 admet un inverse modulo #3.\\De plus on a $#2 \times \resinvmod = \xinteval{#2*\resinvmod} \equiv \xintiirem{\xinteval{#2*\resinvmod}}{#3}\:[#3]$, donc \resinvmod\ est l'inverse de #2 modulo #3.\relax}%
    }%
    {}%
}
\NewCommandCopy\pflinvmod\InverseModulo

\defKV[chiffaffine]{a=\xdef\tmpcoeffa{#1},b=\xdef\tmpcoeffb{#1},modulo=\xdef\tmpmodulo{#1}}
\setKVdefault[chiffaffine]{a=3,b=12,Dechiffr=false,modulo=26}

\NewDocumentCommand\ChiffrementAffine{ O{} m }{%
  \restoreKV[chiffaffine]%
  \setKV[chiffaffine]{#1}%
  %pgcd stocké
  \xdef\PGCDD{\xinteval{gcd(\tmpcoeffa,\tmpmodulo)}}%
  \pfllenstr{#2}[\tmpnbcaract]%
  \ifboolKV[chiffaffine]{Dechiffr}%
  {%
    \xdef\resinvmod{0}%
    \xintFor* ##1 in {\xintSeq{1}{\tmpmodulo}}\do{%
      \xintifboolexpr{\xintiiRem{\xinteval{(\tmpcoeffa)*(##1)}}{\tmpmodulo} == 1}%
      {\xdef\resinvmod{##1}}{}%
    }%
  }{}%
  \foreach \i in {1,...,\tmpnbcaract}{%
    \StrChar{#2}{\i}[\tmpchar]%
    \IfStrEq{\tmpchar}{ }%
      {~}%
      {%
        %majuscule minuscule
        \IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
          {%
            \StrPosition{\aLPHaBeTMajusc}{\tmpchar}[\tmpcoeffx]%
          }%
          {%
            \StrPosition{\aLPHaBeTminusc}{\tmpchar}[\tmpcoeffx]%
          }%
        \xdef\tmpcoeffx{\xinteval{\tmpcoeffx-1}}%
        \ifboolKV[chiffaffine]{Dechiffr}%
          {%
            \xintifboolexpr{\PGCDD == 1}%
              {%
                \xdef\tmpres{\xintiiRem{\xinteval{(\resinvmod)*(\tmpcoeffx)-(\resinvmod)*(\tmpcoeffb)}}{\tmpmodulo}}%
                \xdef\tmpres{\xinteval{\tmpres+1}}%
                \IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
                  {%
                    \StrChar{\aLPHaBeTMajusc}{\tmpres}%
                  }%
                  {%
                    \StrChar{\aLPHaBeTminusc}{\tmpres}%
                  }%
              }%
              {}%
          }%
          {%
            \xdef\tmpres{\xintiiRem{\xinteval{\tmpcoeffa*\tmpcoeffx+\tmpcoeffb}}{\tmpmodulo}}%
            \xdef\tmpres{\xinteval{\tmpres+1}}%
            \IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
              {%
                \StrChar{\aLPHaBeTMajusc}{\tmpres}%
              }%
              {%
                \StrChar{\aLPHaBeTminusc}{\tmpres}%
              }%
          }%
      }%
  }%
  \ifboolKV[chiffaffine]{Dechiffr}%
  {\xintifboolexpr{\PGCDD != 1}{Le message ne peut pas être déchiffré car $\text{PGCD}(\tmpcoeffa;\tmpmodulo)\neq1$ !\relax}{}}%
  {}%
}
\NewCommandCopy\pflchiffraff\ChiffrementAffine

\defKV[chiffhill]{Matrice=\xdef\tmpcoeffmat{#1},Modulo=\xdef\tmpmodulo{#1}}
\setKVdefault[chiffhill]{Matrice={1,2,3,5},Dechiffr=false,modulo=26}

\NewDocumentCommand\ChiffrementHill{ O{} m }{%
  \restoreKV[chiffhill]%
  \setKV[chiffhill]{#1}%
  %passage de la châine en nb pair
  \pfllenstr{#2}[\tmpnbcar]%
  \xintifboolexpr{\xintiirem{\tmpnbcar}{2} == 1}{\xdef\tmpchaine{#2A}}{\xdef\tmpchaine{#2}}%
  \pfllenstr{\tmpchaine}[\tmpnbblocs]%
  \xdef\tmpnbblocs{\xintieval{\tmpnbblocs/2}}%
  %extraction des coeffs de la matrice + déterminant
  \setsepchar{,}%
  \readlist*\coeffmathill{\tmpcoeffmat}%
  \itemtomacro\coeffmathill[1]{\tmpcoeffa}%
  \itemtomacro\coeffmathill[2]{\tmpcoeffb}%
  \itemtomacro\coeffmathill[3]{\tmpcoeffc}%
  \itemtomacro\coeffmathill[4]{\tmpcoeffd}%
  \xdef\detmathill{\xintieval{(\tmpcoeffa)*(\tmpcoeffd)-(\tmpcoeffb)*(\tmpcoeffc)}}%
  %chiffrement / déchiffrement
  \ifboolKV[chiffhill]{Dechiffr}%
  {%
    %le cas où la matrice n'est pas inversible
    \xintifboolexpr{\detmathill == 0}%
    {%
      La matrice $\begin{pmatrix} \tmpcoeffa & \tmpcoeffb \\ \tmpcoeffc & \tmpcoeffd \end{pmatrix}$ n'est pas inversible, donc pas de déchiffrement possible !\relax
    }%
    {%
      %inversibilité du déterminant
      \xdef\resinvmod{0}%
      \xintFor* ##1 in {\xintSeq{1}{\tmpmodulo}}\do{%
        \xintifboolexpr{\xintiiRem{\xinteval{(\detmathill)*(##1)}}{\tmpmodulo} == 1}%
        {\xdef\resinvmod{##1}}{}%
      }%
      \xintifboolexpr{\resinvmod == 0}%si det non inversible modulo
      {%
        Le déterminant de la matrice $\begin{pmatrix} \tmpcoeffa & \tmpcoeffb \\ \tmpcoeffc & \tmpcoeffd \end{pmatrix}$ (qui vaut $\detmathill$) n'est pas inversible modulo \tmpmodulo, donc pas de déchiffrement !\relax
      }%
      {%
        %on peut déchiffrer !!
        \foreach \i in {1,...,\tmpnbblocs}{%
          \xdef\tmpindicea{\xintieval{2*(\i-1)+1}}\xdef\tmpindiceb{\xintieval{2*(\i)}}%
          \StrChar{\tmpchaine}{\tmpindicea}[\tmpchara]%
          \IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
          {%
            \StrPosition{\aLPHaBeTMajusc}{\tmpchara}[\tmpcoeffx]%
          }%
          {%
            \StrPosition{\aLPHaBeTminusc}{\tmpchara}[\tmpcoeffx]%
          }%
          \xdef\tmpcoeffx{\xintieval{\tmpcoeffx-1}}%
          \StrChar{\tmpchaine}{\tmpindiceb}[\tmpcharb]%
          \IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
          {%
            \StrPosition{\aLPHaBeTMajusc}{\tmpcharb}[\tmpcoeffy]%
          }%
          {%
            \StrPosition{\aLPHaBeTminusc}{\tmpcharb}[\tmpcoeffy]%
          }%
          \xdef\tmpcoeffy{\xintieval{\tmpcoeffy-1}}%
          %1ère lettre
          \xdef\tmpresa{\xintiiRem{\xintieval{(\resinvmod)*(\tmpcoeffd)*(\tmpcoeffx)-(\resinvmod)*(\tmpcoeffb)*(\tmpcoeffy)}}{\tmpmodulo}}%
          \xdef\tmpresa{\xinteval{\tmpresa+1}}%
          %2ème lettre
          \xdef\tmpresb{\xintiiRem{\xintieval{-(\resinvmod)*(\tmpcoeffc)*(\tmpcoeffx)+(\resinvmod)*(\tmpcoeffa)*(\tmpcoeffy)}}{\tmpmodulo}}%
          \xdef\tmpresb{\xinteval{\tmpresb+1}}%
          %affichage des deux caractères
          \IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
          {%
            \StrChar{\aLPHaBeTMajusc}{\tmpresa}%
          }%
          {%
            \StrChar{\aLPHaBeTminusc}{\tmpresa}%
          }%
          \IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
          {%
            \StrChar{\aLPHaBeTMajusc}{\tmpresb}%
          }%
          {%
            \StrChar{\aLPHaBeTminusc}{\tmpresb}%
          }%
        }%
      }%
    }%
  }%
  {%
    \foreach \i in {1,...,\tmpnbblocs}{%
      \xdef\tmpindicea{\xintieval{2*(\i-1)+1}}\xdef\tmpindiceb{\xintieval{2*(\i)}}%
      \StrChar{\tmpchaine}{\tmpindicea}[\tmpchara]%
      \IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
      {%
        \StrPosition{\aLPHaBeTMajusc}{\tmpchara}[\tmpcoeffx]%
      }%
      {%
        \StrPosition{\aLPHaBeTminusc}{\tmpchara}[\tmpcoeffx]%
      }%
      \xdef\tmpcoeffx{\xintieval{\tmpcoeffx-1}}%
      \StrChar{\tmpchaine}{\tmpindiceb}[\tmpcharb]%
      \IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
      {%
        \StrPosition{\aLPHaBeTMajusc}{\tmpcharb}[\tmpcoeffy]%
      }%
      {%
        \StrPosition{\aLPHaBeTminusc}{\tmpcharb}[\tmpcoeffy]%
      }%
      \xdef\tmpcoeffy{\xintieval{\tmpcoeffy-1}}%
      %1ère lettre
      \xdef\tmpresa{\xintiiRem{\xintieval{(\tmpcoeffa)*(\tmpcoeffx)+(\tmpcoeffb)*(\tmpcoeffy)}}{\tmpmodulo}}%
      \xdef\tmpresa{\xinteval{\tmpresa+1}}%
      %2ème lettre
      \xdef\tmpresb{\xintiiRem{\xintieval{(\tmpcoeffc)*(\tmpcoeffx)+(\tmpcoeffd)*(\tmpcoeffy)}}{\tmpmodulo}}%
      \xdef\tmpresb{\xinteval{\tmpresb+1}}%
      %affichage des deux caractères
      \IfSubStr{\aLPHaBeTMajusc}{\tmpchara}%
      {%
        \StrChar{\aLPHaBeTMajusc}{\tmpresa}%
      }%
      {%
        \StrChar{\aLPHaBeTminusc}{\tmpresa}%
      }%
      \IfSubStr{\aLPHaBeTMajusc}{\tmpcharb}%
      {%
        \StrChar{\aLPHaBeTMajusc}{\tmpresb}%
      }%
      {%
        \StrChar{\aLPHaBeTminusc}{\tmpresb}%
      }%
    }%
  }%
}
\NewCommandCopy\pflchiffrhill\ChiffrementHill

