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A pr\303\255mek el oszl\303\241sa, szit\303\241l\303\241s" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 63 "2. Egyszer\305\261 faktoriz\303\241l\303\241si m\303\263 dszerek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 64 "3. Egyszer\305\261 pr\303\255mtesztel\303\251si m\303\26 3dszerek" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 18 "4. Lucas-sorozatok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 23 "5. Alkalmaz\30 3\241sok " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 36 "6. Sz\303\241mok \303\251s polinomok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 45 "7. Gyors Four ier-transzform\303\241ci\303\263" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 38 "8. Elliptikus f\303\274ggv\303\2 51nyek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 59 "9. Sz\303\241mol\303\241s elliptikus g\303\266rb\303\251 ken" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 65 "10. Faktoriz\303\241l\303\241s elliptikus g\303\274rb\303\251 kkel" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 55 "11. Pr\303\255mteszt elliptikus g\303\266rb\303\251kkel" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 37 "12. Polinomfaktoriz\303\241l \303\241s" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 16 "13. Az AKS-teszt" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 36 "14. A szita m\303\263dszerek \+ alapjai" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "restart; with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfracGF$I)cfracpolGF$I+cyclot omicGF$I)divisorsGF$I)factorEQGF$I*factorsetGF$I'fermatGF$I)imagunitGF $I&indexGF$I/integral_basisGF$I)invcfracGF$I'invphiGF$I*issqrfreeGF$I' jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendreGF$I)mcombineGF$I)mersenne GF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%mlogGF$I'mobiusGF$I&mrootGF $I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthdenomGF$I)nthnumerGF$I'nthp owGF$I&orderG%*protectedGI)pdexpandGF$I$phiGF$I#piGF$I*pprimrootGF$I)p rimrootGF$I(quadresGF$I+rootsunityGF$I*safeprimeGF$I&sigmaGF$I*sq2fact orGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 13 "14.1. Dixon v" }{TEXT 206 8 "\303\251" }{TEXT 206 8 "letlen n" }{TEXT 206 8 "\303\251" }{TEXT 206 7 "gyzet m" }{TEXT 206 8 "\303\263 " }{TEXT 206 7 "dszere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "n:=nextprime(7*10^2)*prevprime(14*1 0^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'*p!)*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 15 "B:=20; F:=[];\n" }{MPLTEXT 1 0 27 "for j from \+ 2 while j" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rnd:=rand(2..n-2); rnd(); ifactors(%^2 mod n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6\"F#F#F#,&-f*F#F#6#I(builtinGF#F#\"$$ RF#F#F#I%FAILG%*protectedG6%\"\"'\"''p!)*\"#?\"\"\"\"\"#F0F#F#F#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"'-N6" }}{PARA 11 "" 1 "" {XPPMATH 20 " 7$\"\"\"7%7$\"\"#F&7$\"\"&F#7$\"&(47F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "R:=[];\n" }{MPLTEXT 1 0 28 "while nops(R)=B then next e lse R:=[op(R),[x,y]] fi;\n" }{MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }} {PARA 11 "" 1 "" {XPPMATH 20 "7/7$\"'d[%)7'7$\"\"#F'7$\"\"&\"\"\"7$\" \"(F'7$\"#8F*7$\"#F*7$\"'&>d#7&7$F'\"\"*7$\"\"$F*7$F,F*F-7$\"'Cu()7&7$F'F,FB7$F)FCF- 7$\"$?