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Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(verbose proc=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "print(fermat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"nG6\"-I#OrGF&6$I*nonnegintG%*protectedG-I$NotGF&6#I)consta ntGF+I&_infoGF&6$I\"fGF&I%infoGF&6#IaoCopyright~(c)~1992~by~the~Univer sity~of~Waterloo.~All~rights~reserved.GF&F&@)/%&nargsG\"\"!O6$-I$catGF +6$IJThe~primality~character~of~the~following~GF&IBFermat~numbers~is~k nown~to~Maple:GF&-I%sortGF+6#-I(convertGF+6$I5known_fermat_numbersGF&. 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Egy" }{TEXT 206 8 "\303\251" }{TEXT 206 7 "b speci" } {TEXT 206 8 "\303\241" }{TEXT 206 8 "lis alak" }{TEXT 206 8 "\303\272" }{TEXT 206 3 " pr" }{TEXT 206 8 "\303\255" }{TEXT 206 4 "mek." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 7 "5.17. K" }{TEXT 206 8 "\303\241" }{TEXT 206 13 "tai egy probl" }{TEXT 206 8 "\303\251" } {TEXT 206 1 "m" }{TEXT 206 8 "\303\241" }{TEXT 206 3 "ja." }}{PARA 0 " " 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{SECT 0 {PARA 4 "" 0 "" {TEXT 206 7 "5.18. P" }{TEXT 206 8 "\303\251" }{TEXT 206 4 "lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 34 "# This is a simple f actorization\n" }{MPLTEXT 1 0 35 "# procedure using trial division.\n" }{MPLTEXT 1 0 37 "# The result is a sequence of pairs\n" }{MPLTEXT 1 0 37 "# [p,e] where the p's are the prime\n" }{MPLTEXT 1 0 42 "# facto rs and the e's are the exponents.\n" }{MPLTEXT 1 0 47 "# The factors a re anyway in increasing order.\n" }{MPLTEXT 1 0 41 "# Only primes <= P are tried, hence the\n" }{MPLTEXT 1 0 38 "# last \"factor\" may compo site, if \n" }{MPLTEXT 1 0 27 "# it is greater then P^2;\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 55 "trialdiv:=proc(n::p osint,P::posint) local L,p,i,d,nn;\n" }{MPLTEXT 1 0 15 "L:=[]; nn:=n; \n" }{MPLTEXT 1 0 32 "if type(nn,even) and 2<=P then\n" }{MPLTEXT 1 0 53 " for i from 0 while type(nn,even) do nn:=nn/2; od;\n" }{MPLTEXT 1 0 15 " L:=[[2,i]];\n" }{MPLTEXT 1 0 5 "fi;\n" }{MPLTEXT 1 0 29 "if \+ nn mod 3=0 and 3<=P then\n" }{MPLTEXT 1 0 50 " for i from 0 while nn \+ mod 3=0 do nn:=nn/3; od;\n" }{MPLTEXT 1 0 21 " L:=[op(L),[3,i]];\n" } {MPLTEXT 1 0 5 "fi;\n" }{MPLTEXT 1 0 13 "d:=2; p:=5;\n" }{MPLTEXT 1 0 27 "while p<=P and nn>=p^2 do\n" }{MPLTEXT 1 0 22 " if nn mod p=0 the n\n" }{MPLTEXT 1 0 52 " for i from 0 while nn mod p=0 do nn:=nn/p; \+ od;\n" }{MPLTEXT 1 0 23 " L:=[op(L),[p,i]];\n" }{MPLTEXT 1 0 7 " f i;\n" }{MPLTEXT 1 0 19 " p:=p+d; d:=6-d;\n" }{MPLTEXT 1 0 5 "od;\n" } {MPLTEXT 1 0 36 "if nn>1 then L:=[op(L),[nn,1]] fi;\n" }{MPLTEXT 1 0 4 "L;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I \"nG6\"I'posintG%*protectedG'I\"PGF&F'6'I\"LGF&I\"pGF&I\"iGF&I\"dGF&I# nnGF&F&F&C+>F,7\">F0F%@$3-I%typeGF(6$F0I%evenGF(1\"\"#F*C$?(F.\"\"!\" \"\"F&F7>F0,$*&#F@FF,7#7$FF0,$*&#F@FNF@F0F@F@>F,7$-I#opGF(6#F,7$FNF.>F/F<>F-\"\"&?( F&F@F@F&31F-F*1*$)F-FF0*&F0F@F-! \"\">F,7$FX7$F-F.>F-,&F-F@F/F@>F/,&\"\"'F@F/Fho@$2F@F0>F,7$FX7$F0F@F,F &F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "trialdiv(2^107-2^5 4+1,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%7$\"\"&\"\"\"7$\"$d)F%7$ \">(R`#>UEd+mh!4o'y$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n 0:=%[3][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\">(R`#>UEd+mh!4o'y$" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n0-1),n0);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "trialdiv(n0-1,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'7$\"\"#F$7$\"#>\"\"\"7$\"$2\"F'7$\"$`$F'7$\"8,$\\gA3Tvq7>8F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n1:=%[5][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"8,$\\gA3Tvq7>8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n1-1),n1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"8*y3SO:?