\defKV[chiffcesar]{Decal=\xdef\tmpdecalcesar{#1}}
\setKVdefault[chiffcesar]{Decal=5,Dechiffr=false}

\NewDocumentCommand\ChiffrementCesar{ O{} m }{%
  \restoreKV[chiffcesar]%
  \setKV[chiffcesar]{#1}%
  \pfllenstr{#2}[\tmpnbcaract]%
  \foreach \i in {1,...,\tmpnbcaract}{%
    \StrChar{#2}{\i}[\tmpchar]%
    %majuscule minuscule
    \IfStrEq{\tmpchar}{ }%
      {~}%
      {%
        \IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
          {%
            \StrPosition{\aLPHaBeTMajusc}{\tmpchar}[\tmpcoeffx]%
          }%
          {%
            \StrPosition{\aLPHaBeTminusc}{\tmpchar}[\tmpcoeffx]%
          }%
        \ifboolKV[chiffcesar]{Dechiffr}%
          {\xdef\tmpcoeffx{\xintiirem{\xinteval{\tmpcoeffx-\tmpdecalcesar}}{26}}}%
          {\xdef\tmpcoeffx{\xintiirem{\xinteval{\tmpcoeffx+\tmpdecalcesar}}{26}}}%
        \IfSubStr{\aLPHaBeTMajusc}{\tmpchar}%
          {%
            \StrChar{\aLPHaBeTMajusc}{\tmpcoeffx}%
          }%
          {%
            \StrChar{\aLPHaBeTminusc}{\tmpcoeffx}%
          }%
      }%
  }%
}
\NewCommandCopy\pflchiffrcesar\ChiffrementCesar

%====DIV EUCL (OK), 2 versions, mises à jour ;-)
\setKVdefault[diveucl]{%
  Quotient=true,%
  Reste=true,%
  Vide=false,%
  Pointilles=\ldots
}
\NewDocumentCommand\DivEucl{ s O{} m m }{%
  \restoreKV[diveucl]%
  \setKV[diveucl]{#2}%
  \ifboolKV[diveucl]{Vide}%
    {%
      \setKV[diveucl]{Quotient=false,Reste=false}%
    }%
    {}%
  \xdef\tmpAA{\xinteval{#3}}\xdef\tmpBB{\xinteval{#4}}%
  \xdef\tmpQuotient{\xintiiQuo{\tmpAA}{\tmpBB}}\xdef\tmpReste{\xintiiRem{\tmpAA}{\tmpBB}}%
  \ensuremath{\num{\xinteval{#3}}=\num{\xinteval{#4}}\times\ifboolKV[diveucl]{Quotient}{\xintifboolexpr{\tmpQuotient < 0}{(\num{\tmpQuotient})}{\num{\tmpQuotient}}}{\useKV[diveucl]{Pointilles}}+\ifboolKV[diveucl]{Reste}{\num{\tmpReste}}{\useKV[diveucl]{Pointilles}}}%
  \IfBooleanT{#1}{~\labelavec\ $0 \pflleq \num{\xintiiRem{\tmpAA}{\tmpBB}} < \xintifboolexpr{\tmpBB < 0}{\lvert\num{\xinteval{#4}}\rvert}{\num{\xinteval{#4}}}$}%
}
\NewCommandCopy\pfldiveucl\DivEucl

\NewDocumentCommand\DivisionEucl{ m m }{%
  \xdef\tmpAA{\xinteval{#1}}\xdef\tmpBB{\xinteval{#2}}%
  \xdef\tmpQuotient{\xintiiQuo{\tmpAA}{\tmpBB}}\xdef\tmpReste{\xintiiRem{\tmpAA}{\tmpBB}}%
  \ensuremath{\num{\tmpBB}\times\xintifboolexpr{\tmpQuotient < 0}{(\num{\tmpQuotient})}{\num{\tmpQuotient}}+\num{\xintiiRem{\tmpAA}{\tmpBB}}}
}
\NewCommandCopy\pflpresdiveucl\DivisionEucl

%====ADDITIONS POSÉES ?
\newlength{\colspecbinadd}
\setlength{\colspecbinadd}{2pt}

\NewDocumentCommand\PosRetenue{ O{red} m }{%
  %\tmptextcircled[#1]{#2}%
  \textcolor{#1}{\scalebox{\addbinposescret}[\addbinposescret]{#2}}%
}

\defKV[AddBinaire]{%
  CouleurRetenue=\def\addbinposecol{#1},%
  Espacement=\setlength{\colspecbinadd}{#1},%
  Police=\def\addbinposefont{#1},%
  EchelleRetenue=\def\addbinposescret{#1},%
  Base=\def\addbinposebase{#1}
}