#7%7$F'\"\"%7$F)F'7$F6F'7$\"'Z(3*7%F47$F6FC7$F " 0 "" {MPLTEXT 1 0 558 "Rc:=[[21475, [[3, 2], [5, 1], [17, 2], [19, 1]]], [855183, [[2, 4], [3, 4], [5, 1], [7, 2]]], [164912, [ [3, 1], [5, 2], [11, 1], [13, 1], [19, 1]]], [728436, [[2, 1], [3, 2], [11, 1], [13, 1], [17, 1]]], [362222, [[3, 2], [19, 1]]], [297430, [[ 5, 1], [19, 4]]], [744161, [[3, 3], [5, 1], [11, 1], [13, 1], [19, 1]] ], [495370, [[2, 8], [3, 6], [5, 1]]], [577106, [[2, 1], [11, 1], [13, 2], [19, 1]]], [699549, [[7, 4]]], [689811, [[5, 4], [13, 1], [17, 1] ]], [695704, [[3, 2], [5, 1], [11, 4]]], [315384, [[2, 4], [3, 1], [5, 1], [11, 1], [13, 1], [19, 1]]]];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7/ 7$\"&v9#7&7$\"\"$\"\"#7$\"\"&\"\"\"7$\"#F+7$\"'$=b)7&7$F(\"\" %7$F'F4F)7$\"\"(F(7$\"'7\\;7'7$F'F+7$F*F(7$\"#6F+7$\"#8F+F.7$\"'O%G(7' 7$F(F+F&F=F?7$F-F+7$\"'AAO7$F&F.7$\"'IuH7$F)7$F/F47$\"'hTu7'7$F'F'F)F= F?F.7$\"'q`\\7%7$F(\"\")7$F'\"\"'F)7$\"'1rd7&FDF=7$F@F(F.7$\"'\\&*p7#7 $F7F47$\"'6)*o7%7$F*F4F?FE7$\"'/dp7%F&F)7$F>F47$\"'%Q:$7(F3F;F)F=F?F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7^rI.BlockDiagonalG6\"I,GramSchmidtGF$I,Jordan BlockGF$I)LUdecompGF$I)QRdecompGF$I*WronskianGF$I'addcolGF$I'addrowGF$ I$adjGF$I(adjointGF$I&angleGF$I(augmentGF$I(backsubGF$I%bandGF$I&basis GF$I'bezoutGF$I,blockmatrixGF$I(charmatGF$I)charpolyGF$I)choleskyGF$I$ colGF$I'coldimGF$I)colspaceGF$I(colspanGF$I*companionGF$I'concatGF$I%c ondGF$I)copyintoGF$I*crossprodGF$I%curlGF$I)definiteGF$I(delcolsGF$I(d elrowsGF$I$detGF$I%diagGF$I(divergeGF$I(dotprodGF$I*eigenvalsGF$I,eige nvaluesGF$I-eigenvectorsGF$I+eigenvectsGF$I,entermatrixGF$I&equalGF$I, exponentialGF$I'extendGF$I,ffgausselimGF$I*fibonacciGF$I+forwardsubGF$ I*frobeniusGF$I*gausselimGF$I*gaussjordGF$I(geneqnsGF$I*genmatrixGF$I% gradGF$I)hadamardGF$I(hermiteGF$I(hessianGF$I(hilbertGF$I+htransposeGF $I)ihermiteGF$I*indexfuncGF$I*innerprodGF$I)intbasisGF$I(inverseGF$I'i smithGF$I*issimilarGF$I'iszeroGF$I)jacobianGF$I'jordanGF$I'kernelGF$I* laplacianGF$I*leastsqrsGF$I)linsolveGF$I'mataddGF$I'matrixGF$I&minorGF $I(minpolyGF$I'mulcolGF$I'mulrowGF$I)multiplyGF$I%normGF$I*normalizeGF $I*nullspaceGF$I'orthogGF$I*permanentGF$I&pivotGF$I*potentialGF$I+rand matrixGF$I+randvectorGF$I%rankGF$I(ratformGF$I$rowGF$I'rowdimGF$I)rows paceGF$I(rowspanGF$I%rrefGF$I*scalarmulGF$I-singularvalsGF$I&smithGF$I ,stackmatrixGF$I*submatrixGF$I*subvectorGF$I)sumbasisGF$I(swapcolGF$I( swaprowGF$I*sylvesterGF$I)toeplitzGF$I&traceGF$I*transposeGF$I,vanderm ondeGF$I*vecpotentGF$I(vectdimGF$I'vectorGF$I*wronskianGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "RM:=matrix(nops(R),nops(F),0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q#RMF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6/-I$mtrGF$6 *-I$mtdGF$6#-I#mnGF$6$Q\"0F'F9F]oF]oF]oF]oF]oF]oF]oFjnFjnFjnFjnFjnFjnF jnFjnFjnFjnFjnFjn/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i to nops(Rc) do\n" }{MPLTEXT 1 0 16 " v:= Rc[i][2];\n" }{MPLTEXT 1 0 23 " for j to nops(v) do\n" }{MPLTEXT 1 0 15 " vv:=v[j];\n" }{MPLTEXT 1 0 66 " for k to nops(F) do if vv[1 ]=F[k] then RM[i,k]:=vv[2] fi od;\n" }{MPLTEXT 1 0 7 " od;\n" } {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "prin t(RM);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\" I,TypesettingGI(_syslibGF'6%-I%mrowGF$6#-I'mtableGF$6/-I$mtrGF$6*-I$mt dGF$6#-I#mnGF$6$Q\"0F'/%,mathvariantGQ'normalF'-F56#-F86$Q\"2F'F;-F56# -F86$Q\"1F'F;F4F4F4F>FC-F26*-F56#-F86$Q\"4F'F;FJFCF>F4F4F4F4-F26*F4FCF >F4FCFCF4FC-F26*FCF>F4F4FCFCFCF4-F26*F4F>F4F4F4F4F4FC-F26*F4F4FCF4F4F4 F4FJ-F26*F4-F56#-F86$Q\"3F'F;FCF4FCFCF4FC-F26*-F56#-F86$Q\"8F'F;-F56#- F86$Q\"6F'F;FCF4F4F4F4F4-F26*FCF4F4F4FCF>F4FC-F26*F4F4F4FJF4F4F4F4-F26 *F4F4FJF4F4FCFCF4-F26*F4F>FCF4FJF4F4F4-F26*FJFCFCF4FCFCF4FC/%%openGQ\" [F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "x:=Rc [10][1]; y:=7^2; x^2-y^2 mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"' \\&*p" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "igcd( n,x-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%*R\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "RM:=addrow(RM,4,9,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q#RMF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#: =F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symm etricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/ %'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)i nfinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6/-I$mtrGF$6*-I$mtdGF$6#-I#mn GF$6$Q\"0F'F9-F^o6#-Fao6$Q\"2F'F9-F^o6#-Fao6$FTF9F]oF]oF]oFdoFio-F[o6* -F^o6#-Fao6$Q\"4F'F9F_pFioFdoF]oF]oF]oF]o-F[o6*F]oFioFdoF]oFioFioF]oFi o-F[o6*FioFdoF]oF]oFioFioFioF]o-F[o6*F]oFdoF]oF]oF]oF]oF]oFio-F[o6*F]o F]oFioF]oF]oF]oF]oF_p-F[o6*F]o-F^o6#-Fao6$Q\"3F'F9FioF]oFioFioF]oFio-F [o6*-F^o6#-Fao6$Q\"8F'F9-F^o6#-Fao6$Q\"6F'F9FioF]oF]oF]oF]oF]o-F[o6*Fd oFdoF]oF]oFdoF^qFioFio-F[o6*F]oF]oF]oF_pF]oF]oF]oF]o-F[o6*F]oF]oF_pF]o F]oFioFioF]o-F[o6*F]oFdoFioF]oF_pF]oF]oF]o-F[o6*F_pFioFioF]oFioFioF]oF io/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "RM:=addrow(RM,3,7,1): RM:=addrow(RM,3,13,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6%Q#RMF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60 Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*s ymmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infix F'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeG Q)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6/-I$mtrGF$6*-I$mtdGF$6#-I #mnGF$6$Q\"0F'F9-F^o6#-Fao6$Q\"2F'F9-F^o6#-Fao6$FTF9F]oF]oF]oFdoFio-F[ o6*-F^o6#-Fao6$Q\"4F'F9F_pFioFdoF]oF]oF]oF]o-F[o6*F]oFioFdoF]oFioFioF] oFio-F[o6*FioFdoF]oF]oFioFioFioF]o-F[o6*F]oFdoF]oF]oF]oF]oF]oFio-F[o6* F]oF]oFioF]oF]oF]oF]oF_p-F[o6*F]oF_p-F^o6#-Fao6$Q\"3F'F9F]oFdoFdoF]oFd o-F[o6*-F^o6#-Fao6$Q\"8F'F9-F^o6#-Fao6$Q\"6F'F9FioF]oF]oF]oF]oF]o-F[o6 *FdoFdoF]oF]oFdoF^qFioFio-F[o6*F]oF]oF]oF_pF]oF]oF]oF]o-F[o6*F]oF]oF_p F]oF]oFioFioF]o-F[o6*F]oFdoFioF]oF_pF]oF]oF]o-F[o6*F_pFdoF^qF]oFdoFdoF ]oFdo/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "RM:=addrow(RM,2,1,1): RM:=addrow(RM,2,6,1): RM:=addro w(RM,2,7,1):\n" }{MPLTEXT 1 0 66 "RM:=addrow(RM,2,8,1): RM:=addrow(RM, 2,12,1):RM:=addrow(RM,2,13,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6%Q#RMF'/%'it alicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%& fenceGQ&falseF'/%*separatorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopG F=/%.movablelimitsGF=/%'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmat hspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mfenced GF$6%-F#6#-I'mtableGF$6/-I$mtrGF$6*-I$mtdGF$6#-I#mnGF$6$Q\"4F'F9-F^o6# -Fao6$Q\"6F'F9-F^o6#-Fao6$Q\"2F'F9Fio-F^o6#-Fao6$Q\"0F'F9F^pFio-F^o6#- Fao6$FTF9-F[o6*F]oF]oFcpFioF^pF^pF^pF^p-F[o6*F^pFcpFioF^pFcpFcpF^pFcp- F[o6*FcpFioF^pF^pFcpFcpFcpF^p-F[o6*F^pFioF^pF^pF^pF^pF^pFcp-F[o6*F]oF] oFioFioF^pF^pF^pF]o-F[o6*F]o-F^o6#-Fao6$Q\"8F'F9F]oFioFioFioF^pFio-F[o 6*-F^o6#-Fao6$Q#12F'F9-F^o6#-Fao6$Q#10F'F9FioFioF^pF^pF^pF^p-F[o6*FioF ioF^pF^pFio-F^o6#-Fao6$Q\"3F'F9FcpFcp-F[o6*F^pF^pF^pF]oF^pF^pF^pF^p-F[ o6*F^pF^pF]oF^pF^pFcpFcpF^p-F[o6*F]oFdoFioFioF]oF^pF^pF^p-F[o6*FcqFdoF ]oFioFioFioF^pFio/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "RM:=addrow(RM,1,5,1): RM:=addrow(RM,1,9,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6%Q#RMF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)s tretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/ %%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\" 1F'/%(maxsizeGQ)infinityF'-I(mfencedGF$6%-F#6#-I'mtableGF$6/-I$mtrGF$6 *-I$mtdGF$6#-I#mnGF$6$Q\"4F'F9-F^o6#-Fao6$Q\"6F'F9-F^o6#-Fao6$Q\"2F'F9 Fio-F^o6#-Fao6$Q\"0F'F9F^pFio-F^o6#-Fao6$FTF9-F[o6*F]oF]oFcpFioF^pF^pF ^pF^p-F[o6*F^pFcpFioF^pFcpFcpF^pFcp-F[o6*FcpFioF^pF^pFcpFcpFcpF^p-F[o6 *F]o-F^o6#-Fao6$Q\"8F'F9FioFioF^pF^pFioFio-F[o6*F]oF]oFioFioF^pF^pF^pF ]o-F[o6*F]oF_qF]oFioFioFioF^pFio-F[o6*-F^o6#-Fao6$Q#12F'F9-F^o6#-Fao6$ Q#10F'F9FioFioF^pF^pF^pF^p-F[o6*FdoF_qFioFioFio-F^o6#-Fao6$Q\"3F'F9Ffr Fio-F[o6*F^pF^pF^pF]oF^pF^pF^pF^p-F[o6*F^pF^pF]oF^pF^pFcpFcpF^p-F[o6*F ]oFdoFioFioF]oF^pF^pF^p-F[o6*F_qFdoF]oFioFioFioF^pFio/%%openGQ\"[F'/%& closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "x:=Rc[5][1] *(Rc[1][1]*Rc[2][1]) mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'B2s" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "y:=2^2*3^4*5*7*17*19 mod n ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'B2s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x^2-y^2 mod n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" !" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "igcd(n,x-y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'*p!)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "interface(echo=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read(\"../../old/factor/fa ctor\");" }}{PARA 6 "" 1 "" {TEXT 207 18 "> with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfracGF$I)cfracpol GF$I+cyclotomicGF$I)divisorsGF$I)factorEQGF$I*factorsetGF$I'fermatGF$I )imagunitGF$I&indexGF$I/integral_basisGF$I)invcfracGF$I'invphiGF$I*iss qrfreeGF$I'jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendreGF$I)mcombineGF $I)mersenneGF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%mlogGF$I'mobiusG F$I&mrootGF$I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthdenomGF$I)nthnum erGF$I'nthpowGF$I&orderG%*protectedGI)pdexpandGF$I$phiGF$I#piGF$I*ppri mrootGF$I)primrootGF$I(quadresGF$I+rootsunityGF$I*safeprimeGF$I&sigmaG F$I*sq2factorGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}{PARA 6 "" 1 "" {TEXT 207 4 "> \n" }{TEXT 207 3 "#\n" }{TEXT 207 42 "# This procedure set u p the factor base.\n" }{TEXT 207 35 "# The factor base is a list which \n" }{TEXT 207 39 "# always start with -1, has length N,\n" }{TEXT 207 36 "# and contains in incerasing order\n" }{TEXT 207 35 "# the pri mes p for which (n|p)=1.\n" }{TEXT 207 3 "#\n" }{TEXT 207 4 "> \n" } {TEXT 207 39 "> setfactorbase:=proc(n,N) local F,p;\n" }{TEXT 207 20 " > F:=[-1];\n" }{TEXT 207 17 "> p:=2;\n" }{TEXT 207 30 "> while nops(F) if j acobi(n,p)=1 then F:=[op(F),p]; fi;\n" }{TEXT 207 36 "> \+ p:=nextprime(p);\n" }{TEXT 207 15 "> od;\n" }{TEXT 207 14 "> F;\n" }{TEXT 207 6 "> end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f *6$I\"nG6\"I\"NGF%6$I\"FGF%I\"pGF%F%F%C&>F(7#!\"\">F)\"\"#?(F%\"\"\"F1 F%2-I%nopsG%*protectedG6#F(F&C$@$/-_I*numtheoryG6$F5I(_syslibGF%I'jaco biGF%6$F$F)F1>F(7$-I#opGF5F6F)>F)-I*nextprimeGF%6#F)F(F%F%F%" }}{PARA 6 "" 1 "" {TEXT 207 4 "> \n" }{TEXT 207 3 "#\n" }{TEXT 207 45 "# The f ollowing procedure is a naive way to\n" }{TEXT 207 46 "# look for a fu ll over a given factor base F\n" }{TEXT 207 41 "# starting with -1. Si mple the number x\n" }{TEXT 207 25 "# is trial divided with\n" }{TEXT 207 41 "# the members of the factor base up to \n" }{TEXT 207 42 "# th e end of the factor base. The result\n" }{TEXT 207 41 "# is a list wit h two members. The first\n" }{TEXT 207 45 "# is the unfactored part, t he other a list.\n" }{TEXT 207 49 "# The members of this list are two- member lists\n" }{TEXT 207 45 "# containing index into the factor base and\n" }{TEXT 207 45 "# exponent of the given factor base member \n" }{TEXT 207 25 "# in the factorization.