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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 203 66 "Sz\303\241m\303\255t\303
\263g\303\251pes sz\303\241melm\303\251let" }}}{EXCHG {PARA 19 "" 0 ""
 {TEXT 204 18 "J\303\241rai Antal" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 
204 68 "Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\2
41lnak" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 59 "1. 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" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 64 "3. Egyszer\305\261 p
r\303\255mtesztel\303\251si m\303\263dszerek" }}{EXCHG {PARA 0 "" 0 ""
 {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 18 "4. \+
Lucas-sorozatok" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 23 "5. Alkalmaz
\303\241sok " }}{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 1 {PARA 4 "" 0 "" {TEXT 
206 14 "5.1. Fermat-sz" }{TEXT 206 8 "\303\241" }{TEXT 206 4 "mok." }}
}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 13 "5.2. Feladat." }}{PARA 0 "" 0 "
" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }
{MPLTEXT 1 0 59 "# This procedure try to find factors i*2^(n+2)+1, 1<=
i<=k\n" }{MPLTEXT 1 0 35 "# of the Fermat number 2^(2^n)+1.\n" }
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 53 "fermatfact
ors:=proc(n::posint,k::posint) local i,f;\n" }{MPLTEXT 1 0 8 "f:=[];\n
" }{MPLTEXT 1 0 51 "for i from 1+2^(n+2) by 2^(n+2) to k*2^(n+2)+1 do
\n" }{MPLTEXT 1 0 46 "  if 2&^(2^n)+1 mod i=0 then f:=[op(f),i] fi\n" 
}{MPLTEXT 1 0 10 "od; f end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"
nG6\"I'posintG%*protectedG'I\"kGF&F'6$I\"iGF&I\"fGF&F&F&C%>F-7\"?(F,,&
\"\"\"F3)\"\"#,&F%F3F5F3F3F4,&*&F*F3F4F3F3F3F3I%trueGF(@$/-I$modGF&6$,
&-I#&^GF&6$F5)F5F%F3F3F3F,\"\"!>F-7$-I#opGF(6#F-F,F-F&F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "fermatfactors(5,10); fermatfactors(
5,100000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"$T'" }}{PARA 11 "" 1 "
" {XPPMATH 20 "7$\"$T'\"(</q'" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 
13 "5.3. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 25 "interface(verboseproc=2);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pr
int(fermat);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$'I\"nG6\"-I#OrGF&6$I
*nonnegintG%*protectedG-I$NotGF&6#I)constantGF+I&_infoGF&6$I\"fGF&I%in
foGF&6#IaoCopyright~(c)~1992~by~the~University~of~Waterloo.~All~rights
~reserved.GF&F&@)/%&nargsG\"\"!O6$-I$catGF+6$IJThe~primality~character
~of~the~following~GF&IBFermat~numbers~is~known~to~Maple:GF&-I%sortGF+6
#-I(convertGF+6$I5known_fermat_numbersGF&.I%listGF+-I%typeGF+6$F%.I%na