%3LYB\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "triald iv(n1,100000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"&8=*\"\"\"7$\"3x p>dOTvO9F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n2:=%[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"3xp>dOTvO9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n2-1),n2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "tria ldiv(n2-1,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&7$\"\"#\"\"%7$\"\" $F$7$\"$Z&\"\"\"7$\".daxKS#=F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n3:=%[4][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\".daxKS#=" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n3-1),n3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\".FsD_y_\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "trialdiv(n3,10000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\"%.6\"\"\"7$\"+>:q`;F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n4:=%[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"+>:q`;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n4-1),n4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "trialdiv(n4-1,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7(7$\"\"#\"\"\"7$\"\"(F%7$\"#>F%7$\"#BF%7$\"$P\"F%7$\"%t>F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "trialdiv(2^107+2^54+1,1000); " }}{PARA 11 "" 1 "" {XPPMATH 20 "7#7$\"B8,x>l(fS\"Q8#HoFfA;\"\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n5:=%[1][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"B8,x>l(fS\"Q8#HoFfA;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^(n5-1),n5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"A$ea**H " 0 "" {MPLTEXT 1 0 31 "t rialdiv(2^107+2^54+1,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7$\" '*eV)\"\"\"7$\"<<>\\'4$Hx-9$*RM#>F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "n6:=%[2][1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"<<> \\'4$Hx-9$*RM#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^ (n6-1),n6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"<'Q>0@M];Yv*eq\"=" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 206 14 "5.19. Feladat." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT 206 8 "5.20. Pr" }{TEXT 206 8 "\303\255" }{TEXT 206 3 "msz" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "mk" }{TEXT 206 8 "\303\263" }{TEXT 206 3 "dol" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "s." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 "5.21. Feladat." }} {PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "p:=safeprime(15634567888142561788867656615552613429876453213456 6543212345678877209127512109876123212233332123343412343212334445432123 4320948725467845467788859812342365); log[2.](p);\n" }{MPLTEXT 1 0 190 "q:=safeprime(29841524475159001676561453467890987651234254321456490998 7676267881825143256789099872365142341542323965878747789933777220049883 76667882767156363888377626677728888); log[2.](q);\n" }{MPLTEXT 1 0 9 " n:=p*q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 52 "e:=28763541324536789 09987653432123409887635423125;\n" }{MPLTEXT 1 0 50 "igcdex(e,(p-1)*(q- 1),'d'); d; d*e mod (p-1)*(q-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"et zJO7)f))yna%yYD([4KM7KaWMB@VBTVL7KLB7K7w)4@^F\"4s()ycM7KamX8KXw)HMh_bh cw'))yhD9))ycMc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+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 "$\"+k\"e3L&!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"e^lLB8J\\Ph>[P-FpRm.k=E@eI\"yn(pfM%Qb(Gj@Ecd1r;AGreb5l[pymnoEPLh J4x\"3#p#oNV1^jG(3]*yV!G%ym!)3:D:*=P;69(o^J)[0moA%[La\\.%fuJl6A-'Rd4!4 5WUPguEiX*>5N`%\\M\"yKtF*G6s/Sw[A#=CCTDHS$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*fd MKsB2er$*y&Q'oNFBT3K\\@#oS/kMUp'y<;(\\1S'HB:%ea$G1#=O^7n?b%y'))>k@6( \\J*R#z(*[L:]sDd6mP'GyCiHr&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 "5.22. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{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\241znak-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 "kiket 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..nops(%));\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`~al atti,~`elmer|^w|gvlt`~harangok|+|+`hint|^w|\\uznak`-e~`hajnalonk|^w|du nt`~`|^w|\\ugyadn|^w|\\ul`|+|+a~tizennyolc~`|^w|duves`~`iskol|^w|\\uso k`|+|+kiket~`felakasztatt|^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$F+F/F%F%FA u5G6n*f%)>]DNDJQ\"fIWf4Y(>ioa*GEOt]53G)Hya\"[x%RJc,aq`N3CyVuh!p6At7EB6 @TY%y*Q#eM#Rg4*=6uS061CY+oG&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"c^l=K bP(*)[CV:%)['eEK\"QKL[T<\"H70&3%H'yf:)3whw1kPJ2>i6JZM\\1-/')fd LXN(oGHIzjzgu_5?*pCZ(4U(oiApFZZ!z\\1RY%[@\"=kr1M+'Gq^/A0I]!=()H\\Y1&G7 &\\ms:Qot/_d#\\e)*)4MQ " 0 " " {MPLTEXT 1 0 52 "c&^d mod n; convert(%,base,256); convert(%,'bytes') ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"b^l$*)=cl\"z&)oh7Il*zqJn\")\\`[Rx Vx[pfF`(4y\\&*[!fw*[54)oDJ:!px#yr-Dz$o@[b3Au5G6n*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 "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%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 }