\n" }{TEXT 207 3 "#\n" }{TEXT 207 4 "> \n" }{TEXT 207 41 "> trialfull:=proc(x,F) local y,p,e,L,i;\n" }{TEXT 207 18 "> L:=[];\n" }{TEXT 207 17 "> y:=x;\n" }{TEXT 207 46 "> if y<0 then L:=[[1,1]]; y:=-y; fi;\n" }{TEXT 207 48 "> for i from 2 to nops(F) while y>1 do\n" }{TEXT 207 28 "> p:=F[i];\n" }{TEXT 207 39 "> if \+ y mod p = 0 then\n" }{TEXT 207 33 "> e:=0;\n" }{TEXT 207 48 "> while y mod p = 0 do\n" } {TEXT 207 43 "> e:=e+1;\n" }{TEXT 207 43 "> y:=y/p;\n" }{TEXT 207 31 "> \+ od;\n" }{TEXT 207 45 "> L: =[op(L),[i,e]];\n" }{TEXT 207 23 "> fi;\n" }{TEXT 207 15 "> od;\n" }{TEXT 207 18 "> [y,L];\n" }{TEXT 207 6 " > end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"FGF%6'I\"yGF%I \"pGF%I\"eGF%I\"LGF%I\"iGF%F%F%C'>F+7\">F(F$@$2F(\"\"!C$>F+7#7$\"\"\"F 8>F(,$F(!\"\"?(F,\"\"#F8-I%nopsG%*protectedG6#F&2F8F(C$>F)&F&6#F,@$/-I $modGF%6$F(F)F3C%>F*F3?(F%F8F8F%FHC$>F*,&F*F8F8F8>F(*&F(F8F)F;>F+7$-I# opGF@6#F+7$F,F*7$F(F+F%F%F%" }}{PARA 6 "" 1 "" {TEXT 207 4 "> \n" } {TEXT 207 3 "#\n" }{TEXT 207 45 "# The following procedure is a naive \+ way to\n" }{TEXT 207 37 "# find full's for a given composite\n" }{TEXT 207 33 "# number n over a factor base F\n" }{TEXT 207 40 "# starting \+ with -1. Simple the numbers\n" }{TEXT 207 37 "# around sqrt(n) trial d ivided with\n" }{TEXT 207 40 "# the members of the factor base until\n " }{TEXT 207 42 "# N full factorizations have been found.\n" }{TEXT 207 3 "#\n" }{TEXT 207 4 "> \n" }{TEXT 207 42 "> trialfulls:=proc(n,F, N) local x,i,R,L;\n" }{TEXT 207 20 "> R:=\{\};\n" }{TEXT 207 37 "> x:=floor(evalf(sqrt(n)));\n" }{TEXT 207 34 "> L: =trialfull(x^2-n,F);\n" }{TEXT 207 46 "> if L[1]=1 then R:=\{[ x,L[2]]\} fi;\n" }{TEXT 207 17 "> i:=1;\n" }{TEXT 207 30 "> \+ while nops(R) L:=trialf ull((x+i)^2-n,F);\n" }{TEXT 207 64 "> if L[1]=1 then R :=R union \{[x+i,L[2]]\} fi;\n" }{TEXT 207 55 "> if i> 0 then i:=-i else i:=-i+1; fi;\n" }{TEXT 207 15 "> od;\n" } {TEXT 207 14 "> R;\n" }{TEXT 207 6 "> end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"nG6\"I\"FGF%I\"NGF%6&I\"xGF%I\"iGF%I\"RGF%I\"LGF% F%F%C)>F+<\">F)-I&floorGF%6#-I&evalfG%*protectedG6#-I%sqrtGF%6#F$>F,-I *trialfullGF%6$,&*$)F)\"\"#\"\"\"FCF$!\"\"F&@$/&F,6#FCFC>F+<#7$F)&F,6# FB>F*FC?(F%FCFCF%2-I%nopsGF66#F+F'C%>F,-F=6$,&*$),&F)FCF*FCFBFCFCF$FDF &@$FF>F+-I&unionGF66$F+<#7$FenFL@%2\"\"!F*>F*,$F*FD>F*,&F*FDFCFCF+F%F% F%" }}{PARA 6 "" 1 "" {TEXT 207 12 "> \n" }{TEXT 207 3 "#\n" } {TEXT 207 45 "# This procedure set up the equation system\n" }{TEXT 207 44 "# from the fulls. The cardinality N of the\n" }{TEXT 207 43 "# factor base have to given. R is the set\n" }{TEXT 207 22 "# of full r elations.\n" }{TEXT 207 3 "#\n" }{TEXT 207 4 "> \n" }{TEXT 207 41 "> s etequations:=proc(R,N) local T,L,LL;\n" }{TEXT 207 24 "> T:=ar ray(N);\n" }{TEXT 207 24 "> LL:=[op(L)];\n" }{TEXT 207 32 "> \+ for i to nops(LL) do\n" }{TEXT 207 18 "> " }} {PARA 8 "" 1 "" {TEXT 208 42 "Error, on line 90, unexpected end of inp ut" }}{PARA 8 "" 1 "" {TEXT 208 48 "Error, while reading ``../../old/f actor/factor``" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" } {MPLTEXT 1 0 42 "# This procedure set up the factor base.\n" }{MPLTEXT 1 0 35 "# The factor base is a list which\n" }{MPLTEXT 1 0 39 "# alwa ys start with -1, has length N,\n" }{MPLTEXT 1 0 36 "# and contains in incerasing order\n" }{MPLTEXT 1 0 35 "# the primes p for which (n|p)= 1.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 37 "set factorbase:=proc(n,N) local F,p;\n" }{MPLTEXT 1 0 16 "F:=[-1]; p:=2;\n " }{MPLTEXT 1 0 20 "while nops(F)F(7#!\"\">F)\"\"#?(F%\"\"\"F1F %2-I%nopsG%*protectedG6#F(F&C$@$/-_I*numtheoryG6$F5I(_syslibGF%I'jacob iGF%6$F$F)F1>F(7$-I#opGF5F6F)>F)-I*nextprimeGF=6#F)F(F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 45 "# The \+ following procedure is a naive way to\n" }{MPLTEXT 1 0 46 "# look for \+ a full over a given factor base F\n" }{MPLTEXT 1 0 41 "# starting with -1. Simple the number x\n" }{MPLTEXT 1 0 25 "# is trial divided with \n" }{MPLTEXT 1 0 41 "# the members of the factor base up to \n" } {MPLTEXT 1 0 42 "# the end of the factor base. The result\n" }{MPLTEXT 1 0 41 "# is a list with two members. The first\n" }{MPLTEXT 1 0 45 " # is the unfactored part, the other a list.