meGF+O,&)\"\"#)FRF%\"\"\"FTFT-FK6$F%.F*C%@'4-I'memberGF+6$F%FGOI2chara
cter~unknownGF&52F%\"#A/I,_EnvTryHardGF&I%trueGF+>F2FP>F2I/object~too~
bigGF&@$/F8FR@=1F%\"\"%>F0I,it~is~primeGF&/F%\"\"&>F06$I;it~is~complet
ely~factored~GF&*&-I!GF&6#,&-I(ifactorGF&6#\"$S'FTFTFTFT-Fap6#,&-Fep6#
\"(;/q'FTFTFTFT/F%\"\"'>F06$F^p*&-Fap6#,&-Fep6#\"'wTFFTFTFTFT-Fap6#,&-
Fep6#\"/?2J@/GnFTFTFTFT/F%\"\"(>F06$F^p*&-Fap6#,&-Fep6#\"2;s\\F\"*e\\'
fFTFTFTFT-Fap6#,&-Fep6#\"7?Z0H^o+#*o/dFTFTFTFT/F%\"\")>F06$F^p*&-Fap6#
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p6#\"[q2$plj@GBUnlqv]SO_9V^\"*3'HF?&f/$eD2a!=jr\\-#=C!\\)Ho$*G@OFTF]tF
TFTFTFTFT/F%\"#5>F06$F^p**-Fap6#,&-Fep6#\")wDfXFTFTFTFT-Fap6#,&-Fep6#
\"+3=.(['FTFTFTFT,&*&-Fap6#\"F^!H_;+%*>k'*3\"[jDdSw8\"FT)F^t\"#7FTFTFT
FTFT,&*&-Fap6#,&*&#FT\"inKwB$e70L+H'\\.Iw8\\_fMa)H]&GUk#4.!H6FTF2FTFT#
FT\"%#>)!\"\"FT)F^t\"#8FTFTFTFTFT/F%F`t>F06$F^p*,-Fap6#,&-Fep6#\"')[>$
FTFTFTFT-Fap6#,&-Fep6#\"'[[(*FTFTFTFT,&*&-Fap6#\"2z7`%y2KD5FT)F^t\"#9F
TFTFTFTFT,&*&-Fap6#\"36(QHG%3tYVFTF\\xFTFTFTFTFT,&*&-Fap6#,&*&#FT\"ens
#)*44l=s)H2M.Mu,G#RW5SIwPrc9i_\"FTF2FTFTFiwF[xFTF\\xFTFTFTFTFT/F%Fdy>F
0I1it~is~composite~GF&/F%\"#?>F0I0it~is~compositeGF&/F%F\\oFgz/F%\"#CF
gzFenC$@%-I)assignedGF+6#&I+fermat_tabGF&6#F%>F3Fa[lYQFnumber~recogniz
ed~but~no~factor~knownF&>F06$-I#ifGF+6%/-I%nopsGF+6#F3FTI:it~has~this~
prime~factor~GF&I<it~has~these~prime~factors~GF&-I#opGF+F_\\lOICno~com
plete~factorization~is~knownGF&OF2.-%)procnameG6#%%argsGF&F&6&FGFGFb[l
Fb[l" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 16 "5.4. Mersenne-sz" }
{TEXT 206 8 "\303\241" }{TEXT 206 4 "mok." }}{PARA 0 "" 0 "" {TEXT 
201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 287 "mersennes:=[2,3,
5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,4423,96
89,9941,11213,19937,21701,23209,44497,86243,110503,132049,216091,75683
9,859833,1257787,1398269,2976221,3021377,6972593,13466917,20996011,240
36583,25964951,30402457,32582657,37156667,42643801,43112609];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7Q\"\"#\"\"$\"\"&\"\"(\"#8\"#<\"#>\"#J\"
#h\"#*)\"$2\"\"$F\"\"$@&\"$2'\"%z7\"%.A\"%\"G#\"%<K\"%`U\"%BW\"%*o*\"%
T**\"&87\"\"&P*>\"&,<#\"&4K#\"&(\\W\"&Vi)\"'.06\"'\\?8\"'\"4;#\"'Rov\"
'L)f)\"((yd7\"(p#)R\"\"(@i(H\"(x8-$\"($fsp\")<pY8\")6g*4#\")$eOS#\")^
\\'f#\")dCSI\")dEeK\")nm:P\"),QkU\")4E6V" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "map(p->p mod 4, mersennes);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "7Q\"\"#\"\"$\"\"\"F$F%F%F$F$F%F%F$F$F%F$F$F$F%F%F%F$F%F%F
%F%F%F%F%F$F$F%F$F$F%F$F%F%F%F%F%F$F$F$F%F%F$F%F%" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 62 "logm:=evalf([[i,log[2.](mersennes[i])]$i=1..