\setKVdefault[AddBinaire]{%
  CouleurRetenue=red,%
  Espacement=2pt,%
  Police=\normalsize\normalfont,%
  EchelleRetenue=0.4,%
  Base=dec,%
  Egal=true
}

\NewDocumentCommand\PoseAddition{ O{} m m }{%
  \restoreKV[AddBinaire]%
  \setKV[AddBinaire]{#1}%
  %le résultat
  \IfStrEq{\addbinposebase}{dec}%
    {\xdef\tmpresultat{\xinteval{#2+#3}}}%
    {}%
  \IfStrEq{\addbinposebase}{bin}%
    {\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{#2}+\xintBinToDec{#3}}}}}%
    {}%
  \IfStrEq{\addbinposebase}{hex}%
    {\xdef\tmpresultat{\xintDecToHex{\xinteval{\xintHexToDec{#2}+\xintHexToDec{#3}}}}}%
    {}%
  %\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{#2}+\xintBinToDec{#3}}}}%
  %les calculs
  \pfllenstr{#2}[\longueurA]\pfllenstr{#3}[\longueurB]\pfllenstr{\tmpresultat}[\longueurC]%
  \xdef\longueurMax{\xinteval{max(\longueurA,\longueurB,\longueurC)}}%\longueurMax
  %le tableau
  \xintifboolexpr{\longueurB > \longueurA}%
    {%
      \xdef\ArgLigneHaut{#3}\xdef\ArgLigneBas{#2}%
      \xdef\addposeehaut{#3}\xdef\addposeebas{#2}%
    }%
    {%
      \xdef\ArgLigneHaut{#2}\xdef\ArgLigneBas{#3}%
      \xdef\addposeehaut{#2}\xdef\addposeebas{#3}%
    }%
  %création de la ligne du haut
  \pfllenstr{\ArgLigneHaut}[\LgHaut]%
  \StrChar{\ArgLigneHaut}{\LgHaut}[\LigneBinAdd]%
  \foreach \i in {1,...,\inteval{\LgHaut-1}}%
    {%
      \xdef\tmpcoeff{\xinteval{\LgHaut-\i}}%
      \StrChar{\ArgLigneHaut}{\tmpcoeff}[\TmpHaut]%
      \xdef\LigneBinAdd{\TmpHaut & \LigneBinAdd}%
    }%
  \xintifboolexpr{\longueurMax > \LgHaut}%
    {%
      \foreach \i in {1,...,\inteval{\longueurMax-\LgHaut}}%
        {\xdef\LigneBinAdd{& \LigneBinAdd}}%
    }%
    {}%
  %création de la ligne du bas
  \pfllenstr{\ArgLigneBas}[\LgBas]%
  \StrChar{\ArgLigneBas}{\LgBas}[\LigneBinAddB]%
  \foreach \i in {1,...,\inteval{\LgBas-1}}%
    {%
      \xdef\tmpcoeff{\xinteval{\LgBas-\i}}%
      \StrChar{\ArgLigneBas}{\tmpcoeff}[\TmpBas]%
      \xdef\LigneBinAddB{\TmpBas & \LigneBinAddB}%
    }%
  \xintifboolexpr{\longueurMax > \LgBas}%
    {%
      \foreach \i in {1,...,\inteval{\longueurMax-\LgBas}}%
      {\xdef\LigneBinAddB{& \LigneBinAddB}}%
    }%
    {}%
  %création de la ligne du résultat
  \pfllenstr{\tmpresultat}[\LgRes]%
  \StrChar{\tmpresultat}{\LgRes}[\LigneBinRes]%
  \foreach \i in {1,...,\inteval{\LgRes-1}}%
    {%
      \xdef\tmpcoeff{\xinteval{\LgRes-\i}}%
      \StrChar{\tmpresultat}{\tmpcoeff}[\TmpRes]%
      \xdef\LigneBinRes{\TmpRes & \LigneBinRes}%
    }%
  %création des retenues...
  \xdef\ArgBasZeros{\ArgLigneBas}%
  \xintifboolexpr{\LgHaut > \LgBas}%
    {%
      \foreach \i in {1,...,\inteval{\LgHaut-\LgBas}}%
      {\xdef\ArgBasZeros{0\ArgBasZeros}}%
    }%
    {}%
  %\ArgBasZeros\par
  \xdef\listeretenues{0}%
  \xdef\longueurmax{\xinteval{max(\longueurA,\longueurB)}}%\longueurMax
  \foreach \i in {1,...,\longueurmax}%
    {%
      \setsepchar{,}%
      \readlist*\lstretenues{\listeretenues}%
      \xdef\tmpcoeff{\xinteval{\longueurmax-\i+1}}%
      \itemtomacro\lstretenues[-\i]{\TmPNbR}%
      \StrChar{\ArgLigneHaut}{\tmpcoeff}[\TmPNbH]%
      \StrChar{\ArgBasZeros}{\tmpcoeff}[\TmPNbB]%
      %cas de la base
      \IfStrEq{\addbinposebase}{dec}%
        {\xdef\tmpresultat{\xinteval{\TmPNbR+\TmPNbH+\TmPNbB}}}%
        {}%
      \IfStrEq{\addbinposebase}{bin}%
        {\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{\TmPNbR}+\xintBinToDec{\TmPNbH}+\xintBinToDec{\TmPNbB}}}}}%
        {}%
      \IfStrEq{\addbinposebase}{hex}%
        {\xdef\tmpresultat{\xintDecToHex{\xinteval{\xintHexToDec{\TmPNbR}+\xintHexToDec{\TmPNbH}+\xintHexToDec{\TmPNbB}}}}}%
        {}%
      %suite
      %\xdef\tmpresultat{\xintDecToBin{\xinteval{\xintBinToDec{\TmPNbR}+\xintBinToDec{\TmPNbH}+\xintBinToDec{\TmPNbB}}}}%
      \pfllenstr{\tmpresultat}[\LgRetenue]%
      \xintifboolexpr{\LgRetenue == 1}%si on met une retenue...
        {%
          \xdef\listeretenues{0,\listeretenues}%
        }%
        {%
          \StrGobbleRight{\tmpresultat}{1}[\tmpretenue]%
          \xdef\listeretenues{\tmpretenue,\listeretenues}%
        }%
    }%
  %On compte le nombre de zéros dans la ligne des retenues
  \StrCount{\listeretenues}{0}[\nbretenues]%\listeretenues\par\nbretenues\par%
  \setsepchar{,}%
  \readlist*\lstretenues{\listeretenues}%
  %\showitems*{\lstretenues}\par
  \xdef\LigneBinAddRet{\xintifboolexpr{\lstretenues[-1] == 0}{}{\PosRetenue[\addbinposecol]{\lstretenues[-1]}}}%
  \foreach \i in {2,...,\longueurMax}%
    {%
      \xdef\LigneBinAddRet{\xintifboolexpr{\lstretenues[-\i] == 0}{}{\PosRetenue[\addbinposecol]{\lstretenues[-\i]}} & \LigneBinAddRet}%
    }%
  %et on passe au tableau
  \xdef\longueurcasesope{\xinteval{\longueurMax-1}}%
  \xdef\longueurtotale{\xinteval{\longueurMax+1}}%
  \setlength{\tabcolsep}{\colspecbinadd}%
  %\begin{tabular}{@{\hskip0.5\colspecbinadd}c@{\hskip0.5\colspecbinadd}*{\longueurcasesope}{c@{\hskip\colspecbinadd}}c@{\hskip0.5\colspecbinadd}}
  \begin{NiceTabular}{c*{\longueurMax}{c}}
    \xintifboolexpr{\nbretenues > \longueurmax}{}{\RowStyle[cell-space-limits=1pt]{\addbinposefont} & \LigneBinAddRet \\}
    \RowStyle[nb-rows=*,cell-space-limits=2pt]{\addbinposefont}
    &\LigneBinAdd \\
    +&\LigneBinAddB \\
    \hline 
    \ifboolKV[AddBinaire]{Egal}{=}{}&\LigneBinRes \\
  \end{NiceTabular}
  %\end{tabular}%
}
\NewCommandCopy\pflposadd\PoseAddition