\n" }{MPLTEXT 1 0 49 "# The members of this list are two-member lists\n" }{MPLTEXT 1 0 45 "# cont aining index into the factor base and\n" }{MPLTEXT 1 0 45 "# exponent \+ of the given factor base member \n" }{MPLTEXT 1 0 25 "# in the factori zation.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "trialfull:=proc(x,F) local y,p,e,L,i;\n" }{MPLTEXT 1 0 14 "L:=[]; \+ y:=x;\n" }{MPLTEXT 1 0 36 "if y<0 then L:=[[1,1]]; y:=-y; fi;\n" } {MPLTEXT 1 0 38 "for i from 2 to nops(F) while y>1 do\n" }{MPLTEXT 1 0 12 " p:=F[i];\n" }{MPLTEXT 1 0 29 " if y mod p = 0 then e:=0;\n" } {MPLTEXT 1 0 46 " while y mod p = 0 do e:=e+1; y:=y/p; od;\n" } {MPLTEXT 1 0 23 " L:=[op(L),[i,e]];\n" }{MPLTEXT 1 0 7 " fi;\n" } {MPLTEXT 1 0 15 "od; [y,L]; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$ I\"xG6\"I\"FGF%6'I\"yGF%I\"pGF%I\"eGF%I\"LGF%I\"iGF%F%F%C'>F+7\">F(F$@ $2F(\"\"!C$>F+7#7$\"\"\"F8>F(,$F(!\"\"?(F,\"\"#F8-I%nopsG%*protectedG6 #F&2F8F(C$>F)&F&6#F,@$/-I$modGF%6$F(F)F3C%>F*F3?(F%F8F8F%FHC$>F*,&F*F8 F8F8>F(*&F(F8F)F;>F+7$-I#opGF@6#F+7$F,F*7$F(F+F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 45 "# The following pro cedure is a naive way to\n" }{MPLTEXT 1 0 37 "# find full's for a give n composite\n" }{MPLTEXT 1 0 33 "# number n over a factor base F\n" } {MPLTEXT 1 0 40 "# starting with -1. Simple the numbers\n" }{MPLTEXT 1 0 37 "# around sqrt(n) trial divided with\n" }{MPLTEXT 1 0 40 "# the members of the factor base until\n" }{MPLTEXT 1 0 42 "# N full factor izations have been found.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n " }{MPLTEXT 1 0 40 "trialfulls:=proc(n,F,N) local x,i,R,L;\n" } {MPLTEXT 1 0 36 "R:=\{\}; x:=floor(evalf(sqrt(n)));\n" }{MPLTEXT 1 0 24 "L:=trialfull(x^2-n,F);\n" }{MPLTEXT 1 0 36 "if L[1]=1 then R:=\{[x ,L[2]]\} fi;\n" }{MPLTEXT 1 0 7 "i:=1;\n" }{MPLTEXT 1 0 20 "while nops (R)0 then i:=-i else i:=-i+1; fi;\n" }{MPLTEXT 1 0 11 "od; R; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"nG6\"I\"FGF% I\"NGF%6&I\"xGF%I\"iGF%I\"RGF%I\"LGF%F%F%C)>F+<\">F)-I&floorG6$%*prote ctedGI(_syslibGF%6#-I&evalfGF46#-I%sqrtGF36#F$>F,-I*trialfullGF%6$,&*$ )F)\"\"#\"\"\"FEF$!\"\"F&@$/&F,6#FEFE>F+<#7$F)&F,6#FD>F*FE?(F%FEFEF%2- I%nopsGF46#F+F'C%>F,-F?6$,&*$),&F)FEF*FEFDFEFEF$FFF&@$FH>F+-I&unionGF4 6$F+<#7$FgnFN@%2\"\"!F*>F*,$F*FF>F*,&F*FFFEFEF+F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n; F:=setfactorbase(n,10);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"'*p!)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "7,!\"\"\"\"&\"#6\"#B\"#H\"#J\"#T\"#V\"#`\"#f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "trialfulls(n,F,11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<-7$\"%o67%7$\"\"#F'7$\"\"%F'7$\"\"&\"\"\"7$\"%K:7&F&7$\" \"'F,7$\"\"(F,7$\"\")F,7$\"$K&7'7$F,F,F&7$\"\"$F,F47$\"#5F,7$\"$G*7'F9 7$F'F,F:F27$\"\"*F,7$\"$o(7&F97$F'F;FBF<7$\"%557$7$F1F'F27$\"$w)7&F97$ F;F'F2F47$\"$%**7%F:7$F)F,F*7$\"%A67&FAFSF2F<7$\"$7*7%F9FA7$F1F;7$\"$[ *7&F9FAF(F0" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 7 "14.2. L" }{TEXT 206 8 "\303\241" }{TEXT 206 3 "nct" }{TEXT 206 8 "\303\266" }{TEXT 206 5 "rtek." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 26 "14.3. Kvadratik us irracion" }{TEXT 206 8 "\303\241" }{TEXT 206 6 "lis sz" }{TEXT 206 8 "\303\241" }{TEXT 206 5 "mok l" }{TEXT 206 8 "\303\241" }{TEXT 206 3 "nct" }{TEXT 206 8 "\303\266" }{TEXT 206 10 "rt alakja." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "14.4. Fak" }{TEXT 206 33 "toriz\303\241l \303\241s l\303\241" }{TEXT 206 3 "nct" }{TEXT 206 8 "\303\266" }{TEXT 206 8 "rtekkel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 7 "14. 5. N" }{TEXT 206 8 "\303\251" }{TEXT 206 14 "gyzetes szita." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "n: =nextprime(7*10^5)*prevprime(14*10^5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"-****p++)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n mod 8; \+ n:=23*n; n mod 8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"/x**4;+aA" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "m:=2000; B:=50; T:=2. ; b:=ceil(sqrt(n));\n" }{MPLTEXT 1 0 39 "F:=[[2,floor(0.5+2.*log[2.](2 )),1]]; \n" }{MPLTEXT 1 0 27 "for j from 3 while jF* \"\"'7%\"#HF+\"#77%\"#JF+F17%\"#PF+\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "S:= rtable(0.. 2*m,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6%-I#miGF$6%Q\"SF'/%'italicGQ%trueF'/%,mathvar iantGQ'italicF'-I#moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separ atorGF=/%)stretchyGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/% 'accentGF=/%%formGQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%( minsizeGQ\"1F'/%(maxsizeGQ)infinityF'-I(mactionGF$6$-I(mfencedGF$6%-I' mtableGF$6&-I$mtrGF$6#-I$mtdGF$6#-F#6$-F,6%Q,~0~..~4000~F'F/F2-F,6%Q&A rrayF'F/F2-F\\o6#-F_o6#-F#6$-F,6%Q,Data~Type:~F'F/F2-F,6%Q)anythingF'F /F2-F\\o6#-F_o6#-F#6$-F,6%Q*Storage:~F'F/F2-F,6%Q,rectangularF'F/F2-F \\o6#-F_o6#-F#6$-F,6%Q(Order:~F'F/F2-F,6%Q.Fortran_orderF'F/F2/%%openG Q\"[F'/%&closeGQ\"]F'/%+actiontypeGQ-browsertableF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "p:=F[1][1]; lp:=F[1][2]; x:=modp(m-b+F[1][ 3],p);\n" }{MPLTEXT 1 0 42 "while x<=2*m do S[x]:=S[x]+lp; x:=x+p; od: " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for j from 2 to nops(F) do\n" }{MPLTEXT 1 0 37 " ppp:=F[j]; p:=ppp[1]; lp:=ppp[2];\n" }{MPLTEXT 1 0 26 " x:=modp (m-b+ppp[3],p);\n" }{MPLTEXT 1 0 46 " while x<=2*m do S[x]:=S[x]+lp; \+ x:=x+p; od:\n" }{MPLTEXT 1 0 26 " x:=modp(m-b-ppp[3],p);\n" }{MPLTEXT 1 0 46 " while x<=2*m do S[x]:=S[x]+lp; x:=x+p; od:\n" }{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "R:=[]; TT:=floo r(log[2.](2*m*b)/T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for j from 0 to 2*m do if S[j]>=TT then R:=[op(R),j-m] fi od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 "7,!%+=!%/6!$%e!$K%!$;\"!\"#\"%m8\"%1:\"%9<\"%M<" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "map(y->ifactors((y+b)^2-n),R );" }}{PARA 11 "" 1 "" {XPPMATH 20 "7,7$!\"\"7)7$\"\"#\"\"$7$F(\"\"\"7 $\"#>F*7$\"#HF*7$\"#JF*7$\"#rF*7$\"$(eF*7$F$7(F&F)7$\"#8F*F+F-7$\"&`4' F*7$F$7)F&F)F7F+F/7$\"#`F*7$\"$p&F*7$F$7(F&F)F+F-7$\"#PF*7$\"%x$)F*7$F $7(7$F'\"\"(F)F7F+FC7$\"$8$F*7$F$7(F&7$F(F'F7F+F-F/7$F*7)F&FOF7F+F-7$ \"$R\"F*7$\"$\"=F*7$F*7)7$F'\"\"&F)F7F+F-F17$\"$$HF*7$F*7(7$F'\"\"'F)F 7F-FC7$\"%zgF*7$F*7(F&7$F(F(F+F/FC7$\"%*\\$F*" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 7 "14.6. T" }{TEXT 206 8 "\303\266" }{TEXT 206 13 "bbpo linomos n" }{TEXT 206 8 "\303\251" }{TEXT 206 14 "gyzetes szita." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "n:=nextprime(7*10^7)*prevprime(14*10^7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"1d(***\\J++)*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n mod 8; n:=37*n; n mod 8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" &" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"345**\\l6+EO" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "m:=1 0000; B:=100; T:=1.5;\n" }{MPLTEXT 1 0 39 "F:=[[2,floor(0.5+2.