nops(mersennes)]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7Q7$$\"\"\"\"\"!$
\"+++++5!\"*7$$\"\"#F&$\"+,D'\\e\"F)7$$\"\"$F&$\"+%4G>K#F)7$$\"\"%F&$
\"+A\\N2GF)7$$\"\"&F&$\"+=(R/q$F)7$$\"\"'F&$\"+TGY(3%F)7$$\"\"(F&$\"+9
v#zC%F)7$$\"\")F&$\"+5j>a\\F)7$$\"\"*F&$\"+QttIfF)7$$\"#5F&$\"+KMtvkF)
7$$\"#6F&$\"+')pYTnF)7$$\"#7F&$\"+(o%o))pF)7$$\"#8F&$\"+j&R^-*F)7$$\"#
9F&$\"+2FbX#*F)7$$\"#:F&$\"+b+3K5!\")7$$\"#;F&$\"+y`_56Fjo7$$\"#<F&$\"
+t]a:6Fjo7$$\"#=F&$\"+A+:l6Fjo7$$\"#>F&$\"+9lU07Fjo7$$\"#?F&$\"+`4367F
jo7$$\"#@F&$\"+1K@C8Fjo7$$\"#AF&$\"+Fv\"zK\"Fjo7$$\"#BF&$\"+q%)GX8Fjo7
$$\"#CF&$\"+sgJG9Fjo7$$\"#DF&$\"+!RZ0W\"Fjo7$$\"#EF&$\"+u'R-X\"Fjo7$$
\"#FF&$\"+X?9W:Fjo7$$\"#GF&$\"+u>hR;Fjo7$$\"#HF&$\"+,EPv;Fjo7$$\"#IF&$
\"+%Qr5q\"Fjo7$$\"#JF&$\"+Yz7s<Fjo7$$\"#KF&$\"+\"piH&>Fjo7$$\"#LF&$\"+
&pp8(>Fjo7$$\"#MF&$\"+@cCE?Fjo7$$\"#NF&$\"+]5_T?Fjo7$$\"#OF&$\"+A]]]@F
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\"#SF&$\"+$>hBV#Fjo7$$\"#TF&$\"+\\G(=X#Fjo7$$\"#UF&$\"+<i+jCFjo7$$\"#V
F&$\"+f%od[#Fjo7$$\"#WF&$\"+#4gd\\#Fjo7$$\"#XF&$\"+w<r9DFjo7$$\"#YF&$
\"+rKeMDFjo7$$\"#ZF&$\"+`1;ODFjo" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 33 "with(plots); plotm:=plot(logm):\n" }{MPLTEXT 1 0 60 "plotline
:=plot(evalf(exp(-gamma))*x,x=1..nops(mersennes)):\n" }{MPLTEXT 1 0 
33 "pointm:=plot(logm,style=POINT):\n" }{MPLTEXT 1 0 35 "display(\{plo
tline,plotm,pointm\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "7gnI,Interacti
veG6\"I(animateGF$I*animate3dGF$I-animatecurveGF$I&arrowGF$I-changecoo
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g/<kjkF;7$$\"34++]\\rV]7Ffp$\"3bg_Kizp?qF;7$$\"33++]#Hg+N\"Ffp$\"3_
\\\"flaT+e(F;7$$\"36++]]c1Y9Ffp$\"3T\"G9Qts!>\")F;7$$\"3nm;/8xCL:Ffp$
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m;/jw9*\\#Ffp$\"3'4_Jj:qJS\"Ffp7$$\"3'***\\i#[Kue#Ffp$\"3W)\\Wb]QFX\"F
fp7$$\"3tmmT1Dy#o#Ffp$\"3G:Qm!oti]\"Ffp7$$\"3))*****4T)G\"y#Ffp$\"3NCF
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Ffp$\"3)eFC0'oRy<Ffp7$$\"3w****\\s\"ynE$Ffp$\"3R,ww&ejT$=Ffp7$$\"3Ym;/
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nkhxa$Ffp$\"3/Yn4AW#>*>Ffp7$$\"3R+]([PQXk$Ffp$\"3u+x\"Rjgi/#Ffp7$$\"30
LL$eJb\"RPFfp$\"3/yzs7UQ*4#Ffp7$$\"3'***\\P%[5#QQFfp$\"39$H0ln**\\:#Ff
p7$$\"3Mnmm%Q7O$RFfp$\"3W6v<yRc3AFfp7$$\"3lmm\"pgu6.%Ffp$\"3Qk3587MjAF
fp7$$\"3%QL3Z'*Gz7%Ffp$\"3tBE'['[m<BFfp7$$\"3j+++rh$o@%Ffp$\"3Rha**eEe
nBFfp7$$\"3SLLeiQt=VFfp$\"3gr=H%3%zCCFfp7$$\"3'pmmEPs)4WFfp$\"3dL$*4lY
'fZ#Ffp7$$\"3H+]PL]/2XFfp$\"3Us1+xJ_IDFfp7$$\"3>+]io;0+YFfp$\"3r5&*RME
u#e#Ffp7$Fcz$\"3')***>Hdf)QEFfpF[[l-%%VIEWG6$;F_[lFcz;$\"1;+!)oen\"\\%