\ifluatex

\RequirePackage{lualinalg}

\defKV[addbinlua]{%
  CouleurRetenue=\def\addbinluacol{#1},%
  Espacement=\def\addbinluaspace{#1},%
  Police=\def\addbinluafonte{#1},%
  Base=\def\addbinluabase{#1},%
  EchelleRetenue=\def\addbinluascret{#1}
}

\setKVdefault[addbinlua]{%
  CouleurRetenue=red,%
  Espacement=2pt,%
  Police=\normalsize\normalfont,%
  Egal=true,%
  Base=dec,%
  EchelleRetenue=0.4
}

\NewDocumentCommand\SommeElementsMatriceBin{ m m m m O{\poseretenue} }{%
  %1=matrice
  %2=colonne
  %3=ligne début
  %4=ligne arrivée
  \xdef\tmpsomme{0}%
  \foreach \lig in {#3,...,#4}{%
    \xdef\tmpsomme{\xinteval{\tmpsomme+\xintBinToDec{\matrixGetElement{#1}{\lig}{#2}}}}%calcul intermédiaire en binaire
  }%
  \xdef#5{\xintDecToBin{\tmpsomme}}%
}

\NewDocumentCommand\SommeElementsMatriceDec{ m m m m O{\poseretenue} }{%
  %1=matrice
  %2=colonne
  %3=ligne début
  %4=ligne arrivée
  \xdef\tmpsomme{0}%
  \foreach \lig in {#3,...,#4}{%
    \xdef\tmpsomme{\xinteval{\tmpsomme+\matrixGetElement{#1}{\lig}{#2}}}%calcul intermédiaire en binaire
  }%
  \xdef#5{\tmpsomme}%
}

\NewDocumentCommand\PoseAdditionLua{ O{} m }{%
  \restoreKV[addbinlua]%
  \setKV[addbinlua]{#1}%
  \setsepchar{+}%
  \readlist*\addbinlstoper{#2}%
  \xdef\nboperandes{\addbinlstoperlen}%
  \xdef\nboperandesmat{\xinteval{\addbinlstoperlen+1}}%
  %le résultat du calcul en binaire
  \xdef\tmpres{0}%
  \foreach \i in {1,...,\addbinlstoperlen}{%
    \itemtomacro\addbinlstoper[\i]{\tmpinteroper}%
    \IfStrEq{\addbinluabase}{dec}%
      {\xdef\tmpres{\xinteval{\tmpres+\tmpinteroper}}}%
      {}%
    \IfStrEq{\addbinluabase}{bin}%
      {\xdef\tmpres{\xinteval{\tmpres+\xintBinToDec{\tmpinteroper}}}}%
      {}%
    %\xdef\tmpres{\xinteval{\tmpres+\xintBinToDec{\tmpinteroper}}}%
  }%
  \IfStrEq{\addbinluabase}{dec}%
    {\xdef\tmpresultat{\tmpres}}%
    {}%
  \IfStrEq{\addbinluabase}{bin}%
    {\xdef\tmpresultat{\xintDecToBin{\tmpres}}}%
    {}%
  %\xdef\tmpresultat{\xintDecToBin{\tmpres}}%\tmpresultat%
  \pfllenstr{\tmpresultat}[\nbtotalchiffresres]%\nbtotalchiffresres%
  \xdef\nblignesmatriceintertmp{\xinteval{1+\nboperandes}}%
  \xdef\nblignesmatricetmp{\xinteval{2+\nboperandes}}%
  \matrixNew{MATINTER}{\nblignesmatricetmp,\nbtotalchiffresres,'zero'}%
  %création des lignes intermédiaires
  \foreach \nblig [count=\xi] in {2,...,\nblignesmatriceintertmp}{%
    \itemtomacro\addbinlstoper[\xi]{\tmpinteroper}%on extrait l'opérande
    \pfllenstr{\tmpinteroper}[\nbchiiffres]%on extrait son nb de chiffres
    \xdef\tmpoffset{\xinteval{\nbtotalchiffresres-\nbchiiffres+1}}%on calcule la position du 1er chiffre
    \foreach \poschar [count=\xii] in {\tmpoffset,...,\nbtotalchiffresres}{%
      \StrChar{\tmpinteroper}{\xii}[\tmpcoeffaddbin]%
      \matrixSetElement{MATINTER}{\nblig}{\poschar}{\tmpcoeffaddbin}%
    }%
  }%
  %création de la ligne résultat
  \foreach \j in {1,...,\nbtotalchiffresres}{%
    \StrChar{\tmpresultat}{\j}[\tmpcoeffaddbin]%
    \matrixSetElement{MATINTER}{\nblignesmatricetmp}{\j}{\tmpcoeffaddbin}%
  }%
  %gestion des retenues, de la droite vers la gauche
  \foreach \nbcol in {\nbtotalchiffresres,...,2}{%
    \IfStrEq{\addbinluabase}{dec}%
      {\SommeElementsMatriceDec{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]}%
      {}%
    \IfStrEq{\addbinluabase}{bin}%
      {\SommeElementsMatriceBin{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]}%
      {}%
    %\SommeElementsMatriceBin{MATINTER}{\nbcol}{1}{\nboperandesmat}[\poseretenue]%
    \xdef\nbcolmun{\xinteval{\nbcol-1}}%
    \pfllenstr{\poseretenue}[\nbchiffresretenue]%
    \xintifboolexpr{\nbchiffresretenue > 1}%
      {\StrGobbleRight{\poseretenue}{1}[\RETENUE]}%
      {\xdef\RETENUE{\poseretenue}}%
    \matrixSetElement{MATINTER}{1}{\nbcolmun}{\RETENUE}%
  }%
  %ligne des retenues pour savoir si il y en a !
  \matrixSubmatrix{MATINTERRETENUE}{MATINTER}{1}{1}{1}{\nbtotalchiffresres}%
  %création de la ligne des retenues
  \xdef\LigneBinAddRet{}%
  \foreach \i in {1,...,\nbtotalchiffresres}{%
    \xintifboolexpr{\matrixGetElement{MATINTER}{1}{\i} == 0}%
      {\xdef\LigneBinAddRet{\LigneBinAddRet &}}%
      {\xdef\LigneBinAddRet{\LigneBinAddRet & \textcolor{\addbinluacol}{\scalebox{\addbinluascret}[\addbinluascret]{\matrixGetElement{MATINTER}{1}{\i}}}}}%
  }%
  %création des lignes intermédiaires
  \xdef\LignesOperande{}%
  \foreach \i in {1,...,\nboperandes}{%
    \xdef\xxi{\xinteval{\i+1}}%
    \itemtomacro\addbinlstoper[\i]{\tmpinteroper}%on extrait l'opérande
    \pfllenstr{\tmpinteroper}[\nbchiiffres]%on extrait son nb de chiffres
    \xdef\tmpoffset{\xinteval{\nbtotalchiffresres-\nbchiiffres}}%on calcule le nombre de cases vides
    \xintifboolexpr{\i == 1}{}{\xdef\LignesOperande{\LignesOperande + }}%
    \xintifboolexpr{\tmpoffset == 0}%
      {}%
      {
        \foreach \k in {1,...,\tmpoffset}{%
          \xdef\LignesOperande{\LignesOperande &}%
        }%
      }%
    \foreach \j in {\inteval{\tmpoffset+1},...,\nbtotalchiffresres}{%
      \xdef\LignesOperande{\LignesOperande & \matrixGetElement{MATINTER}{\xxi}{\j}}%
    }%
    \xdef\LignesOperande{\LignesOperande \\}%
  }%
  %et le tableau !!
  \setlength{\tabcolsep}{\addbinluaspace}%
  \begin{NiceTabular}{c*{\nbtotalchiffresres}{c}}
    \xintifboolexpr{\matrixNormOne{MATINTERRETENUE} == 0}{}{\RowStyle[cell-space-limits=1pt]{\addbinluafonte} \LigneBinAddRet \\}
    \RowStyle[nb-rows=*,cell-space-limits=2pt]{\addbinluafonte}
    \LignesOperande
    \hline
    \ifboolKV[addbinlua]{Egal}{=}{} \xintFor* ##1 in {\xintSeq{1}{\nbtotalchiffresres}}\do{& \matrixGetElement{MATINTER}{\nblignesmatricetmp}{##1}} \\
  \end{NiceTabular}%
  %\(\matrixPrint{MATINTER}\)
}
\NewCommandCopy\pflposaddlua\PoseAdditionLua

