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We want to develop a 3x3 linear system with \+ nice results. Therefore we need an integer 3x3-matrix with determinant +-1." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7br%#&xG%$AddG%(AdjointG%3Backwa rdSubstituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFormG%-Bi linearFormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%'ColumnG %0ColumnDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMatrixG%0 ConditionNumberG%/ConstantMatrixG%/ConstantVectorG%%CopyG%2CreatePermu tationG%-CrossProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%)Diago nalG%/DiagonalMatrixG%*DimensionG%+DimensionsG%+DotProductG%6EigenCond itionNumbersG%,EigenvaluesG%-EigenvectorsG%&EqualG%2ForwardSubstituteG %.FrobeniusFormG%4GaussianEliminationG%2GenerateEquationsG%/GenerateMa trixG%2GetResultDataTypeG%/GetResultShapeG%5GivensRotationMatrixG%,Gra mSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTransposeG%/Hessenber gFormG%.HilbertMatrixG%2HouseholderMatrixG%/IdentityMatrixG%2Intersect ionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*IsUnitaryG%2JordanB lockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositionG%-LeastSquaresG%,L inearSolveG%$MapG%%Map2G%*MatrixAddG%2MatrixExponentialG%/MatrixFuncti onG%.MatrixInverseG%5MatrixMatrixMultiplyG%+MatrixNormG%,MatrixPowerG% 5MatrixScalarMultiplyG%5MatrixVectorMultiplyG%2MinimalPolynomialG%&Min orG%(ModularG%)MultiplyG%,NoUserValueG%%NormG%*NormalizeG%*NullSpaceG% 3OuterProductMatrixG%*PermanentG%&PivotG%*PopovFormG%0QRDecompositionG %-RandomMatrixG%-RandomVectorG%%RankG%6RationalCanonicalFormG%6Reduced RowEchelonFormG%$RowG%-RowDimensionG%-RowOperationG%)RowSpaceG%-Scalar MatrixG%/ScalarMultiplyG%-ScalarVectorG%*SchurFormG%/SingularValuesG%* SmithFormG%*SubMatrixG%*SubVectorG%)SumBasisG%0SylvesterMatrixG%/Toepl itzMatrixG%&TraceG%*TransposeG%0TridiagonalFormG%+UnitVectorG%2Vanderm ondeMatrixG%*VectorAddG%,VectorAngleG%5VectorMatrixMultiplyG%+VectorNo rmG%5VectorScalarMultiplyG%+ZeroMatrixG%+ZeroVectorG%$ZipG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "M:=RandomMatrix(3,3,generator=-5..5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'RTABLEG6%\")wzSW-%'MATR IXG6#7%7%\"\"!!\"\"\"\"&7%\"\"\"F.\"\"#7%F3!\"%F2%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "while not(Determinant(M)=1 or Deter minant(M)=-1) do M:=RandomMatrix(3,3,generator=-5..5): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "M;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6%\")O*ou$-%'MATRIXG6#7%7%!\"&\"\"!\"\"\"7%!\"$F.F07%\"\"$ F.!\"%%'MatrixG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determin ant(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 80 "We solve the linear system given as M.x=<1,2,3> wh ich now has an integer result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "LinearSolve(M,<1,2,3>);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' RTABLEG6%\")k&[]%-%'MATRIXG6#7%7#\"\"#7#\"#T7#\"#6&%'VectorG6#%'column G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "If you need a matrix whose e igenvalues are integers, you can do as follows:" }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(M);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6%\")/IAN-%'MATRIXG6#7%7#,(*&\"\"' !\"\",&\"%O5F/*&^#\"#O\"\"\"\"$V'#F5\"\"#F5#F5\"\"$F5*(\"#iF5F:F/F0#F/ F:F5#\"\")F:F/7#,**&\"#7F/F0F9F/*(\"#JF5F:F/F0F=F/#F?F:F/*(^#F7F5F:F7, &*&F.F/F0F9F5*(F " 0 "" {MPLTEXT 1 0 165 "f:=factor(CharacteristicPolynomial(M,t)):\nwhil e not(type(f,`*`) and nops(f)=3) do M:=RandomMatrix(3,3,generator=-5.. 5): f:=factor(CharacteristicPolynomial(M,t)): od:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 2 "M;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTAB LEG6%\")oK&)R-%'MATRIXG6#7%7%\"\"\"F,F,7%!\"\"\"\"&F,7%F,F,F/%'MatrixG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "factor(CharacteristicPo lynomial(M,t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&%\"tG\"\"\"F&! \"\"F&,&F%F&\"\"%F'F&,&F%F&\"\"'F'F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Eigenvalues(M);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% 'RTABLEG6%\")kM1O-%'MATRIXG6#7%7#\"\"\"7#\"\"%7#\"\"'&%'VectorG6#%'col umnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Eigenvectors(M);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTABLEG6%\"))G,h$-%'MATRIXG6#7%7# \"\"\"7#\"\"%7#\"\"'&%'VectorG6#%'columnG-F$6%\")[aXU-F(6#7%7%!\"$\"\" !#F,\"\"$7%!\"\"FA#\"\"#F?7%F,F,F,%'MatrixG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 78 "With computer algebra, one generates Fortran code etc. \+ for better performance." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " p:=randpoly(x,terms=8,degree=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"pG,2\"#i!\"\"*&\"#:\"\"\")%\"xG\"#5F*F**&\"#\\F*)F,\"\"(F*F'*&\"#PF *)F,\"\"'F*F'*&F&F*)F,\"\"&F*F'*&\"#`F*)F,\"\"%F*F'*&\"#FF*)F,\"\"$F*F '*&\"#!*F*F,F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "codegen [horner](p,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"#i!\"\"*&,&\"#!* F%*&,&\"#FF%*&,&\"#`F%*&,&F$F%*&,&\"#PF%*&,&\"#\\F%*&\"#:\"\"\")%\"xG \"\"$F9F9F9F;F9F9F9F;F9F9F9F;F9F9F9F;F9F9F9)F;\"\"#F9F9F9F;F9F9" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Robertson's Conjecture (1989): Do es the function sqrt((exp(x)-1)/x) have nonnegative Taylor coefficient s?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f:=sqrt((exp(x)-1)/x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*$*&,&-%$expG6#%\"xG\"\"\"F ,!\"\"F,F+F-#F,\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "ser ies(f,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1%\"xG\"\"\"\"\"!#F%\" \"%F%#\"\"&\"#'*\"\"##F%\"$G\"\"\"$#\"#z\"&g@*F(#F/\"&g4%F*-%\"OG6#F% \"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "series(f,x=0,15); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+C%\"xG\"\"\"\"\"!#F%\"\"%F%#\"\"& \"#'*\"\"##F%\"$G\"\"\"$#\"#z\"&g@*F(#F/\"&g4%F*#\"#r\")/jQ7\"\"'#\"$8 \"\"*!3ExC\"\"(#\"%`I\"-+%=&3*=\"\"\")#F%\",+;C\\E#\"\"*#\"#<\"-S]&)G, $*\"#5#\"#>\"-KS3.Tu\"#6#\"'f1Fy Q%\"#8#!(PBX#\"6+[3bT;*p:1@\"#9-%\"OG6#F%\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "How is relativistic energy related to classical kineti c energy?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "E:=m*c^2*(1/sq rt(1-v^2/c^2)-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG*(%\"mG\"\" \")%\"cG\"\"#F',&*&F'F'*$,&F'F'*&%\"vGF*F)!\"#!\"\"#F'F*F2F'F'F2F'" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "series(E,v=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+)%\"vG,$*&\"\"#!\"\"%\"mG\"\"\"F*F',$**\"\"$F* \"\")F(F)F*%\"cG!\"#F*\"\"%-%\"OG6#F*\"\"'" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "read(\"FPS.mpl\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%GPackage~Formal~Power~Series,~Maple~V-8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TCopyright~1995,~Dominik~Gruntz,~University~of~BaselG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%^oCopyright~2002,~Detlef~M|gzller ~&~Wolfram~Koepf,~University~of~KasselG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "FPS(E,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\" \"\"\"#F&*&%\"mGF&-%$SumG6$*.)\"\"%,$%\"kG!\"\"F&)*&F&F&*$)%\"cGF'F&F2 F1F&-%*factorialG6#,&F&F&*&F'F&F1F&F&F&-F96#F1!\"#,&F1F&F&F&F2)%\"vG,& *&F'F&F1F&F&F'F&F&/F1;\"\"!%)infinityGF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "Progra mming" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 32 "Pascal-like Programming \+ Language" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "if then else construc t" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "SIN :=proc(x) if x>0 then sin(x) else 0 fi end:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 103 "The following is a typical example of the importance o f the order of evaluation in symbolic computation" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "plot(SIN(x),x=-Pi..Pi);" }}{PARA 8 "" 1 "" {TEXT -1 76 "Error, (in SIN) cannot determine if this expression is tr ue or false: 0 < x\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plo t('SIN(x)',x=-Pi..Pi);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6%-%'CURVESG6$7bo7$$!3*****4tk#fTJ!#<$\"\"!F, 7$$!3w\"*pr)3PY+$F*F+7$$!3?3E[*ysa)GF*F+7$$!3gX&))>2g9v#F*F+7$$!3)4577 .flh#F*F+7$$!3u\"QM::*H#[#F*F+7$$!3y\\AhcI#yN#F*F+7$$!3&e=d-FN*GAF*F+7 $$!3u%*GBP$Rc4#F*F+7$$!3mCj&f)3xi>F*F+7$$!3(or4AN*4E=F*F+7$$!3QQQ*3**= dq\"F*F+7$$!3e!>jg@*>q:F*F+7$$!3`;m,%)H7M9F*F+7$$!3v(4F,K))HI\"F*F+7$$ !3S*4!R#[0R=\"F*F+7$$!3pY@QqWIU5F*F+7$$!3Wp3w$R)\\B#*!#=F+7$$!3EJc8o39 GyFjnF+7$$!3'=[BP-8If'FjnF+7$$!3CB3<@?)yB&FjnF+7$$!3_*)RF+7$$!38,z7VK J,J!#?F+7$$\"3p0t!3)GF;mF`p$\"3)zUWu!oW6mF`p7$$\"39SFf3xEa8Fjn$\"37AR] l=8]8Fjn7$$\"312**yIK@d>Fjn$\"3lMlFjn7$$\"3'R2()Hve,c#Fjn$\"3+F! fa.$GKDFjn7$$\"38lB>NkN6KFjn$\"3))f7*e]Vk:$Fjn7$$\"3IcwRC;**Fjn7$$\"3)G#[O8J*GZ\"F*$\"3%*R'y@28@&**Fjn7$ $\"3=6*yzQ3X]\"F*$\"3i]On*eP!y**Fjn7$$\"3[**HfiO7O:F*$\"3#)Hs&Rj*)R*** Fjn7$$\"3!y32s$*Qxc\"F*$\"3Y\\4)=E`*****Fjn7$$\"3(e61(3gy+;F*$\"3yS,#> V.b***Fjn7$$\"3&R90-3LQj\"F*$\"3'>2!z;%Q,)**Fjn7$$\"3-sTq^,)om\"F*$\"3 #y%RB&*\\(Q&**Fjn7$$\"35+K?Bs#**p\"F*$\"3nG)er%=u;**Fjn7$$\"39YK*)Gjak 7M'z!o'*Fjn7$$\"3%>>C Z'eYk>F*$\"3-'f-+?x]B*Fjn7$$\"3kYuyfix%4#F*$\"33yU$=iZ$e')Fjn7$$\"3T4q ))fu.GAF*$\"3Jjs%pI2o\"zFjn7$$\"3y?w'**=&>gBF*$\"3ycCmz?sUqFjn7$$\"3M? UD/[\"4U#F*$\"3k\")[f98%*)f'Fjn7$$\"3!*>3a=Wj\"[#F*$\"3B,&=XdQ38'Fjn7$ $\"31?#=([fA^DF*$\"3w3N$\\&plmbFjn7$$\"3A?c*)yu\"3i#F*$\"3[6)p&**p_v\\ Fjn7$$\"35TH4t41$o#F*$\"3'[ZF^P>jU%Fjn7$$\"3-i-HnWIXFF*$\"3_^@G2'o*fQF jn7$$\"3Q!4nV,p;\"GF*$\"3'R)[@pyqRKFjn7$$\"3u=RWhN.yGF*$\"3H4EwtQ=0EFj n7$$\"3tX=#4!HbTHF*$\"3Y-J**3H3()>Fjn7$$\"3ss(*RSA20IF*$\"3%eLwJMn4O\" Fjn7$$\"3/')[UXCLtIF*$\"3.X>OF6s?oF`p7$$\"3!)***\\/l#fTJF*$\"3pawpOMzR J!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%+AXESLABELSG6$Q\"x6\"Q!F]_l-%%VI EWG6$;$!+aEfTJ!\"*$\"+aEfTJFe_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "for and while loops: Computation of factorial" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "x:=1: for i from 1 to 100 do x:=x*i od: x;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"it++++ ++++++++ko\"4@&=^#ePAF3#zp`iG=l:wRYT*3c\"*HK***f<_*Q'Hfo9i\"Qk#ofr!\\+ nEc)Q#*p\"o_T%RW:iK$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "i: =1: x:=1: while i<=100 do x:=x*i; i:=i+1; od: x;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"it++++++++++++ko\"4@&=^#ePAF3#zp`iG=l:wRYT*3c\"*HK*** f<_*Q'Hfo9i\"Qk#ofr!\\+nEc)Q#*p\"o_T%RW:iK$*" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 69 "However, in many instances there are high level constru cts available:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "100!;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"it+++++++++ +++ko\"4@&=^#ePAF3#zp`iG=l:wRYT*3c\"*HK***f<_*Q'Hfo9i\"Qk#ofr!\\+nEc)Q #*p\"o_T%RW:iK$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "product (i,i=1..100);" }}{PARA 8 "" 1 "" {TEXT -1 102 "Error, (in product) pro duct variable previously assigned, second argument evaluates to 101 = \+ 1 .. 100\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "i;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$,\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "product('i','i'=1..100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"it++++++++++++ko\"4@&=^#ePAF3#zp`iG=l:wRYT*3c\"*HK*** f<_*Q'Hfo9i\"Qk#ofr!\\+nEc)Q#*p\"o_T%RW:iK$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "i:='i':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "product(i,i=1..100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"it++++ ++++++++ko\"4@&=^#ePAF3#zp`iG=l:wRYT*3c\"*HK***f<_*Q'Hfo9i\"Qk#ofr!\\+ nEc)Q#*p\"o_T%RW:iK$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "x:= 'x':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Recursive Program" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Euclidean Algorithm" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "GCD:=(a,b)->\nif a<0 or b<0 then GCD(abs (a),abs(b))\nelif a " 0 "" {MPLTEXT 1 0 22 "GCD(-123456 7,1604137);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$F\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "random := rand(1..10^200):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a:=12373*random();" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"aG\"gwU&R\">;Q7wM\"GwH(pj3j\"Q]@hTl\\.J3 Bp;S&R?&pS\"=-rW>&G%=4Dpn()Qzt-a65`PT#)\\tt:)ohp3ruN;EY)R*)4[7j(zos)o^ ^DFMXM6xj$fGp_$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "b:=12373 *random();" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"bG\"gwCOfbas:CZZ**pB QIM'oW1i$)[xEJ#R)ePd0@.Ugo]dGka47HkRaxh3]bf.J#y %oK=BMhT=O%)>KvyWMx8RX*\\^" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "GCD(a,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&YZ#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Modular Powers" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "powermod:=proc(a,n,p)\nopti on remember;\nif n=0 then 1\nelif type(n,even) then modp(powermod(a,n/ 2,p)^2,p)\nelse modp(powermod(a,n-1,p)*a,p) \nfi: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "powermod(a,b,b);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#\"gwk5!