F;$\"/7TXQ^!p#!#7" 1 2 2 1 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" 
"Curve 2" "Curve 3" }}{TEXT 207 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT
 206 13 "5.5. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 56 "# This proced
ure do a pretest for the Mersenne numbers\n" }{MPLTEXT 1 0 44 "# M[n]=
2^n-1. The exponents are taken from\n" }{MPLTEXT 1 0 60 "# the sequenc
e L. The result is the sequence of sequences,\n" }{MPLTEXT 1 0 57 "# w
hich contains all exponents n which is prime and the\n" }{MPLTEXT 1 0 
63 "# prime divisors having the form 2*k*n+1 for k until K, where\n" }
{MPLTEXT 1 0 33 "# k congruent to 0 or -n mod 4.\n" }{MPLTEXT 1 0 3 "#
\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 50 "mersennepretest:=proc(L::li
st(posint),K::posint)\n" }{MPLTEXT 1 0 33 "local nL,D,M,n,k,d1,d2; nL:
=[];\n" }{MPLTEXT 1 0 15 "for n in L do\n" }{MPLTEXT 1 0 22 "  if ispr
ime(n) then\n" }{MPLTEXT 1 0 44 "    D:=[n]; d1:=modp(n,4); d2:=modp(-
n,4);\n" }{MPLTEXT 1 0 33 "    for k from d2 while k<=K do\n" }
{MPLTEXT 1 0 19 "      M:=2*k*n+1;\n" }{MPLTEXT 1 0 26 "      if ispri
me(M) then\n" }{MPLTEXT 1 0 52 "        if pmod(2&^n-1,M)=0 then D:=[o
p(D),M]; fi;\n" }{MPLTEXT 1 0 11 "      fi;\n" }{MPLTEXT 1 0 28 "     \+
 k:=k+d1; M:=2*k*n+1;\n" }{MPLTEXT 1 0 26 "      if isprime(M) then\n"
 }{MPLTEXT 1 0 49 "        if 2&^n-1 mod M=0 then D:=[op(D),M] fi;\n" 
}{MPLTEXT 1 0 11 "      fi;\n" }{MPLTEXT 1 0 16 "      k:=k+d2;\n" }
{MPLTEXT 1 0 9 "    od;\n" }{MPLTEXT 1 0 21 "    nL:=[op(nL),D];\n" }
{MPLTEXT 1 0 7 "  fi;\n" }{MPLTEXT 1 0 19 "od;         nL end;" }}
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2*7#\"$6*7#\"$>*7#\"$H*7#\"$P*7#\"$Z*7#\"$`*7#\"$r*7#\"$$)*7#\"$\"**7#
\"$(**" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$O\"" }}}}{SECT 0 {PARA 4 ""
 0 "" {TEXT 206 13 "5.6. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "interface(verboseproc=2);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "print(mersenne);" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6
#'I\"nG6\"<$I'posintG%*protectedG7#F(6#I\"wGF&6#IaoCopyright~(c)~1992~
by~the~University~of~Waterloo.~All~rights~reserved.GF&F&C%@$0%&nargsG
\"\"\"YQ:wrong~number~of~argumentsF&>F,7M\"\"#\"\"$\"\"&\"\"(\"#8\"#<
\"#>\"#J\"#h\"#*)\"$2\"\"$F\"\"$@&\"$2'\"%z7\"%.A\"%\"G#\"%<K\"%`U\"%B
W\"%*o*\"%T**\"&87\"\"&P*>\"&,<#\"&4K#\"&(\\W\"&Vi)\"'.06\"'\\?8\"'\"4
;#\"'Rov\"'L%f)\"((yd7\"(p#)R\"\"(@i(H\"(x8-$\"($fsp\")<pY8\")6g*4#\")
$eOS#\")^\\'f#I%NULLGF)@%-I%typeGF)6$F%7#.F(C$@$2-I%nopsGF)F+&F%6#F3YQ
0index~too~largeF&,&)F8&F,6#FhoF3F3!\"\"@'-I'memberGF)6$F%F,,&)F8F%F3F