*log[2.] (2)),1]]; \n" }{MPLTEXT 1 0 27 "for j from 3 while jF-F*7%\"#BF'\"#67%\"#HF'F27%\" #TF'\"#97%\"#Z\"\"'F-7%\"#`F:F27%\"#fF:\"#A7%\"#rF:\"\")7%\"#(*F*\"#n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dd:=floor((2*n/m^2)^(1/4.)): if type(dd,odd) then dd: =dd+1; fi:\n" }{MPLTEXT 1 0 11 "dd:=dd..dd;" }}{PARA 11 "" 1 "" {XPPMATH 20 ";\"$#HF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "if abs(op(1,dd)-(2*n/m^2)^(1/4.))3 or jacobi(d,n)<>1 do\n" }{MPLTEXT 1 0 22 " d:=prev prime(d);\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 20 " dd:=d..op(2 ,dd);\n" }{MPLTEXT 1 0 6 "else\n" }{MPLTEXT 1 0 27 " d:=nextprime(op( 2,dd));\n" }{MPLTEXT 1 0 43 " while modp(d,4)<>3 or jacobi(d,n)<>1 do \n" }{MPLTEXT 1 0 22 " d:=nextprime(d);\n" }{MPLTEXT 1 0 7 " od;\n " }{MPLTEXT 1 0 20 " dd:=op(1,dd)..d;\n" }{MPLTEXT 1 0 5 "fi:\n" } {MPLTEXT 1 0 6 "d; dd;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$6$" }} {PARA 11 "" 1 "" {XPPMATH 20 ";\"$r#\"$6$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a:=d^2; h0:=n&^((d-3)/4) mod d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&@n*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#`" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "h1:=n*h0 mod d; (n-h1^2)/d; h2:=%*h 0*((d+1)/2) mod d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$B#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"1![sMx;f;\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" #D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "b:=mods(h1+h2*d,a); b ^2-n mod a; c:=(b^2-n)/a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%)*z" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "! .09`G*[P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "S:= rtable(0..2 *m,0); p:=F[1][1]; lp:=F[1][2]; x:=modp(m+(-b+F[1][3])/a,p);\n" } {MPLTEXT 1 0 42 "while x<=2*m do S[x]:=S[x]+lp; x:=x+p; od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6%-I#miGF$6%Q\"SF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I #moGF$60Q#:=F'/F3Q'normalF'/%&fenceGQ&falseF'/%*separatorGF=/%)stretch yGF=/%*symmetricGF=/%(largeopGF=/%.movablelimitsGF=/%'accentGF=/%%form GQ&infixF'/%'lspaceGQ/thickmathspaceF'/%'rspaceGFO/%(minsizeGQ\"1F'/%( maxsizeGQ)infinityF'-I(mactionGF$6$-I(mfencedGF$6%-I'mtableGF$6&-I$mtr GF$6#-I$mtdGF$6#-F#6$-F,6%Q-~0~..~20000~F'F/F2-F,6%Q&ArrayF'F/F2-F\\o6 #-F_o6#-F#6$-F,6%Q,Data~Type:~F'F/F2-F,6%Q)anythingF'F/F2-F\\o6#-F_o6# -F#6$-F,6%Q*Storage:~F'F/F2-F,6%Q,rectangularF'F/F2-F\\o6#-F_o6#-F#6$- F,6%Q(Order:~F'F/F2-F,6%Q.Fortran_orderF'F/F2/%%openGQ\"[F'/%&closeGQ \"]F'/%+actiontypeGQ-browsertableF'" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "for \+ j from 2 to nops(F) do\n" }{MPLTEXT 1 0 37 " ppp:=F[j]; p:=ppp[1]; lp :=ppp[2];\n" }{MPLTEXT 1 0 31 " x:=modp(m+(-b+ppp[3])/a,p);\n" } {MPLTEXT 1 0 46 " while x<=2*m do S[x]:=S[x]+lp; x:=x+p; od:\n" } {MPLTEXT 1 0 31 " x:=modp(m+(-b-ppp[3])/a,p);\n" }{MPLTEXT 1 0 46 " \+ while x<=2*m do S[x]:=S[x]+lp; x:=x+p; od:\n" }{MPLTEXT 1 0 3 "od:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "R:=[]; TT:=floor(log[2.](a* m^2/2.)/T); " }}{PARA 11 "" 1 "" {XPPMATH 20 "7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for j \+ from 0 to 2*m do if S[j]>=TT then R:=[op(R),j-m] fi od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "R;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 7%\"%:7\"%DE\"%CS" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "map(y- >ifactors(a*y^2+2*b*y+c),R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%7$!\" \"7*7$\"\"#\"\"$7$\"\"&\"\"\"7$\"\"(F+7$\"#>F+7$\"#TF+7$\"#ZF+7$\"#`F+ 7$\"%PmF+7$F$7)7$F'\"#7F)7$\"#BF+7$\"#PF+F2F47$\"#rF+7$F$7*F)F,F.F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "" "%#%?G" }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }