{VERSION 7 1 "Linux" "7.1" } {USTYLETAB {PSTYLE "Ordered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "MS Serif" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 3 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "MS S erif" 1 14 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 12 40 120 40 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered \+ List 4" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "H eading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 12 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered Li st 2" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "Equation Label" -1 200 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "MS Serif" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 0 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 202 "Couri er" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Ser if" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 203 38 "Bevezet\303\251s a matem atik\303\241ba" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 204 18 "J\303\241rai \+ Antal" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 201 69 "Ezek a programok csak \+ szeml\303\251ltet\303\251sre szolg\303\241lnak." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 11 "1. Halmazok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 35 "2. Term\303\251s zetes sz\303\241mok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 53 "3. A sz\303\241mfogalom b\305\221v\303\2 55t\303\251se" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 24 "4. V\303\251ges halmazok" }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 27 "5. V\303\251gtelen halmazok" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 28 "6 . Sz\303\241melm\303\251let" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 30 "6 .1. Oszthat\303\263s\303\241g" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 90 "6.1.1. Oszthat\303\263s\303\241g a term\303\251szetes sz\303\241mok k\303\266r\303\251ben." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "res tart; with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I )bigomegaGF$I&cfracGF$I)cfracpolGF$I+cyclotomicGF$I)divisorsGF$I)facto rEQGF$I*factorsetGF$I'fermatGF$I)imagunitGF$I&indexGF$I/integral_basis GF$I)invcfracGF$I'invphiGF$I*issqrfreeGF$I'jacobiGF$I*kroneckerGF$I'la mbdaGF$I)legendreGF$I)mcombineGF$I)mersenneGF$I(migcdexGF$I*minkowskiG F$I(mipolysGF$I%mlogGF$I'mobiusGF$I&mrootGF$I&msqrtGF$I)nearestpGF$I*n thconverGF$I)nthdenomGF$I)nthnumerGF$I'nthpowGF$I&orderG%*protectedGI) pdexpandGF$I$phiGF$I#piGF$I*pprimrootGF$I)primrootGF$I(quadresGF$I+roo tsunityGF$I*safeprimeGF$I&sigmaGF$I*sq2factorGF$I(sum2sqrGF$I$tauGF$I% thueGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "divisors(60);" } }{PARA 11 "" 1 "" {XPPMATH 20 "<.\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"#5\" #7\"#:\"#?\"#I\"#g" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->6.1.2. Feladat. " }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*6.1.3. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "*6.1.4. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "6.1.5. Az o" }{TEXT 207 25 "szthat\303\263s\303\241g " }{TEXT 207 9 "tulajdons" }{TEXT 207 8 "\303\241" }{TEXT 207 4 "gai " }{TEXT 207 57 "a term\303\251szet es sz\303\241mok k\303\266r\303\251ben." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 68 "6.1. 6. T\303\266rzssz\303\241mok \303\251s pr\303\255msz\303\241mok." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "isprime(16); isprime(17); divisors(17); nextprime(15); prevprime (18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$\"\"\"\"#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#<" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#<" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6. 1.7. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.1.8. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.1.9. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->6.1.10. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.11. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 93 "6.1.12. Oszthat\303\263s\303\241g egys\3 03\251gelemes integrit\303\241si tartom\303\241nyban." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 117 "6.1.13. Az oszthat\303\263s\303\241g tulajdons\303\241gai egy s\303\251gelemes integrit\303\241si tartom\303\241nyban." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 54 "6.1.14. Asszoci\303\241ltak \303\251s egys\303\251gek." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 " " 0 "" {TEXT 207 55 "6.1.15. Felbonthatatlan elem \303\251s pr\303\255 melem." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 86 "6.1.1 6. Oszthat\303\263s\303\241g az eg\303\251sz sz\303\241mok k\303\266r \303\251ben." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "divisors(-60); igcd(12,18); igcd(-12,18); ilcm(12 ,18); ilcm(-12,-18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.\"\"\"\"\"#\" \"$\"\"%\"\"&\"\"'\"#5\"#7\"#:\"#?\"#I\"#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#O" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 43 "6.1.17. P\303\251lda: Gauss-eg\303\251szek." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(GaussInt); GInormal(2-3*I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7@I(GIbasisG6\"I(GIchremGF$I*GIdivisorGF$I*GIfacpolyGF$I)GIfacsetGF$ I)GIfactorGF$I*GIfactorsGF$I&GIgcdGF$I(GIgcdexGF$I*GIhermiteGF$I(GIiss qrGF$I&GIlcmGF$I*GImcmbineGF$I&GImodGF$I*GInearestGF$I(GInodivGF$I'GIn ormGF$I)GInormalGF$I(GIorderGF$I&GIphiGF$I(GIprimeGF$I*GIquadresGF$I&G IquoGF$I&GIremGF$I(GIrootsGF$I(GIsieveGF$I(GIsmithGF$I*GIsqrfreeGF$I'G IsqrtGF$I-GIunitnormalGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\" \"*&\"\"#F$^#F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 131 "6.1.18. Legnagyobb k\303\2 66z\303\266s oszt\303\263, legkisebb k\303\266z\303\266s t\303\266bbsz \303\266r\303\266s, relativ primek." }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->6.1.19. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 " " {TEXT 207 19 "->6.1.20. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->6.1.21. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 19 "->6.1.22. Feladat. " }{TEXT 207 0 "" } }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.23. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 115 "6.1.24. Legnagyobb k\30 3\266z\303\266s oszt\303\263, legkisebb k\303\266z\303\266s t\303\266b bsz\303\266r\303\266s " }{TEXT 207 52 "az eg\303\251sz sz\303\241mok k \303\266r\303\251ben." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 60 "6.1.25. B\305\221v\303\255tett euklid\303\251szi algorit mus." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "exgcd:=proc(a::integer,b::integer) local n,x,y,r,q;\n " }{MPLTEXT 1 0 55 "x[0]:=1;y[0]:=0;r[0]:=a;x[1]:=0;y[1]:=1;r[1]:=b;n: =0;\n" }{MPLTEXT 1 0 47 "do if r[n+1]=0 then return x[n],y[n],r[n] fi; \n" }{MPLTEXT 1 0 58 " q[n+1]:=floor(r[n]/r[n+1]);r[n+2]:=r[n]-q[n+1] *r[n+1];\n" }{MPLTEXT 1 0 58 " x[n+2]:=x[n]-q[n+1]*x[n+1];y[n+2]:=y[n ]-q[n+1]*y[n+1];\n" }{MPLTEXT 1 0 11 " n:=n+1;\n" }{MPLTEXT 1 0 8 "od ; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"aG6\"I(integerG%*prote ctedG'I\"bGF&F'6'I\"nGF&I\"xGF&I\"yGF&I\"rGF&I\"qGF&F&F&C*>&F-6#\"\"! \"\"\">&F.F4F5>&F/F4F%>&F-6#F6F5>&F.F=F6>&F/F=F*>F,F5?(F&F6F6F&I%trueG F(C(@$/&F/6#,&F,F6F6F6F5O6%&F-6#F,&F.FN&F/FN>&F0FI-I&floorGF&6#*&FPF6F H!\"\">&F/6#,&F,F6\"\"#F6,&FPF6*&FRF6FHF6FW>&F-FZ,&FMF6*&FRF6&F-FIF6FW >&F.FZ,&FOF6*&FRF6&F.FIF6FW>F,FJF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "exgcd(12,18); exgcd(-12,-18); igcdex(12,18,'x','y'); \+ x; y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\"\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\"\"\"\"!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "6.1.26. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "exgcd:=proc(a::integer,b::integer) local x0,x1,x2,y0,y1,y2,r0, r1,r2,q;\n" }{MPLTEXT 1 0 38 "x0:=1;y0:=0;r0:=a;x1:=0;y1:=1;r1:=b;\n" }{MPLTEXT 1 0 37 "do if r1=0 then return x0,y0,r0 fi;\n" }{MPLTEXT 1 0 32 " q:=floor(r0/r1);r2:=r0-q*r1;\n" }{MPLTEXT 1 0 28 " x2:=x0-q*x 1;y2:=y0-q*y1;\n" }{MPLTEXT 1 0 46 " r0:=r1;r1:=r2;x0:=x1;x1:=x2;y0:= y1;y1:=y2;\n" }{MPLTEXT 1 0 8 "od; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"aG6\"I(integerG%*protectedG'I\"bGF&F'6,I#x0GF&I#x1GF&I#x2G F&I#y0GF&I#y1GF&I#y2GF&I#r0GF&I#r1GF&I#r2GF&I\"qGF&F&F&C)>F,\"\"\">F/ \"\"!>F2F%>F-F:>F0F8>F3F*?(F&F8F8F&I%trueGF(C-@$/F3F:O6%F,F/F2>F5-I&fl oorG6$F(I(_syslibGF&6#*&F2F8F3!\"\">F4,&F2F8*&F5F8F3F8FM>F.,&F,F8*&F5F 8F-F8FM>F1,&F/F8*&F5F8F0F8FM>F2F3>F3F4>F,F->F-F.>F/F0>F0F1F&F&F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "exgcd(12,18);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"\"\"\"\"\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 67 "6.1. 27. P\303\251lda: b\305\221v\303\255tett euklideszi algoritmus." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "exgcd(172,62);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%!\"*\"#D\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 35 "6.1.28. K\303\266vetkezm\303\251ny." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "igc d(12,18,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 21 "6.1.29. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 " " "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "6.1.30. Megjegyz\3 03\251s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 54 "6.1.31. A sz\303\241melm\303\251let ala pt\303\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "ifactor(720);" }}{PARA 11 "" 1 "" {XPPMATH 20 "*()-I!G6\"6#\"\"#\"\"%\"\"\")-F%6#\"\"$F(F*-F%6#\"\"&F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.32. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.33. Feladat. " }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.34. Feladat. " }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.35. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.36. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.37. Feladat. " } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.38. Felada t. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.1.39. \+ Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6. 1.40. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 19 "->6.1.41. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "6.1.42. Feladat: Lam" }{TEXT 207 8 "\303\251" }{TEXT 207 2 " t" }{TEXT 207 8 "\303\251" }{TEXT 207 5 "tele." }{TEXT 207 0 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.1.43. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.1.44. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "6.1.45. Felada t: bin" }{TEXT 207 8 "\303\241" }{TEXT 207 9 "ris lnko." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 39 "6.1.46. Euklid\303\251sz t \303\251tele." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "P:=2; ifactor(P+1); P:=P*3; ifactor(P+1); P:=P* 5; ifactor(P+1); P:=P*7; ifactor(P+1); P:=P*11; ifactor(P+1); P:=P*13; ifactor(P+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I!G6\"6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"' " }}{PARA 11 "" 1 "" {XPPMATH 20 "-I!G6\"6#\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I!G6\"6#\"#J" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"$5#" }}{PARA 11 "" 1 "" {XPPMATH 20 "- I!G6\"6#\"$6#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%5B" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I!G6\"6#\"%6B" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&I+$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I!G6\"6#\"#f\"\"\"-F$6#\"$4&F(" } }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 26 "6.1.47. Megjegyz\303\251s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "ifactor(P+2 ); ifactor(P+3); ifactor(P+4); ifactor(P+5); ifactor(P+6); ifactor(P+7 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&)-I!G6\"6#\"\"#\"\"%\"\"\"-F%6# \"%x=F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*()-I!G6\"6#\"\"$\"\"#\"\"\"- F%6#\"#ZF*-F%6#\"#rF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I!G6\"6#\"\" #\"\"\"-F$6#\"&<]\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&-I!G6\"6#\"\" &\"\"\"-F$6#\"%2gF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "*()-I!G6\"6#\"\"# F(\"\"\"-F%6#\"\"$F)-F%6#\"%.DF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&)- I!G6\"6#\"\"(\"\"#\"\"\"-F%6#\"$8'F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "6.1. 48. 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Kanonikus alak." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ifactor(-720); ifactors(-720);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*()-I!G6\"6#\"\"#\"\"%\"\"\")-F&6#\"\" $F)F+-F&6#\"\"&F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"\"7%7$\" \"#\"\"%7$\"\"$F&7$\"\"&\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "6.1.50. K\303\26 6vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "6.1.51. K\303\266vetkezm\303 \251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "6.1.52. K\303\266vetkezm\303\251ny." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 35 "6.1.53. K\303\266vetkezm\303\251ny." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 26 "6.1.54. 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Kongruenci\303\241k" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 27 "6.2.1. Kon gruenci\303\241k." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }} }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.2. Feladat. " }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 38 "6.2.3. 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Gyo rs hatv\303\241nyoz\303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "fastexp:=proc(g,n::posint,mu lt::procedure) local j,x,b;\n" }{MPLTEXT 1 0 29 "b:=convert(n,base,2); x:=g;\n" }{MPLTEXT 1 0 36 "for j from nops(b)-1 to 1 by -1 do\n" } {MPLTEXT 1 0 49 " x:=mult(x,x); if b[j]=1 then x:=mult(x,g) fi;\n" } {MPLTEXT 1 0 11 "od; x; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"g G6\"'I\"nGF%I'posintG%*protectedG'I%multGF%I*procedureGF)6%I\"jGF%I\"x GF%I\"bGF%F%F%C&>F0-I(convertGF)6%F'I%baseGF%\"\"#>F/F$?(F.,&-I%nopsGF )6#F0\"\"\"F>!\"\"F?F>I%trueGF)C$>F/-F+6$F/F/@$/&F06#F.F>>F/-F+6$F/F$F /F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fastexp(2,11,(x, y)->x*y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%[?" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 42 "6.2.10. Diffie-Hellmann-Merkle-kulcscsere." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(numthe ory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfr acGF$I)cfracpolGF$I+cyclotomicGF$I)divisorsGF$I)factorEQGF$I*factorset GF$I'fermatGF$I)imagunitGF$I&indexGF$I/integral_basisGF$I)invcfracGF$I 'invphiGF$I*issqrfreeGF$I'jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendre GF$I)mcombineGF$I)mersenneGF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%m logGF$I'mobiusGF$I&mrootGF$I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthd enomGF$I)nthnumerGF$I'nthpowGF$I&orderG%*protectedGI)pdexpandGF$I$phiG F$I#piGF$I*pprimrootGF$I)primrootGF$I(quadresGF$I+rootsunityGF$I*safep rimeGF$I&sigmaGF$I*sq2factorGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "q:=safeprime(435987601265432 78905668798765412345657890987654234186);\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 31 "p:=(q-1)/2; isprime(p); g:=3;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"Vn0Eaw)4*ylXBTl()zoc!*yKaE,w)fV" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"V$GIr#Q\\X*GG<1FQ*RMGXR;Fj+Q*z@" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "T:=convert(floor(time( )*1000)+3141592 mod (111^3),base,111);while nops(T)<3 do T:=[op(TT),0] od;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"$0\"\"$4\"\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "N:=3; T:=vector(N+3,T); n:=0; m:=0; c:=\"a\";" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I #miGF$6%Q\"TF'/%'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%-I'mtableGF$6#-I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$ Q$105F'F9-F\\o6#-F_o6$Q$109F'F9-F\\o6#-F_o6$Q#32F'F9-F\\o6#-I%msubGF$6 %F+-F#6#-F_o6$Q\"4F'F9/%/subscriptshiftGQ\"0F'-F\\o6#-F_p6%F+-F#6#-F_o 6$Q\"5F'F9Ffp-F\\o6#-F_p6%F+-F#6#-F_o6$Q\"6F'F9Ffp/%%openGQ\"[F'/%&clo seGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "Q\"a6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "nexttime:=proc() local i,S; \+ global T,TT,N,m,n,c;\n" }{MPLTEXT 1 0 49 " S:=\"0123456789-abcdefghij klmnopqrstuvwxyz\";\n" }{MPLTEXT 1 0 24 " m:=1+(T[3+n] mod 3);\n" } {MPLTEXT 1 0 19 " i:=T[m] mod 37;\n" }{MPLTEXT 1 0 20 " if S[i+1]=c \+ then\n" }{MPLTEXT 1 0 66 " n:=n+1; T[3+n]:=floor(time()*1000) mod 3 *37*16; T[m]:=T[3+n]\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 36 " \+ if n=N then print(T); return fi;\n" }{MPLTEXT 1 0 24 " m:=1+(T[3+n] m od 3);\n" }{MPLTEXT 1 0 19 " i:=T[m] mod 37;\n" }{MPLTEXT 1 0 20 " p rint(T,S[i+1]);\n" }{MPLTEXT 1 0 26 " c:=readline(terminal);\n" } {MPLTEXT 1 0 26 " while true do i:=0 od;\n" }{MPLTEXT 1 0 4 "end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "f*6\"6$I\"iGF#I\"SGF#F#F#C,>F&QF01234567 89-abcdefghijklmnopqrstuvwxyzF#>I\"mGF#,&\"\"\"F--I$modGF#6$&I\"TGF#6# ,&\"\"$F-I\"nGF#F-F5F->F%-F/6$&F26#F+\"#P@$/&F&6#,&F%F-F-F-I\"cGF#C%>F 6,&F6F-F-F->F1-F/6$-I&floorG6$%*protectedGI(_syslibGF#6#,$*&\"%+5F--I% timeGFLF#F-F-\"%w<>F:F1@$/F6I\"NGF#C$-I&printGFL6#F2OF#F*F7-Fen6$F2F?> FB-I)readlineGF#6#I)terminalGF#?(F#F-F-F#I%trueGFL>F%\"\"!F#6(F2I#TTGF #FXF+F6FBF#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nexttime();" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Types ettingGI(_syslibGF'6%-I'mtableGF$6#-I$mtrGF$6(-I$mtdGF$6#-I#mnGF$6$Q%1 271F'/%,mathvariantGQ'normalF'-F26#-F56$Q$110F'F8-F26#-F56$Q%1245F'F8- F26#-F56$Q#20F'F8F@F1/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "convert(T,list); %[4..N+3]; map(t->t mod 16,%);\n" }{MPLTEXT 1 0 52 "x:=add(%[i]*16^(i-1),i=1..nops(%)); X:=g&^x mod q;\n" }}{PARA 11 " " 1 "" {XPPMATH 20 "7(\"$0\"\"$4\"\"#K&I\"TG6\"6#\"\"%&F'6#\"\"&&F'6# \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%&I\"TG6\"6#\"\"%&F$6#\"\"&&F$ 6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%&I\"TG6\"6#\"\"%&F$6#\"\"&& F$6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(&I\"TG6\"6#\"\"%\"\"\"*& \"#;F(&F$6#\"\"&F(F(*&\"$c#F(&F$6#\"\"'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I#&^G6\"6$\"\"$,(&I\"TGF$6#\"\"%\"\"\"*&\"#;F,&F)6#\"\"& F,F,*&\"$c#F,&F)6#\"\"'F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "x:=76813406921544738654231750679890923210164916436739; X:=g&^x mod q;\n" }{MPLTEXT 1 0 71 "y:=476143609917846513247659876667890987665423 33432543; Y:=g&^y mod q;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"SRnV;\\; 5K#4*)z1vJUlQZa@pS8o(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"VoVu?_F!G$H@: t]:`gPHx9U]:D7l#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"SVDVLBam()4*ymw)fw C8l%y\"*4O9w%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"VERg1F)>\"psx3<4:ipyk [V&yWWk?E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "X&^y mod q; Y& ^x mod q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"UY(GbcnV\"3G$)Q1\")>:8PB^ !*R(>*R)\\#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"UY(GbcnV\"3G$)Q1\")>:8P B^!*R(>*R)\\#" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.11. Feladat." }{TEXT 207 0 "" }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "[[3^i mod 17,i]$i=1..16]; sort(map(x->[x[1] mod 17,x[2]],%),(x,y)->x[1]F' 7$\"&\\!f\"#57$\"'Zr<\"#67$\"'T9`\"#77$\"(BVf\"\"#87$\"(pHy%\"#97$\")2 *[V\"\"#:7$\")@n/V\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "727$\"\"\"\"#; 7$\"\"#\"#97$\"\"$F$7$\"\"%\"#77$\"\"&F/7$\"\"'\"#:7$\"\"(\"#67$\"\") \"#57$\"\"*F'7$F8F*7$F5F47$F-\"#87$F>F,7$F(F:7$F2F17$F%F7" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "6.2.12. Az Euler-f" }{TEXT 207 29 "\303\251le \317\206 f \303\274" }{TEXT 207 3 "ggv" }{TEXT 207 8 "\303\251" }{TEXT 207 3 "ny. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "phi(1); phi(2); phi(3); phi(4); phi(5); phi(6); phi(7); phi(8 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"%" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 15 "6. 2.13. Lemma." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "6.2.14. Euler-Fermat t\303\251te l." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 56 "6.2.15. K\303\266vetkezm\303\251ny: Ferm at-t\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}} }{SECT 1 {PARA 5 "" 0 "" {TEXT 207 53 "6.2.16. Line\303\241ris kongrue ncia megold\303\241sa." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%? G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "6.2.17. P\303\251lda." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x:='x'; msolve(172*x=6,62);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\" xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<#/I\"xG6\"\"\"%<#/F%\"#N" }} }{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.18. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.19. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.20. Feladat." }{TEXT 207 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.21. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.22. Feladat." } {TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 61 "6.2.23. Line\3 03\241ris kongruenciarendszer megold\303\241sa." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "y:='y'; mso lve(\{3*x-4*y=1,7*x+y=2\},19);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"yG6 \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I\"yG6\"\"#6/I\"xGF%\"#:" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.24. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 44 "6.2.25. Diofantikus probl\303\251m\303\2 41k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "isolve(x^2+y^2=z^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%/I\"xG6\"*(I$_Z3GF%\"\"\",&*$)I$_Z1GF%\"\"#F(!\"\"*$)I$_Z2GF%F-F(F( F(-I%igcdG%*protectedG6%F),&F*F(F/F(,$*(F-F(F,F(F1F(F.F./I\"zGF%*(F'F( F6F(F2F./I\"yGF%,$*,F-F(F'F(F,F(F1F(F2F.F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "isolve(x^3+y^3=z^3);" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.2. 26. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*6.2.27. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 48 "6.2.28. K\303\255nai marad\303\251kt\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "chrem ([1,2,2],[2,3,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#B" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 26 "6.2.29. Megjegyz\303\251s." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "6.2.30. P\303\251lda." }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.31. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.32. Feladat." }{TEXT 207 0 "" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "42&^600 mod 13;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "-> 6.2.33. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.34. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.35. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.36. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.37. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.38. Feladat." }{TEXT 207 0 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 37 "6.2.39. Az RSA elj\303\241r \303\241s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 183 "p:=safeprime(1563456788814256178886765661555261342 9876453213456654321234567887720912751210987612321223333212334341234321 23344454321234320948725467845467788859812342365); log[2.](p);\n" } {MPLTEXT 1 0 190 "q:=safeprime(298415244751590016765614534678909876512 3425432145649099876762678818251432567890998723651423415423239658787477 8993377722004988376667882767156363888377626677728888); log[2.](q);\n" }{MPLTEXT 1 0 9 "n:=p*q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 52 "e:= 2876354132453678909987653432123409887635423125;\n" }{MPLTEXT 1 0 52 "i gcdex(e,(p-1)*(q-1),'d'); d; d*e mod (p-1)*(q-1);\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"etzJO7)f))yna%yYD([4KM7KaWMB@VBTVL7KLB7K7w)4@^F\"4s() ycM7KamX8KXw)HMh_bhcw'))yhD9))ycMc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " $\"+zt**)3&!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\\uFN!ynEwP))QOcrw# )ymw$))\\+AxP$**yZ(ye'RKU:MU^Os)*4*ycK9D=)yEww)*4\\c9KaUB^w)4*yY`9cw;+ f^ZC:%)H" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+k\"e3L&!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"e^lLB8J\\Ph>[P-FpRm.k=E@eI\"yn(pfM%Qb(Gj@Ecd1r; AGreb5l[pymnoEPLhJ4x\"3#p#oNV1^jG(3]*yV!G%ym!)3:D:*=P;69(o^J)[0moA%[La \\.%fuJl6A-'Rd4!45WUPguEiX*>5N`%\\M\"yKtF*G6s/Sw[A#=CCTDHS$flY" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"ODJUNw))4M7KMl()*4*yOXKTNwG" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!e^lJQ [+U9*oK23g&)zJv?Ft;Dl/75!oVPro:['pO-Ue*3)4CK=dlH:3X\\BM:yi-+k8S^'*[>#f '*>r*H[U*fdMKsB2er$*y&Q'oNFBT3K\\@#oS/kMUp'y<;(\\1S'HB:%ea$G1#=O^7n?b% y'))>k@6(\\J*R#z(*[L:]sDd6mP'GyCiHr&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "M:=\"Mint v\303 \255z alatti, elmer\303\274lt harangok\n" }{MPLTEXT 1 0 63 "hint\303\2 41znak-e hajnalonk\303\251nt \303\241gyadn\303\241l\n" }{MPLTEXT 1 0 43 "a tizennyolc \303\251ves iskol\303\241sok\n" }{MPLTEXT 1 0 32 "kik et felakasztatt\303\241l\";\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 " convert(M,'bytes');\n" }{MPLTEXT 1 0 38 "m:=sum(%[i]*256^(i-1),i=1..no ps(%));\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 14 "c:=m&^e mod n;" }} {PARA 11 "" 1 "" {XPPMATH 20 "Q\\tMint~`v|^w|huz`~alatti,~`elmer|^w|gv lt`~harangok|+|+`hint|^w|\\uznak`-e~`hajnalonk|^w|dunt`~`|^w|\\ugyadn| ^w|\\ul`|+|+a~tizennyolc~`|^w|duves`~`iskol|^w|\\usok`|+|+kiket~`felak asztatt|^w|\\ul`6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7]s\"#x\"$0\"\"$5 \"\"$;\"\"#K\"$=\"\"$&>\"$t\"\"$A\"F'\"#(*\"$3\"F,F&F&F$\"#WF'\"$,\"F- \"$4\"F/\"$9\"F)\"$)=F-F&F'\"$/\"F,F1F,F%\"$.\"\"$6\"\"$2\"\"#5F7F3F$F %F&F)\"$h\"F+F%F,F6\"#XF/F'F3F,\"$1\"F%F,F-F5F%F6F)\"$p\"F%F&F'F)F8F4 \"$@\"F,\"$+\"F%F)F8F-F7F7F,F'F&F$F+F/F%F%FAu5G6n*f%)>]DNDJQ\" fIWf4Y(>ioa*GEOt]53G)Hya\"[x%RJc,aq`N3CyVuh!p6At7EB6@TY%y*Q#eM#Rg4*=6u S061CY+oG&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"c^l=KbP(*)[CV:%)['eEK\" QKL[T<\"H70&3%H'yf:)3whw1kPJ2>i6JZM\\1-/')fdLXN(oGHIzjzgu_5?*p CZ(4U(oiApFZZ!z\\1RY%[@\"=kr1M+'Gq^/A0I]!=()H\\Y1&G7&\\ms:Qot/_d#\\e)* )4MQ " 0 "" {MPLTEXT 1 0 52 "c&^d mod n; convert(%,base,256); convert(%,'bytes');" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"b^l$*)=cl\"z&)oh7Il*zqJn\")\\`[RxVx[pfF`(4y\\&*[!fw *[54)oDJ:!px#yr-Dz$o@[b3Au5G6n*f%)>]DNDJQ\"fIWf4Y(>ioa*GEOt]5 3G)Hya\"[x%RJc,aq`N3CyVuh!p6At7EB6@TY%y*Q#eM#Rg4*=6uS061CY+oG&>" }} {PARA 11 "" 1 "" {XPPMATH 20 "7]s\"#x\"$0\"\"$5\"\"$;\"\"#K\"$=\"\"$&> \"$t\"\"$A\"F'\"#(*\"$3\"F,F&F&F$\"#WF'\"$,\"F-\"$4\"F/\"$9\"F)\"$)=F- F&F'\"$/\"F,F1F,F%\"$.\"\"$6\"\"$2\"\"#5F7F3F$F%F&F)\"$h\"F+F%F,F6\"#X F/F'F3F,\"$1\"F%F,F-F5F%F6F)\"$p\"F%F&F'F)F8F4\"$@\"F,\"$+\"F%F)F8F-F7 F7F,F'F&F$F+F/F%F%F " 0 "" {MPLTEXT 1 0 55 "millerrabin:=proc(n::posint, a::posint) local j,k,q,b;\n" }{MPLTEXT 1 0 50 "if n=2 or n=3 or n=5 or n=7 then return true fi;\n" }{MPLTEXT 1 0 30 "if n<9 then return fals e fi;\n" }{MPLTEXT 1 0 48 "b:=a mod n; if b=0 or b=1 then return FAIL \+ fi;\n" }{MPLTEXT 1 0 57 "k:=0; q:=n-1; while type(q,even) do k:=k+1; q :=q/2; od;\n" }{MPLTEXT 1 0 50 "b:=b&^q mod n; j:=0; if b=1 then retur n true fi;\n" }{MPLTEXT 1 0 70 "while jF/-I$modGF&6$F*F%@$5/F/\"\"!/F/ \"\"\"OI%FAILGF(>F-FK>F.,&F%FMFM!\"\"?(F&FMFMF&-I%typeGF(6$F.I%evenGF( C$>F-,&F-FMFMFM>F.,$*&#FMF6FMF.FMFM>F/-FF6$-I#&^GF&6$F/F.F%>F,FK@$FLF= ?(F&FMFMF&2F,F-C%@$/F/FRF=>F/-FF6$*$)F/F6FMF%>F,,&F,FMFMFMFCF&F&F&" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "millerrabin(9,2); millerrab in(11,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&falseG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 62 "for n do if millerrabin(n,2)<>isprime(n) the n print(n) fi; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%Z?" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%xK" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%LS" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"%\"o%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%@$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&Te\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&T$H" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&*zU" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&T\"\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&L E&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&\"Gl" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&lY(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&\"e!)" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"&*[&)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&d$))" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&^2*" }}{PARA 7 "" 1 "" {TEXT 208 33 "Warning, computation interrupted" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 96 "for n do if millerrabin(n,2)<>isprime(n) and m illerrabin(n,3)<> isprime(n) then print(n) fi; od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"(`OP\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"((yI:" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"(@q)>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"(`WG#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"(2h6$" }}{PARA 7 "" 1 "" {TEXT 208 33 "Warning, computation interrupted" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 2 "n;" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 36 "* 6.2.42. Digital Signature Standard." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "StringTools[Hash](M); h:=c onvert(%,decimal,hex);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "QA6848b65ee4 08b7eccb7521a7f25888956\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"HP%[Z33EC (4B:-9?&eD<'Q\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "q:=nextp rime(convert(\"97654376ad4efcbe43598123daf56c7b386acbda\",decimal,hex) );\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 257 "for i from convert(\"fff fffffeeeeeeeeddddddddccccccccbbbbbbbbaaaaaaaa9999999988888888777777776 6666666555555554444444433333333222222221111111100000000ffffffffeeeeeee eddddddddccccccccbbbbbbbbaaaaaaaa9999999988888888777777776666666655555 555\",decimal,hex) do\n" }{MPLTEXT 1 0 49 " p:=i*q+1; if isprime(p) t hen break fi; od: p;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 25 "log[2.] (q); log[2.](p);\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 30 "a:=2; g:=a& ^((p-1)/q) mod p;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 "x:=4327650 9876576543211245656730909809123093875;\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 16 "y:=g&^x mod p;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"Q P>+q$e;mBP(fqM4PaYHm&eeJk)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"`^lP!*=@ HAG.>C1F!fqMTx;EZ_b0Ym;)>(eu0&)Hbqv]NxEF62Nlg+*fv-e^hV]AeU&yUV)H#G&*o% z4gg^Ps@\\Q;a7(Hcftx6o=#*Rg)pmeHBnNZD'>FSc]O**H5P<.Dk%pKW'GzK8%p+8LyXi /zaz=s$y(eCll$>3fbV\"yNJ1\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+\"z@C f\"!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+z@CB5!\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"_^l9]-0]Y:a 6Z@j*p_F6i`'f!oq$H&e27Ww9n')[)e!pq&G*4lS_(\\9\\R#y3JLis[FZK&p2q`a2O)=] gH)QLYC9lzC4uZTTTj1D>-'og([)=jCk6#fGS6\\&>P))H%Rt%f7Npo>pQVybe?,t>9Xb5 84&3i'42_aHh2L/tl))>+vd(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"MvQ4B\"4)4 4tccC6Kawl()4lFV" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"_^l\"Rf0(QB]-qK/*Q RQb&e()[lZ$ep`u3.9>$z-y==^d(3%R,*eh\\?Iqvp$[dkS+o*pTCBrX)R,%)e][=sm9^% QpT(4#R$RKf$Hi-M4i.?js=St9^')=EO1T#[c&35%[L\"4Ihr]$>\"p2[jxUf9vo>On#z \"RJ.idw7))**[zO " 0 "" {MPLTEXT 1 0 34 "k:=9804563211276543278906543278;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 24 "r:=(g&^k mod p) mod q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 21 "s:=(h+x*r)/k mod q;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"=yKa1*yKaw7@ jX!)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"P`@\\dyv\\p`o.\\Sl9rd.o^^FB$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"Q\")Q]&QqV5$*3Ligi]9c-Oo5#GBY" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "w:=1/s mod q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 16 "u1:=h*w mod q;\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 16 "u2:=r*w mod q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 29 "v:=(g&^u1*y&^u2 mod p) mod q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"Q h57Mzmjh_49(=)*pLgGsHI;#*)e" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"QY#y(Q( )4S\"Q%**)fc\\+*yp+3!*y4P>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"Q\\&pxd^ *>Q$=jam&=5'plo+E!>W9" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"P`@\\dyv\\p`o .\\Sl9rd.o^^FB$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "6.2.43. Feladat: " }{TEXT 207 8 "\303\241" } {TEXT 207 3 "lpr" }{TEXT 207 8 "\303\255" }{TEXT 207 4 "mek." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "6.2.44. Feladat: Car michel-sz" }{TEXT 207 8 "\303\241" }{TEXT 207 4 "mok." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 18 "->6.2.45. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 31 "6.2.46. Feladat: pit agoraszi sz" }{TEXT 207 8 "\303\241" }{TEXT 207 2 "mh" }{TEXT 207 8 " \303\241" }{TEXT 207 7 "rmasok." }{TEXT 207 0 "" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 " 6.2.47. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.48. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.49. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.2.50. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "6.2.51. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 20 "bbi feladatok megold" }{TEXT 207 8 "\303\241" }{TEXT 207 7 "so kkal." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "6.2.52 . Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 13 "bbi feladatok" }{TEXT 207 1 "." }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "% #%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 56 "6.3. Sz\303\241melm\30 3\251leti f\303\274ggv\303\251nyek" }}{PARA 0 "" 0 "" {TEXT 201 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " restart: with(numtheory);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7QI&GIgcdG6\"I)bigomegaGF$I&cfracGF$I )cfracpolGF$I+cyclotomicGF$I)divisorsGF$I)factorEQGF$I*factorsetGF$I'f ermatGF$I)imagunitGF$I&indexGF$I/integral_basisGF$I)invcfracGF$I'invph iGF$I*issqrfreeGF$I'jacobiGF$I*kroneckerGF$I'lambdaGF$I)legendreGF$I)m combineGF$I)mersenneGF$I(migcdexGF$I*minkowskiGF$I(mipolysGF$I%mlogGF$ I'mobiusGF$I&mrootGF$I&msqrtGF$I)nearestpGF$I*nthconverGF$I)nthdenomGF $I)nthnumerGF$I'nthpowGF$I&orderG%*protectedGI)pdexpandGF$I$phiGF$I#pi GF$I*pprimrootGF$I)primrootGF$I(quadresGF$I+rootsunityGF$I*safeprimeGF $I&sigmaGF$I*sq2factorGF$I(sum2sqrGF$I$tauGF$I%thueGF$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 59 "6.3.1. Sz \303\241melm\303\251leti f\303\274ggv\303\251nyek." }}{EXCHG {PARA 0 " " 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 20 "6.3.2. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%# %?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 28 "6.3.3. P\303\251ld\303 \241k." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "divisors(60); tau(60); sigma[0](60); sigma(60);\n" } {MPLTEXT 1 0 27 "sigma[1](60); sigma[2](60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<.\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"#5\"#7\"#:\"#?\"#I\"#g " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$o\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%ga" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "mobius(1); mobius(2); mobius(4); mo bius(20); mobius(21);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "kap pa:=proc(n::posint) nops(ifactors(n)[2]) end; kappa(1); kappa(2); kapp a(3); kappa(4); kappa(5); kappa(6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f *6#'I\"nG6\"I'posintG%*protectedGF&F&F&-I%nopsGF(6#&-I)ifactorsG6$F(I( _syslibGF&6#F%6#\"\"#F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "nu:=proc(n::posint) nops(factorset( n)) end; factorset(60); nu(60);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#' I\"nG6\"I'posintG%*protectedGF&F&F&-I%nopsGF(6#-_I*numtheoryG6$F(I(_sy slibGF&I*factorsetGF&6#F%F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<%\" \"#\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "6.3.4. T\303\251tel." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "phi(60); invphi(16);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "7( \"#<\"#K\"#M\"#S\"#[\"#g" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "% #%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 20 "6.3.5. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "phi(5500);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%+?" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->6.3.6. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->6.3.7. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->6.3.8. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 17 "->6.3.9. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.10. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.11. Feladat." } {TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.12. Felada t." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.3.13. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "-> 6.3.14. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 18 "->6.3.15. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.16. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.17. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.18. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.19. Feladat." }{TEXT 207 0 " " }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 34 "*6.3.20. Konvol\303\272ci\3 03\263." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*6.3.21. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 57 "*6.3.22. \303\226sszegz\303\251si f\303\274ggv\303\251ny." }} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 59 "*6.3.23 M\303\266bius-f\303\251le inverzi\303\263s \+ formula." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*6.3.24. P\303\251lda." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*6.3.25. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*6.3.26. T\303\25 1tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 16 "6.3.27. Feladat." }{TEXT 207 0 "" }}} {SECT 1 {PARA 5 "" 0 "" {TEXT 207 17 "*6.3.28. Feladat." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "6.3.29. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 20 "bbi feladatok megold" }{TEXT 207 8 "\303\241 " }{TEXT 207 7 "sokkal." }{TEXT 207 0 "" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 11 "6.3.30. Tov" }{TEXT 207 8 "\303\241" }{TEXT 207 13 "bbi \+ feladatok" }{TEXT 207 1 "." }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 29 "6.4. L\303\241nct\303\266rtek" }}{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)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$" }}}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 33 "*6.4.1. L\303\2 41nct\303\266rtek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "nextcfrac:=proc(L::list) local a,q,j; j:=no ps(L);\n" }{MPLTEXT 1 0 38 "if j=0 then return FAIL fi; a:=L[j];\n" } {MPLTEXT 1 0 39 "if type(a,integer) then return(L) fi;\n" }{MPLTEXT 1 0 74 "q:=floor(a); a:=a-q; a:=simplify(expand(1/a)); [op(L[1..j-1]),q, a]; end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 65 "[19/7]; nextcfrac(% ); nextcfrac(%); nextcfrac(%); nextcfrac(%);\n" }{MPLTEXT 1 0 2 "\n" } {MPLTEXT 1 0 37 "cfrac(19/7); cfrac(19/7,quotients);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"LG6\"I%listG%*protectedG6%I\"aGF&I\"qGF&I\"j GF&F&F&C*>F,-I%nopsGF(6#F%@$/F,\"\"!OI%FAILGF(>F*&F%6#F,@$-I%typeGF(6$ F*I(integerGF(OF%>F+-I&floorGF&6#F*>F*,&F*\"\"\"F+!\"\">F*-I)simplifyG F&6#-I'expandGF(6#*$F*FG7%-I#opGF(6#&F%6#;FF,&F,FFFFFGF+F*F&F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "7##\"#>\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"##\"\"(\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\" \"#\"\"\"#\"\"&F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"#\"\"\"F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"#\"\"\"F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I&CFRACG6\"6#7&\"\"#\"\"\"F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\"#\"\"\"F#F#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 29 "*6.4.2. 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Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "*6.4.4. Feladat." }{TEXT 207 0 "" }}} {SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "*6.4.5. Feladat." }{TEXT 207 0 " " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 16 "*6.4.6. Feladat." }{TEXT 207 0 "" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 31 "*6.4.7. L\303\241nct \303\266rtk" }{TEXT 207 8 "\303\266" }{TEXT 207 3 "zel" }{TEXT 207 8 " \303\255" }{TEXT 207 32 "t\303\251sek z\303\241rt alakja." }}{PARA 0 " " 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cfra c(3/5+sqrt(29)); nthconver(%,7); nthnumer(%%,7); nthdenom(%%%,7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "-I&CFRACG6\"6#7.\"\"&\"\"\"\"#m\"\"#F*F' \"#5F(F(F*F*I$...G6$%*protectedGI(_syslibGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"'S=7\"&d.#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'S=7" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"&d.#" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 21 "*6.4 .8. 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Megjegyz\303\251s." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#% ?G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 207 22 "*6.4.11. T\303\251tel." }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 " " 0 "" {TEXT 207 22 "*6.4.12. P\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "nextrangecfrac:=p roc(L::list) local a,a1,a2,j,q1,q2; j:=nops(L);\n" }{MPLTEXT 1 0 38 "i f j=0 then return FAIL fi; a:=L[j];\n" }{MPLTEXT 1 0 57 "a1:=op(1,a); \+ a2:=op(2,a); q1:=floor(a1); q2:=floor(a2);\n" }{MPLTEXT 1 0 29 "if q1< >q2 then return L fi;\n" }{MPLTEXT 1 0 46 "[op(L[1..j-1]),q1,1/(a2-q2) ..1/(a1-q1)] end;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 149 "[31415926 5/10^8..314159266/10^8]; nextrangecfrac(%); nextrangecfrac(%); nextran gecfrac(%); nextrangecfrac(%); nextrangecfrac(%); nextrangecfrac(%);\n " }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 17 "evalf(355/113);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#'I\"LG6\"I%listG%*protectedG6(I\"aGF&I#a1GF& I#a2GF&I\"jGF&I#q1GF&I#q2GF&F&F&C+>F--I%nopsGF(6#F%@$/F-\"\"!OI%FAILGF (>F*&F%6#F->F+-I#opGF(6$\"\"\"F*>F,-F?6$\"\"#F*>F.-I&floorG6$F(I(_sysl ibGF&6#F+>F/-FH6#F,@$0F.F/OF%7%-F?6#&F%6#;FA,&F-FAFA!\"\"F.;*$,&F,FAF/ FYFY*$,&F+FAF.FYFYF&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#;#\")`=$G' \")+++?#\"*L'zq:\")+++]" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"$;#\")+ ++]\"(L'zq#\")+++?\"(`=$G" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"$\"\" (;#\"(`=$G\"'Hq<#\"(L'zq\"'pDW" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"\" $\"\"(\"#:;#\"'pDW\"')4T%#\"'Hq<\"'=k<" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"$\"\"(\"#:\"\"\";#\"'=k<\"$6'#\"')4T%\"%r9" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"$\"\"(\"#:\"\"\";#\"'=k<\"$6'#\"')4T%\"%r9" }} {PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"$\"\"(\"#:\"\"\";#\"'=k<\"$6'#\"') 4T%\"%r9" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+?HfTJ!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "cfrac(Pi,100,quotients);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7bq\"\"$\"\"(\"#:\"\"\"\"$#HF&F&F&\"\"#F&F#F& \"#9F(F&F&F(F(F(F(F&\"#%)F(F&F&F%F#\"#8F&\"\"%F(\"\"'F-\"#**F&F(F(F-F# \"\"&F&F&F-\"\")F&F$F&F(F#F$F&F(F&F&\"#7F&F&F&F#F&F&F0F&F&F(F&F-F&F&F/ F(F(F#F&F(F,F,\"#;F&\"$h\"\"#XF&\"#AF&F(F(F&F,F&F(\"#CF&F(F&F#F&F(F&F& \"#5F(I$...G6$%*protectedGI(_syslibG6\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 207 30 "6.4. 13. 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