3F`p52F%&F,6#F`p4-I(isprimeG6$F)I(_syslibGF&6#F%I&falseGF)I%FAILGF)F&F
&F&" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 18 "5.7. h*2^m+-1 alak" }
{TEXT 206 8 "\303\272" }{TEXT 206 3 " pr" }{TEXT 206 8 "\303\255" }
{TEXT 206 9 "mek keres" }{TEXT 206 8 "\303\251" }{TEXT 206 3 "se." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "#\n" }{MPLTEXT 1 0 59 "# A pretest will be done for (h0+c*h)*b^n+
d, where h runs\n" }{MPLTEXT 1 0 59 "# from 0 to H-1 and d in D. All p
rime divisors from P0 to\n" }{MPLTEXT 1 0 61 "# P1 will be tried. P0 m
ust be odd, and b,c must have prime\n" }{MPLTEXT 1 0 52 "# factors sma
ller then P0. The values h, for which\n" }{MPLTEXT 1 0 58 "# (h0+c*h)*
b^n+d not divisible with primes from P0 to P1\n" }{MPLTEXT 1 0 34 "# f
or all d in D are given back.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 
2 "\n" }{MPLTEXT 1 0 70 "pretest:=proc(h0::nonnegint,H::posint,c::posi
nt,b::posint,n::posint,\n" }{MPLTEXT 1 0 40 "D::set(integer),P0::posin
t,P1::posint)\n" }{MPLTEXT 1 0 40 "local T,h,p,d,hh,L,nn,bb,cc,bT,cT,i
,j;\n" }{MPLTEXT 1 0 19 "T:=array(0..H-1);\n" }{MPLTEXT 1 0 36 "for h \+
from 0 to H-1 do T[h]:=1 od;\n" }{MPLTEXT 1 0 69 "bT:=array(0..b-1);  \+
                      # table of inverses for b\n" }{MPLTEXT 1 0 37 "f
or i from 0 to b-1 do bT[i]:=0 od;\n" }{MPLTEXT 1 0 24 "for i from 0 t
o b-1 do\n" }{MPLTEXT 1 0 26 "  for j from 0 to b-1 do\n" }{MPLTEXT 1 
0 41 "    if irem(i*j+1,b)=0 then bT[i]:=j fi\n" }{MPLTEXT 1 0 6 "  od
\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 69 "cT:=array(0..c-1);      \+
                  # table of inverses for c\n" }{MPLTEXT 1 0 37 "for i
 from 0 to c-1 do cT[i]:=0 od;\n" }{MPLTEXT 1 0 24 "for i from 0 to c-
1 do\n" }{MPLTEXT 1 0 26 "  for j from 0 to c-1 do\n" }{MPLTEXT 1 0 
41 "    if irem(i*j+1,c)=0 then cT[i]:=j fi\n" }{MPLTEXT 1 0 6 "  od\n
" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 29 "for p from P0 to P1 by 2 d
o\n" }{MPLTEXT 1 0 22 "  if isprime(p) then\n" }{MPLTEXT 1 0 22 "    n
n:=modp(n,p-1);\n" }{MPLTEXT 1 0 32 "    bb:=(bT[modp(p,b)]*p+1)/b;\n"
 }{MPLTEXT 1 0 25 "    bb:=modp(bb&^nn,p);\n" }{MPLTEXT 1 0 32 "    cc
:=(cT[modp(p,c)]*p+1)/c;\n" }{MPLTEXT 1 0 19 "    for d in D do\n" }
{MPLTEXT 1 0 34 "      hh:=modp((-d*bb-h0)*cc,p);\n" }{MPLTEXT 1 0 48 
"      for h from hh to H-1 by p do T[h]:=0; od\n" }{MPLTEXT 1 0 8 "  \+
  od\n" }{MPLTEXT 1 0 6 "  fi\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 
0 8 "L:=[];\n" }{MPLTEXT 1 0 25 "for h from 0 to H-1 do \n" }{MPLTEXT 
1 0 34 "  if T[h]=1 then L:=[op(L),h] fi\n" }{MPLTEXT 1 0 10 "od; L en