\\%Gd;J^qU1jm?A0(p(*f#H>`*\\_M#pNG6,ZxE)G#H!zU \")orB3<**H&Q%zk-%*pJ=$>Xr!Q9yC+<-$*)[B:4(*fMi8;Q'=%zk]^B?WO8`!yYRI-%f ]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "powermod(b,a,a);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#\"fwcm`%*H)Q_,$Rr\"Rlot'zX(*oJOJT?H>\\ ^?*)*fY@.sh>3/\"z<]')oZBy8$RVkdvM\\9In*>$*H^`b#=wp-Ie3dhq#3pw6P6MzC6RI !zkVM<:v/F)4%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Fermat Test" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#\"gwU&R\">;Q7wM\"GwH(pj3j\"Q]@hTl\\.J3 Bp;S&R?&pS\"=-rW>&G%=4Dpn()Qzt-a65`PT#)\\tt:)ohp3ruN;EY)R*)4[7j(zos)o^ ^DFMXM6xj$fGp_$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "powermod (2,a,a);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#\"gwcY(QhFd]2^>Sd\">`7YCE2 Zv.#QX`h?1Nz#>yCJy&H[Cnajjn1!QRPA*e(*R(=bV0>g4[%e8)**fm&z'3cY7vQ_[q3^k Hxb\\.[^oK`u%\\K:uE_y#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "i sprime(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "powermod(2,b,b);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#\"gw[#*p\"=,TR@v>hq6xSbOmy&Q8V9X#HYv[P!)z5g/(fi5`(o#4 hr7E)=&)\\pqi*)>ep4@QJ!e>q:k)*pJzmJrJ*ezEf?S8sOZ(fOj]5g\"=n()=TOlj&G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "isprime(b);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(a);" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, computati on interrupted\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "RSA Cryptosy stem" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "auxiliary functions" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 514 "rest:= proc(liste) \nlocal k;\n[seq(op(k,liste),k=2..nops(liste))] end:\ncomb :=proc(liste)\nif nops(liste)=1 then op(1,liste) else 1000*comb(ListTo ols[Reverse] (rest(ListTools[Reverse](liste)))) + op(nops(liste),liste ) fi end:\nwith(StringTools):\nConvert:=proc(str)\ncomb(map(Ord,Explod e(str)))\nend:\nmakelist:=proc(zahl)\nif zahl<1000 then [zahl]\nelse [ op(makelist((zahl-modp(zahl,1000))/1000)),modp(zahl,1000)] \nfi end:\n Convertback:=proc(zahl)\nlocal ch;\nch:=map(Char,makelist(zahl));\ncat (seq(op(k,ch),k=1..nops(ch))) end:" }}{PARA 7 "" 1 "" {TEXT -1 39 "War ning, the name Map has been rebound\n" }}{PARA 7 "" 1 "" {TEXT -1 74 " Warning, the assigned names Fibonacci and Group now have a global bind ing\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "EncryptRSA:=proc( nachricht) powermod(Convert(nachricht),e,n) end:\nDecryptRSA:=proc(zah l) Convertback(powermod(zahl,d,n)) end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 216 "InitializeRSA:=proc()\nglobal p,q,n,e,d; local\nphi; \np:=nextprime(random());\nq:=nextprime(random());\nn:=p*q;\nphi:=(p- 1)*(q-1);\ne:=phi;\nwhile not(gcd(e,phi)=1) do e:=nextprime(random()) \+ od;\nd:=e^(-1) mod phi;\nNULL end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Computation of keys" }{MPLTEXT 1 0