\fi

%====RESTE MODULO
\NewDocumentCommand\ResteMod{ s m m }{%
  %étoilé := version négative;#2=nb;#3=base
  \IfBooleanTF{#1}%
    {\num{\xintiieval{irem(#2,#3)-(#3)}}}%
    {\num{\xintiieval{irem(#2,#3)}}}%
}
\NewCommandCopy\pflrestemod\ResteMod
\NewCommandCopy\pflremmod\ResteMod

\NewDocumentCommand\QuoMod{ m m }{%
  \num{\xintiieval{iquo(#1,#2)}}%
}
\NewCommandCopy\pflquomod\QuoMod

%====OPÉRATIONS POSÉES 2/10/16 MULTICOMPILATEUR ?
\RequirePackage{calc}

\newlength{\widestcharwd}
\newlength{\charwd}
\newlength{\heightercharht}

\defKV[poseoperation]{%
  Base=\def\PoseOpeBase{#1},%
  LimiteCapac=\def\PoseOpeLimit{#1},%
  SymbDecal=\def\PoseOpeSymb{#1},%
  Offset=\def\PoseOpeOffset{#1},%
  CouleurRetenue=\def\PoseOpeColReten{#1}
}

\setKVdefault[poseoperation]{%
  Base=dec,%
  SymbDecal=.,%
  Interm=true,%
  Offset=6pt,%
  AffEgal=true,%
  LimiteCapac=0,%
  CouleurRetenue=red,%
  AffRetenues=true
}

\NewDocumentCommand\IntCalcMaxWidth{ m }{%
  \pfllenstr{#1}[\tmplen]%
  \setlength{\widestcharwd}{0pt}
  \xintFor* ##1 in {\xintSeq{1}{\tmplen}}\do{%
    \StrChar{#1}{##1}[\tmpchar]%
    \settowidth{\charwd}{\tmpchar}%
    \setlength{\widestcharwd}{\maxof{\widestcharwd}{\charwd}}%
  }%
}

\NewDocumentCommand\IntCreateBoxNumbers{ O{c} m }{%
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\IntCalcMaxWidth{0123456789}}%
    {bin}{\IntCalcMaxWidth{01}}%
    {hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
  }%
  \addtolength{\widestcharwd}{\PoseOpeOffset/2}%
  \pfllenstr{#2}[\tmplennumber]%
  \xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
    \StrChar{#2}{##1}[\tmpchiff]%
    \makebox[\the\widestcharwd][#1]{\tmpchiff}%
  }%
}

\newcommand\AffRetenue[1]{%
  \textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{#1}}%
}

\NewDocumentCommand\IntCreateBoxNumbersRetenue{ m }{%
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\IntCalcMaxWidth{0123456789}}%
    {bin}{\IntCalcMaxWidth{01}}%
    {hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
  }%
  \addtolength{\widestcharwd}{\PoseOpeOffset/2}%
  \pfllenstr{#1}[\tmplennumber]%
  \xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
    \StrChar{#1}{##1}[\tmpchiff]%
    \IfEq{\tmpchiff}{X}%
      {\makebox[\the\widestcharwd][c]{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{}}}}%
      {\makebox[\the\widestcharwd][c]{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\tmpchiff}}}}%
  }%
}

\NewDocumentCommand\IntPoseAddition{ m m D<>{} }{%
  \pfllenstr{#1}[\nbchiffA]%
  %essai des retenues ??
  \xdef\ListeRetenues{X}%
  \xintFor* ##1 in {\xintSeq{1}{\nbchiffA}}\do{%
    \StrRight{#1}{##1}[\tmpinterA]%
    \StrRight{#2}{##1}[\tmpinterB]%
    \IfStrEqCase{\PoseOpeBase}{%
      {dec}{\xdef\rescalcrete{\xinteval{\tmpinterA+\tmpinterB}}}%
      {bin}{\xdef\rescalcrete{\xintDecToBin{\xinteval{\xintBinToDec{\tmpinterA}+\xintBinToDec{\tmpinterB}}}}}%
      {hex}{\xdef\rescalcrete{\xintDecToHex{\xinteval{\xintHexToDec{\tmpinterA}+\xintHexToDec{\tmpinterB}}}}}%
    }%
    \pfllenstr{\rescalcrete}[\tmpnbchiffresinter]%
    \xintifboolexpr{\tmpnbchiffresinter > ##1}%
      {%
        \StrGobbleRight{\rescalcrete}{##1}[\tmpretenue]%
        \xdef\ListeRetenues{\tmpretenue\ListeRetenues}%
      }%
      {%
        \xintifboolexpr{##1 == \nbchiffA}{}{\xdef\ListeRetenues{X\ListeRetenues}}%
      }%
  }%
  %suite
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\xdef\rescalcbin{\xinteval{#1+#2}}}%
    {bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}+\xintBinToDec{#2}}}}}%
    {hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}+\xintHexToDec{#2}}}}}%
  }%
  \xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
  %\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
  \ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
    \ifboolKV[poseoperation]{AffRetenues}{\RowStyle[cell-space-limits=0pt]{} & \IntCreateBoxNumbersRetenue{\ListeRetenues} \\[-0.5\heightercharht]}{}
    \RowStyle[nb-rows=*,cell-space-limits=1pt]{}  & \IntCreateBoxNumbers{#1} \\
    + & \IntCreateBoxNumbers{#2} \\
    \hline
    \ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
  \end{NiceArray}}%
  %\end{array}}%
}

\NewDocumentCommand\IntPoseSoustraction{ m m D<>{} }{%
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\xdef\rescalcbin{\xinteval{#1-#2}}}%
    {bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}-\xintBinToDec{#2}}}}}%
    {hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}-\xintHexToDec{#2}}}}}%
  }%
  %bourrage de zéros pour les retenues
  \pfllenstr{#1}[\lenA]\pfllenstr{#2}[\lenB]%
  \xdef\BwithZeros{#2}%
  \xintifboolexpr{ \lenA > \lenB }%
    {%
      \xintFor* ##1 in {\xintSeq{1}{\xinteval{\lenA-\lenB}}}\do{\xdef\BwithZeros{0\BwithZeros}}%
    }%
    {}%
  %suite
  \xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
  %\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
  \ifboolKV[poseoperation]{AffRetenues}{%
  \ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset/2}}r@{\,}}
    \RowStyle[nb-rows=*,cell-space-limits=1pt]{}
    & \IfStrEqCase{\PoseOpeBase}{%
      {dec}{\IntCalcMaxWidth{0123456789}}%
      {bin}{\IntCalcMaxWidth{01}}%
      {hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
    }%
    \addtolength{\widestcharwd}{\PoseOpeOffset/2}%
    \pfllenstr{#1}[\tmplennumber]%
    \xintFor* ##1 in {\xintSeq{1}{\tmplennumber}}\do{%
      \xdef\tmplenctr{\inteval{\tmplennumber-##1+1}}%
      \StrChar{#1}{##1}[\tmpchiff]%
      \StrRight{#1}{\tmplenctr}[\tmpA]%
      \StrRight{\BwithZeros}{\tmplenctr}[\tmpB]%
      \IfStrEqCase{\PoseOpeBase}{%
        {dec}{\xdef\rescalctmp{\xinteval{\tmpA-\tmpB}}}%
        {bin}{\xdef\rescalctmp{\xinteval{\xintBinToDec{\tmpA}-\xintBinToDec{\tmpB}}}}%
        {hex}{\xdef\rescalctmp{\xinteval{\xintHexToDec{\tmpA}-\xintHexToDec{\tmpB}}}}%
      }%
      \xintifboolexpr{ \rescalctmp >= 0 }%
        {\makebox[\the\widestcharwd][r]{\tmpchiff}}%
        %{\makebox[\the\widestcharwd][r]{${}_{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}$\kern-0.1em\tmpchiff}}%
        {\makebox[\the\widestcharwd][r]{\raisebox{-0.5\height}{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}\kern-0.1em\tmpchiff}}%
    } \\
    - & \IfStrEqCase{\PoseOpeBase}{%
      {dec}{\IntCalcMaxWidth{0123456789}}%
      {bin}{\IntCalcMaxWidth{01}}%
      {hex}{\IntCalcMaxWidth{0123456789ABCDEF}}%
    }%
    \addtolength{\widestcharwd}{\PoseOpeOffset/2}%
    \pfllenstr{#1}[\tmplennumber]%
    \StrChar{#2}{\lenB}[\tmplastB]%
    \xintFor* ##1 in {\xintSeq{1}{\xinteval{\tmplennumber-1}}}\do{%
      \xdef\tmplenctr{\inteval{\tmplennumber-##1}}%
      \StrChar{\BwithZeros}{##1}[\tmpchiff]%
      \StrRight{#1}{\tmplenctr}[\tmpA]%
      \StrRight{\BwithZeros}{\tmplenctr}[\tmpB]%
      \IfStrEqCase{\PoseOpeBase}{%
        {dec}{\xdef\rescalctmp{\xinteval{\tmpA-\tmpB}}}%
        {bin}{\xdef\rescalctmp{\xinteval{\xintBinToDec{\tmpA}-\xintBinToDec{\tmpB}}}}%
        {hex}{\xdef\rescalctmp{\xinteval{\xintHexToDec{\tmpA}-\xintHexToDec{\tmpB}}}}%
      }%
      \xintifboolexpr{ \rescalctmp >= 0 }%
        {\makebox[\the\widestcharwd][r]{\xintifboolexpr{ ##1 <= \lenA-\lenB }{}{\tmpchiff}}}%
        %{\makebox[\the\widestcharwd][r]{${}_{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}$\xintifboolexpr{ ##1 <= \lenA-\lenB }{}{\kern-0.1em\tmpchiff}}}%
        {\makebox[\the\widestcharwd][r]{\raisebox{-0.5\height}{\textcolor{\PoseOpeColReten}{\scalebox{0.5}[0.5]{\text{1}}}}\xintifboolexpr{ ##1 <= \lenA-\lenB }{\kern0.25em}{\kern-0.1em\tmpchiff}}}%
    }\makebox[\the\widestcharwd][r]{\tmplastB} \\
    
    %\IntCreateBoxNumbers{#2} \\
    \hline
    \ifboolKV[poseoperation]{AffEgal}{=}{} &\IntCreateBoxNumbers[r]{\rescalcbin} \\
  \end{NiceArray}}%
  }%
  {%
  \ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
    \RowStyle[nb-rows=*,cell-space-limits=1pt]{}
    & \IntCreateBoxNumbers{#1} \\
    - & \IntCreateBoxNumbers{#2} \\
    \hline
    \ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
  \end{NiceArray}}%
  }%
  %\end{array}}%
}

\NewDocumentCommand\IntPoseMultiplication{ m m D<>{} }{%
  \pfllenstr{#1}[\nbligninit]%
  \pfllenstr{#2}[\nbligninter]%
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\xdef\rescalcbin{\xinteval{#1*#2}}}%
    {bin}{\xdef\rescalcbin{\xintDecToBin{\xinteval{\xintBinToDec{#1}*\xintBinToDec{#2}}}}}%
    {hex}{\xdef\rescalcbin{\xintDecToHex{\xinteval{\xintHexToDec{#1}*\xintHexToDec{#2}}}}}%
  }%
  \xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbin}{\PoseOpeLimit}[\rescalcbin]}{}%
  %\ensuremath{\begin{array}{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
  \ensuremath{\begin{NiceArray}[#3]{@{\,}r@{\hspace{\PoseOpeOffset}}r@{\,}}
    \RowStyle[nb-rows=*,cell-space-limits=1pt]{}
    & \IntCreateBoxNumbers{#1} \\
    \times & \IntCreateBoxNumbers{#2} \\
    \ifboolKV[poseoperation]{Interm}%
      {%
        \hline
        \xintFor* ##1 in {\xintSeq{1}{\nbligninter}}\do{%
          \xintifboolexpr{##1 == 1}{}{+}&
          \xdef\tmpindice{\inteval{\nbligninter-##1+1}}\StrChar{#2}{\tmpindice}[\tmpchiff]\IfStrEqCase{\PoseOpeBase}{{dec}{\xdef\rescalcbininter{\xinteval{#1*\tmpchiff}}}{bin}{\xdef\rescalcbininter{\xintDecToBin{\xinteval{\xintBinToDec{#1}*\xintBinToDec{\tmpchiff}}}}}{hex}{\xdef\rescalcbininter{\xintDecToHex{\xinteval{\xintHexToDec{#1}*\xintHexToDec{\tmpchiff}}}}}}%
          \IfEq{\tmpchiff}{0}%
            {\xdef\rescalcbininter{}\xintFor* ##2 in {\xintSeq{1}{\nbligninit}}\do{\xdef\rescalcbininter{0\rescalcbininter}}%
            }{}%
          \xintifboolexpr{\PoseOpeLimit > 0 }{\StrRight{\rescalcbininter}{\PoseOpeLimit}[\rescalcbininter]}{}%
          \IntCreateBoxNumbers{\rescalcbininter}
          \xintifboolexpr{##1 == 1}{}{\xintFor* ##2 in {\xintSeq{1}{##1-1}}\do{\makebox[\the\widestcharwd][c]{\PoseOpeSymb}}}\\
        }%
      }%
      {}%
    \hline
    \ifboolKV[poseoperation]{AffEgal}{=}{} & \IntCreateBoxNumbers{\rescalcbin} \\
  \end{NiceArray}}%
  %\end{array}}%
}

\NewDocumentCommand\OperationPosee{ O{} m D<>{} }{%
  \restoreKV[poseoperation]%
  \setKV[poseoperation]{#1}%
  \IfStrEqCase{\PoseOpeBase}{%
    {dec}{\settoheight{\heightercharht}{\hbox{0123456789}}}%
    {bin}{\settoheight{\heightercharht}{\hbox{01}}}%
    {hex}{\settoheight{\heightercharht}{\hbox{0123456789ABCDEF}}}%
  }%
  \IfSubStr{#2}{+}%
    {%
      \StrCut{#2}{+}{\tmpcalcA}{\tmpcalcB}%
      \IntPoseAddition{\tmpcalcA}{\tmpcalcB}<#3>
    }%
    {}%
  \IfSubStr{#2}{-}%
    {%
      \StrCut{#2}{-}{\tmpcalcA}{\tmpcalcB}%
      \IntPoseSoustraction{\tmpcalcA}{\tmpcalcB}<#3>
    }%
    {}%
  \IfSubStr{#2}{*}%
    {%
      \StrCut{#2}{*}{\tmpcalcA}{\tmpcalcB}%
      \IntPoseMultiplication{\tmpcalcA}{\tmpcalcB}<#3>
    }%
    {}%
}
\NewCommandCopy\pflopeposee\OperationPosee

%====FACTORIELLE
\defKV[calcfactorielle]{ChSignif=\def\factochfsign{#1},Sens=\def\factochfsens{#1}}
\setKVdefault[calcfactorielle]{%
  Complet=false,%
  Enonce=false,%
  Partiel=false,%
  Grand=false,%
  ChSignif=9,%
  Espace=\mkern1.5mu\relax,%
  Sens=m
}

\NewDocumentCommand\Factorielle{ s O{} m }{%
  \restoreKV[calcfactorielle]%
  \setKV[calcfactorielle]{#2}%
  \ifboolKV[calcfactorielle]{Grand}%
    {%
      \xdef\tmpres{\xintfloateval[\factochfsign]{factorial(#3)}}%
    }%
    {%
      \xdef\tmpres{\xinteval{factorial(#3)}}%
    }%
  \ensuremath{%
    \ifboolKV[calcfactorielle]{Enonce}%
      {%
        \IfBooleanTF{#1}{#3\useKV[calcfactorielle]{Espace}!}{\num{#3}\useKV[calcfactorielle]{Espace}!}=
      }%
      {}%
    \ifboolKV[calcfactorielle]{Partiel}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1 \times 2 \times \ldots \times \xinteval{#3-1} \times #3 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{#3 \times \xinteval{#3-1} \times \ldots \times 2 \times 1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1 \times 2 \times \ldots \times \num{\xinteval{#3-1}} \times \num{#3} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{\num{#3} \times \num{\xinteval{#3-1}} \times \ldots \times 2 \times 1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
      }%
      {}%
    \ifboolKV[calcfactorielle]{Complet}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1 \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times ##1} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{##1 \times} 1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1 \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times \num{##1}} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{\num{##1} \times} 1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
      }%
      {}%
    \IfBooleanTF{#1}%
      {%
        \tmpres%
      }%
      {%
        \ifboolKV[calcfactorielle]{Grand}%
          {%
            \num[exponent-mode=scientific]{\tmpres}%
          }%
          {%
            \num{\tmpres}%
          }%
      }%
  }%
}
\NewCommandCopy\pflfacto\Factorielle

\NewDocumentCommand\Primorielle{ s O{} m }{%
  \restoreKV[calcfactorielle]%
  \setKV[calcfactorielle]{#2}%
  \def\tmpres{1}%
  \xintFor* ##1 in {\xintSeq{1}{#3}}\do{%
    \pgfmathisprime{##1}\ifnum\pgfmathresult=1\xdef\tmpres{\xinteval{\tmpres*##1}}\fi%
  }%
  \ifboolKV[calcfactorielle]{Grand}%
    {%
      \xdef\tmpres{\xintfloateval[\factochfsign]{\tmpres}}%
    }%
    {}%
  \ensuremath{%
    \ifboolKV[calcfactorielle]{Enonce}%
      {%
        \IfBooleanTF{#1}{#3\#}{\num{#3}\#}=
      }%
      {}%
    \ifboolKV[calcfactorielle]{Complet}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                2 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{\pgfmathisprime{##1}\ifnum\pgfmathresult=1\times##1\fi} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{3}}\do{\pgfmathisprime{##1}\ifnum\pgfmathresult=1##1\times\fi} 2 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                2 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{\pgfmathisprime{##1}\ifnum\pgfmathresult=1\times\num{##1}\fi} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{3}}\do{\pgfmathisprime{##1}\ifnum\pgfmathresult=1\num{##1}\times\fi} 2 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
      }%
      {}%
    \IfBooleanTF{#1}%
      {%
        \tmpres%
      }%
      {%
        \ifboolKV[calcfactorielle]{Grand}%
          {%
            \num[exponent-mode=scientific]{\tmpres}%
          }%
          {%
            \num{\tmpres}%
          }%
      }%
  }%
}
\NewCommandCopy\pflprimor\Primorielle

\NewDocumentCommand\DoubleFactorielle{ s O{} m }{%
  \restoreKV[calcfactorielle]%
  \setKV[calcfactorielle]{#2}%
  \def\tmpres{1}%  
  \xintFor* ##1 in {\xintSeq{1}{#3}}\do{%
    \xintifboolexpr{\xintiiOdd{#3} == \xintiiOdd{##1} 'or' \xintiiEven{#3} == \xintiiEven{##1}}%
      {%
        \xdef\tmpres{\xinteval{\tmpres*##1}}%
      }%
      {}%
  }%
  \ifboolKV[calcfactorielle]{Grand}%
  {%
    \xdef\tmpres{\xintfloateval[\factochfsign]{\tmpres}}%
  }%
  {}%
  \ensuremath{%
    \ifboolKV[calcfactorielle]{Enonce}%
    {%
      \IfBooleanTF{#1}{#3\useKV[calcfactorielle]{Espace}!!}{\num{#3}\useKV[calcfactorielle]{Espace}!!}=
    }%
    {}%
    \ifboolKV[calcfactorielle]{Complet}%
    {%
      \IfBooleanTF{#1}%
      {%
        \IfStrEq{\factochfsens}{m}%
          {%
            \xintifboolexpr{\xintiiOdd{#3} == 1}%
              {%
                1 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{%
                  \xintifboolexpr{\xintiiOdd{##1} == 1}{\times##1}{}%
                }%
              }%
              {%
                2 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{%
                  \xintifboolexpr{\xintiiOdd{##1} != 1}{\times##1}{}%
                }%
              }%
            \ifboolKV[calcfactorielle]{Grand}{\approx}{=}%
          }%
          {%
            \xintifboolexpr{\xintiiOdd{#3} == 1}%
            {%
              \xintFor* ##1 in {\xintSeq{#3}{3}}\do{%
                \xintifboolexpr{\xintiiOdd{##1} == 1}{##1\times}{}%
              }1%
            }%
            {%
              \xintFor* ##1 in {\xintSeq{#3}{3}}\do{%
                \xintifboolexpr{\xintiiOdd{##1} != 1}{##1\times}{}%
              }2%
            }%
            \ifboolKV[calcfactorielle]{Grand}{\approx}{=}%
          }%
      }%
      {%
        \IfStrEq{\factochfsens}{m}%
        {%
          \xintifboolexpr{\xintiiOdd{#3} == 1}%
          {%
            1 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{%
              \xintifboolexpr{\xintiiOdd{##1} == 1}{\times\num{##1}}{}%
            }%
          }%
          {%
            2 \xintFor* ##1 in {\xintSeq{3}{#3}}\do{%
              \xintifboolexpr{\xintiiOdd{##1} != 1}{\times\num{##1}}{}%
            }%
          }%
          \ifboolKV[calcfactorielle]{Grand}{\approx}{=}%
        }%
        {%
          \xintifboolexpr{\xintiiOdd{#3} == 1}%
          {%
            \xintFor* ##1 in {\xintSeq{#3}{3}}\do{%
              \xintifboolexpr{\xintiiOdd{##1} == 1}{\num{##1}\times}{}%
            }1%
          }%
          {%
            \xintFor* ##1 in {\xintSeq{#3}{3}}\do{%
              \xintifboolexpr{\xintiiOdd{##1} != 1}{\num{##1}\times}{}%
            }2%
          }%
          \ifboolKV[calcfactorielle]{Grand}{\approx}{=}%
        }%
      }%
    }%
    {}%
    \IfBooleanTF{#1}%
    {%
      \tmpres%
    }%
    {%
      \ifboolKV[calcfactorielle]{Grand}%
      {%
        \num[exponent-mode=scientific]{\tmpres}%
      }%
      {%
        \num{\tmpres}%
      }%
    }%
  }%
}
\NewCommandCopy\pfldblefacto\DoubleFactorielle

\NewDocumentCommand\HyperFactorielle{ s O{} m }{%
  \restoreKV[calcfactorielle]%
  \setKV[calcfactorielle]{#2}%
  \def\tmpres{1}%
  \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\xdef\tmpres{\xinteval{\tmpres*(##1)^(##1)}}}%
  \ifboolKV[calcfactorielle]{Grand}%
    {%
      \xdef\tmpres{\xintfloateval[\factochfsign]{\tmpres}}%
    }%
    {}%
  \ensuremath{%
    \ifboolKV[calcfactorielle]{Enonce}%
      {%
        \IfBooleanTF{#1}{\text{H}(#3)}{\text{H}(\rm{#3})}=
      }%
      {}%
    \ifboolKV[calcfactorielle]{Partiel}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1^1 \times 2^2 \times \ldots \times \xinteval{#3-1}^{\xinteval{#3-1}} \times #3^{#3} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{#3^{#3} \times \xinteval{#3-1}^{\xinteval{#3-1}} \times \ldots \times 2^2 \times 1^1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1 \times 2 \times \ldots \times \num{\xinteval{#3-1}}^{\num{\xinteval{#3-1}}} \times \num{#3}^{\num{#3}} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{\num{#3}^{\num{#3}} \times \num{\xinteval{#3-1}}^{\num{\xinteval{#3-1}}} \times \ldots \times 2^2 \times 1^1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
      }%
      {}%
    \ifboolKV[calcfactorielle]{Complet}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1^1 \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times ##1^{##1}} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{##1^{##1} \times} 1^1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1^1 \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times \num{##1}^{\num{##1}}} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{\num{##1}^{\num{##1}} \times} 1^1 \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
      }%
      {}%
    \IfBooleanTF{#1}%
      {%
        \tmpres%
      }%
      {%
        \ifboolKV[calcfactorielle]{Grand}%
          {%
            \num[exponent-mode=scientific]{\tmpres}%
          }%
          {%
            \num{\tmpres}%
          }%
      }%
  }%
}
\NewCommandCopy\pflhyperfacto\HyperFactorielle

\NewDocumentCommand\SuperFactorielle{ s O{} m }{%
  \restoreKV[calcfactorielle]%
  \setKV[calcfactorielle]{#2}%
  \def\tmpres{1}%
  \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\xdef\tmpres{\xinteval{\tmpres*factorial(##1)}}}%
  \ifboolKV[calcfactorielle]{Grand}%
    {%
      \xdef\tmpres{\xintfloateval[\factochfsign]{\tmpres}}%
    }%
    {}%
  \ensuremath{%
    \ifboolKV[calcfactorielle]{Enonce}%
      {%
        \IfBooleanTF{#1}{\text{sf}(#3)}{\text{sf}(\num{#3})}=
      }%
      {}%
    \ifboolKV[calcfactorielle]{Partiel}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1\useKV[calcfactorielle]{Espace}! \times 2\useKV[calcfactorielle]{Espace}! \times \ldots \times \xinteval{#3-1}\useKV[calcfactorielle]{Espace}! \times #3\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{#3\useKV[calcfactorielle]{Espace}! \times \xinteval{#3-1}\useKV[calcfactorielle]{Espace}! \times \ldots \times 2\useKV[calcfactorielle]{Espace}! \times 1\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
          {%
            \IfStrEqCase{\factochfsens}{%
              {m}{1\useKV[calcfactorielle]{Espace}! \times 2\useKV[calcfactorielle]{Espace}! \times \ldots \times \num{\xinteval{#3-1}}\useKV[calcfactorielle]{Espace}! \times \num{#3}\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
              {d}{\num{#3}\useKV[calcfactorielle]{Espace}! \times \num{\xinteval{#3-1}}\useKV[calcfactorielle]{Espace}! \times \ldots \times 2\useKV[calcfactorielle]{Espace}! \times 1\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}}%
            }%
          }%
      }%
      {}%
    \ifboolKV[calcfactorielle]{Complet}%
      {%
        \IfBooleanTF{#1}%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1\useKV[calcfactorielle]{Espace}! \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times ##1\useKV[calcfactorielle]{Espace}!} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{##1\useKV[calcfactorielle]{Espace}! \times} 1\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
          {%
            \IfStrEq{\factochfsens}{m}%
              {%
                1\useKV[calcfactorielle]{Espace}! \xintFor* ##1 in {\xintSeq{2}{#3}}\do{\times \num{##1}\useKV[calcfactorielle]{Espace}!} \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
              {%
                \xintFor* ##1 in {\xintSeq{#3}{2}}\do{\num{##1}\useKV[calcfactorielle]{Espace}! \times} 1\useKV[calcfactorielle]{Espace}! \ifboolKV[calcfactorielle]{Grand}{\approx}{=}
              }%
          }%
      }%
      {}%
    \IfBooleanTF{#1}%
      {%
        \tmpres%
      }%
      {%
        \ifboolKV[calcfactorielle]{Grand}%
          {%
            \num[exponent-mode=scientific]{\tmpres}%
          }%
          {%
            \num{\tmpres}%
          }%
      }%
  }%
}
\NewCommandCopy\pflsuperfacto\SuperFactorielle

%====CONVERSIONS ENTRE BASES
\ExplSyntaxOn
%commande interne (stockage ou non)
\NewDocumentCommand\pflbasetobase{ m m m }{%
  %1=init
  %2=fin
  %3=nb
  \ifnum#2=10%
    \int_from_base:nn {#3}{#1}%
  \else%
    \int_to_Base:nn {\int_from_base:nn {#3}{#1}}{#2}%
  \fi%
}

\NewDocumentCommand\tmpresconvbases{ m m m O{\tmpconvres} }{%
  %1=init
  %2=fin
  %3=nb
  \ifnum#2=10%
    \xdef#4{\int_from_base:nn {#3}{#1}}%
  \else%
    \xdef#4{\int_to_Base:nn {\int_from_base:nn {#3}{#1}}{#2}}%
  \fi%
}
\ExplSyntaxOff

%commande interne split adaptée de https://tex.stackexchange.com/questions/171007/split-a-character-string-n-by-n // unbonpetit // CC BY-SA 3.0
\NewDocumentCommand\tmpStrSplit{ O{\,} m m O{\tmpsplitres} }{%
  %1=espace / %2=nb caract / %3=chaîne / %4=macro de stockage
  \xdef\splitstring{#3}\let\splitresult\empty%
  \loop%
    \pfllenstr\splitstring[\tempa]%
    \StrSplit\splitstring{\number\numexpr\tempa-#2}\splitstring\tempb%
    \xdef\splitresult{\unless\ifx\splitstring\empty#1\fi\tempb\splitresult}%
    \unless\ifx\splitstring\empty%
  \repeat%
  \xdef#4{\splitresult}%
}

\NewDocumentCommand\ConversionEntreBases{ s O{\,} m m }{%
  %calculs internes
  \StrCut{#3}{->}{\tmpbaseinit}{\tmpbasefin}%
  \tmpresconvbases{\tmpbaseinit}{\tmpbasefin}{#4}%\tmpconvres
  %affichage
  \ensuremath{%
    \ifnum\tmpbaseinit=2%
      \tmpStrSplit{4}{#4}%
      {\tmpsplitres}\IfBooleanT{#1}{_{\tmpbaseinit}}%
    \else%
      {#4}\IfBooleanT{#1}{_{\tmpbaseinit}}%
    \fi%
    =%
    \ifnum\tmpbasefin=2%
      \tmpStrSplit[#2]{4}{\tmpconvres}%
      {\tmpsplitres}\IfBooleanT{#1}{_{\tmpbasefin}}%
    \else%
      {\tmpconvres}\IfBooleanT{#1}{_{\tmpbasefin}}%
    \fi%
  }%
}
\NewCommandCopy\pflconvbases\ConversionEntreBases

\endinput