d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6*'I#h0G6\"I*nonnegintG%*protect
edG'I\"HGF&I'posintGF('I\"cGF&F+'I\"bGF&F+'I\"nGF&F+'I\"DGF&-I$setGF(6
#I(integerGF('I#P0GF&F+'I#P1GF&F+6/I\"TGF&I\"hGF&I\"pGF&I\"dGF&I#hhGF&
I\"LGF&I#nnGF&I#bbGF&I#ccGF&I#bTGF&I#cTGF&I\"iGF&I\"jGF&F&F&C.>F=-I&ar
rayGF(6#;\"\"!,&F*\"\"\"FR!\"\"?(F>FPFRFQI%trueGF(>&F=6#F>FR>FF-FM6#;F
P,&F/FRFRFS?(FHFPFRFgnFU>&FF6#FHFP?(FHFPFRFgnFU?(FIFPFRFgnFU@$/-I%irem
GF(6$,&*&FHFRFIFRFRFRFRF/FP>FjnFI>FG-FM6#;FP,&F-FRFRFS?(FHFPFRFjoFU>&F
GF[oFP?(FHFPFRFjoFU?(FIFPFRFjoFU@$/-Fao6$FcoF-FP>F]pFI?(F?F9\"\"#F;FU@
$-I(isprimeG6$F(I(_syslibGF&6#F?C'>FC-I%modpGF(6$F1,&F?FRFRFS>FD*&,&*&
&FF6#-F`q6$F?F/FRF?FRFRFRFRFRF/FS>FD-F`q6$-I#&^GF&6$FDFCF?>FE*&,&*&&FG
6#-F`q6$F?F-FRF?FRFRFRFRFRF-FS?&F@F3FUC$>FA-F`q6$*&,&*&F@FRFDFRFSF%FSF
RFEFRF??(F>FAF?FQFU>FWFP>FB7\"?(F>FPFRFQFU@$/FWFR>FB7$-I#opGF(6#FBF>FB
F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 
0 62 "# This procedure do the Proth test for the number n=h*2^m+1.\n" 
}{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 59 "proth:=proc(h::posint,m::posin
t) local a,b,n; n:=h*2^m+1;\n" }{MPLTEXT 1 0 60 "if h>=2^m then error \+
\"first parmeter is too large\",h fi;\n" }{MPLTEXT 1 0 69 "if not type
(h,odd) then error \"first parameter must be odd\",h fi;\n" }{MPLTEXT 
1 0 36 "if m=1 or m=2 then return true fi;\n" }{MPLTEXT 1 0 62 "if m=3
 then if h=5 then return true else return false fi fi;\n" }{MPLTEXT 1 
0 55 "# 2 is not a quadratic residue mod n, we start with 3\n" }
{MPLTEXT 1 0 50 "# (2*a|n)=(a|n), hence enough to try odd bases a\n" }
{MPLTEXT 1 0 22 "for a from 3 by 2 do\n" }{MPLTEXT 1 0 35 "  b:=2&^m m
od a; b:=h*b+1 mod a; \n" }{MPLTEXT 1 0 49 "  # by quadratic reciproci
ty, (a|n)=(n|a)=(b|a)\n" }{MPLTEXT 1 0 38 "  b:=modp(2&^m,a); b:=modp(
h*b+1,a);\n" }{MPLTEXT 1 0 42 "  if jacobi(b,a)=0 then return false fi
;\n" }{MPLTEXT 1 0 26 "  if jacobi(b,a)=-1 then\n" }{MPLTEXT 1 0 72 " \+
   if modp(a&^((n-1)/2),n)=n-1 then return true else return false fi;
\n" }{MPLTEXT 1 0 7 "  fi;\n" }{MPLTEXT 1 0 7 "od end;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "f*6$'I\"hG6\"I'posintG%*protectedG'I\"mGF&F'6%I\"aGF
&I\"bGF&I\"nGF&F&F&C(>F.,&*&F%\"\"\")\"\"#F*F3F3F3F3@$1F4F%Y6$Q<first~
parmeter~is~too~largeF&F%@$4-I%typeGF(6$F%I$oddGF(Y6$Q<first~parameter
~must~be~oddF&F%@$5/F*F3/F*F5OI%trueGF(@$/F*\"\"$@%/F%\"\"&FHOI&falseG
F(?(F,FLF5F&FIC(>F--I$modGF&6$-I#&^GF&6$F5F*F,>F--FV6$,&*&F%F3F-F3F3F3
F3F,>F--I%modpGF(FW>F--F\\oFgn@$/-_I*numtheoryG6$F(I(_syslibGF&I'jacob
iGF&6$F-F,\"\"!FP@$/Fao!\"\"@%/-F\\o6$-FY6$F,,&*&#F3F5F3F.F3F3FdpF[pF.
,&F.F3F3F[pFHFPF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n
" }{MPLTEXT 1 0 57 "# After a pretest the Proth test is used to find p
rimes\n" }{MPLTEXT 1 0 57 "# (h0+c*h)*2^n+1, where h runs from 0 to H-
1. All prime\n" }{MPLTEXT 1 0 62 "# divisors from interval P will be t
ried during the pretest.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n"
 }{MPLTEXT 1 0 78 "prothprimes:=proc(h0::posint,c::posint,H::posint,n:
:posint,P::range(posint))\n" }{MPLTEXT 1 0 15 "local h,L,LL;\n" }
{MPLTEXT 1 0 55 "L:=pretest(h0,H,c,2,n,\{1\},op(1,P),op(2,P)); LL:=[];
\n" }{MPLTEXT 1 0 15 "for h in L do\n" }{MPLTEXT 1 0 45 "  if proth(h0
+c*h,n) then LL:=[op(LL),h] fi\n" }{MPLTEXT 1 0 19 "od;         LL end
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6''I#h0G6\"I'posintG%*protectedG'
I\"cGF&F''I\"HGF&F''I\"nGF&F''I\"PGF&-I&rangeGF(6#F'6%I\"hGF&I\"LGF&I#
LLGF&F&F&C&>F6-I(pretestGF&6*F%F,F*\"\"#F.<#\"\"\"-I#opGF(6$F?F0-FA6$F
=F0>F77\"?&F5F6I%trueGF(@$-I&prothGF&6$,&F%F?*&F*F?F5F?F?F.>F77$-FA6#F
7F5F7F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "prothprimes(
3,30,3000,2000,7..1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7)\"$=&\"$\"
f\"$h)\"%%H\"\"%]:\"%zG\"%@H" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 
13 "5.8. Feladat." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 13 "5.9. Felad
at." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 53 "# This procedure calculate the \+
Lucas sequence terms\n" }{MPLTEXT 1 0 48 "# [V[n],V[n+1]]. If the para
meter N also given\n" }{MPLTEXT 1 0 60 "# then the calculations are do
ne mod N. Here V[n]=a^n+b^n,\n" }{MPLTEXT 1 0 42 "# where a,b are the \+
roots of x^2-Px+Q=0.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 3 " \n" }
{MPLTEXT 1 0 39 "lucasV:=proc(n,P,Q,N) local L,B,i,Qk;\n" }{MPLTEXT 1 
0 23 "B:=convert(n,base,2);\n" }{MPLTEXT 1 0 8 "Qk:=1;\n" }{MPLTEXT 1 
0 27 "if nargs=3 then L:=[2,P];\n" }{MPLTEXT 1 0 36 "  for i from nops
(B) to 1 by -1 do\n" }{MPLTEXT 1 0 20 "    if B[i]=0 then\n" }{MPLTEXT
 1 0 50 "      L:=[L[1]^2-2*Qk,L[2]*L[1]-P*Qk]; Qk:=Qk^2;\n" }{MPLTEXT
 1 0 10 "    else\n" }{MPLTEXT 1 0 54 "      L:=[L[2]*L[1]-P*Qk,L[2]^2
-2*Q*Qk]; Qk:=Qk^2*Q;\n" }{MPLTEXT 1 0 9 "    fi;\n" }{MPLTEXT 1 0 39 
"  od;                                \n" }{MPLTEXT 1 0 28 "else L:=[2
 mod N,P mod N];\n" }{MPLTEXT 1 0 36 "  for i from nops(B) to 1 by -1 \+
do\n" }{MPLTEXT 1 0 20 "    if B[i]=0 then\n" }{MPLTEXT 1 0 70 "      \+
L:=[L[1]&^2-2*Qk mod N,L[2]*L[1]-P*Qk mod N]; Qk:=Qk&^2 mod N;\n" }
{MPLTEXT 1 0 10 "    else\n" }{MPLTEXT 1 0 74 "      L:=[L[2]*L[1]-P*Q
k mod N,L[2]&^2-2*Q*Qk mod N]; Qk:=Qk&^2*Q mod N;\n" }{MPLTEXT 1 0 9 "
    fi;\n" }{MPLTEXT 1 0 7 "  od;\n" }{MPLTEXT 1 0 10 "fi; L end;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"nG6\"I\"PGF%I\"QGF%I\"NGF%6&I\"LG
F%I\"BGF%I\"iGF%I#QkGF%F%F%C&>F+-I(convertG%*protectedG6%F$I%baseGF%\"
\"#>F-\"\"\"@%/%&nargsG\"\"$C$>F*7$F5F&?(F,-I%nopsGF26#F+!\"\"F7I%true
GF2@%/&F+6#F,\"\"!C$>F*7$,&*$)&F*6#F7F5F7F7*&F5F7F-F7FC,&*&&F*6#F5F7FP
F7F7*&F&F7F-F7FC>F-*$)F-F5F7C$>F*7$FS,&*$)FUF5F7F7*(F5F7F'F7F-F7FC>F-*
&FZF7F'F7C$>F*7$-I$modGF%6$F5F(-Fbo6$F&F(?(F,F@FCF7FD@%FFC$>F*7$-Fbo6$
,&-I#&^GF%6$FPF5F7FRFCF(-Fbo6$FSF(>F--Fbo6$-F_p6$F-F5F(C$>F*7$Fap-Fbo6
$,&-F_p6$FUF5F7F[oFCF(>F--Fbo6$*&FfpF7F'F7F(F*F%F%F%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 64 "# This procedure ca
lculate a^h+a^(-h) for a unit a=r+s*sqrt(D)\n" }{MPLTEXT 1 0 62 "# wit
h norm 1. Here r,s rational numbers, D must be a square\n" }{MPLTEXT 
1 0 50 "# free integer. All computations are done mod N.\n" }{MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 41 "rieselsetup:=proc(r
,s,D,h,N) local P,a;\n" }{MPLTEXT 1 0 17 "a:=r+s*sqrt(D);\n" }{MPLTEXT
 1 0 43 "P:=r+s*sqrt(D)+(r-s*sqrt(D))/(r^2-s^2*D);\n" }{MPLTEXT 1 0 
64 "if not type(P,integer) then ERROR(a+1/a, `not an integer`) fi;\n" 
}{MPLTEXT 1 0 24 "lucasV(h,P,1,N)[1]; end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6'I\"rG6\"I\"sGF%I\"DGF%I\"hGF%I\"NGF%6$I\"PGF%I\"aGF%F
%F%C&>F,,&F$\"\"\"*&F&F0-I%sqrtGF%6#F'F0F0>F+,(F$F0F1F0*&,&F$F0F1!\"\"
F0,&*$)F$\"\"#F0F0*&)F&F=F0F'F0F9F9F0@$4-I%typeG%*protectedG6$F+I(inte
gerGFD-I&ERRORGFD6$,&F,F0*$F,F9F0I/not~an~integerGF%&-I'lucasVGF%6&F(F
+F0F)6#F0F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }
{MPLTEXT 1 0 66 "# This procedure use iteration v<-v^2-2 starting with
 v=a^h+a^-h\n" }{MPLTEXT 1 0 59 "# where the quadratic unit a=r+s*sqrt
(D) is given for the\n" }{MPLTEXT 1 0 60 "# primality testing of h*2^n
-1. Here n>1 and h<2^n is odd.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 
2 "\n" }{MPLTEXT 1 0 38 "riesel:=proc(h,n,r,s,D) local N,v,i;\n" }
{MPLTEXT 1 0 61 "if h>=2^n then error \"first parameter is too large\"
,h fi;\n" }{MPLTEXT 1 0 69 "if type(h,even) then error \"first paramet
er have to be odd\",h fi;\n" }{MPLTEXT 1 0 59 "if n<2 then error \"sec
ond parameter is too small\",h fi;\n" }{MPLTEXT 1 0 13 "N:=h*2^n-1;\n"
 }{MPLTEXT 1 0 28 "v:=rieselsetup(r,s,D,h,N);\n" }{MPLTEXT 1 0 36 "for
 i to n-2 do v:=v^2-2 mod N od;\n" }{MPLTEXT 1 0 16 "evalb(v=0); end;"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"hG6\"I\"nGF%I\"rGF%I\"sGF%I\"D
GF%6%I\"NGF%I\"vGF%I\"iGF%F%F%C)@$1)\"\"#F&F$Y6$Q=first~parameter~is~t
oo~largeF%F$@$-I%typeG%*protectedG6$F$I%evenGF9Y6$Q?first~parameter~ha
ve~to~be~oddF%F$@$2F&F2Y6$Q>second~parameter~is~too~smallF%F$>F+,&*&F$
\"\"\"F1FGFGFG!\"\">F,-I,rieselsetupGF%6'F'F(F)F$F+?(F-FGFG,&F&FGF2FHI
%trueGF9>F,-I$modGF%6$,&*$)F,F2FGFGF2FHF+-I&evalbGF96#/F,\"\"!F%F%F%" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 53 "# A
fter a pretest the Riesel test with D=5 and unit\n" }{MPLTEXT 1 0 59 "
# (1+sqrt(5))^2/4 is used to find primes (h0+30*h)*2^n-1,\n" }{MPLTEXT
 1 0 59 "# where h runs from 0 to H-1. Only the pairs [3,0],[9,0],\n" 
}{MPLTEXT 1 0 58 "# [23,0],[29,0],[7,1],[9,1],[19,1],[27,1],[11,2],[17
,2],\n" }{MPLTEXT 1 0 47 "# [21,2],[27,2],[1,3],[3,3],[13,3],[21,3] fo
r\n" }{MPLTEXT 1 0 56 "# [h0 mod 30,n mod 4] gives combinations for wh
ich the\n" }{MPLTEXT 1 0 56 "# test may used and N is not divisible by
 2,3,5, hence\n" }{MPLTEXT 1 0 56 "# this procedure should be used onl
y with these pairs.\n" }{MPLTEXT 1 0 55 "# All prime divisors until P \+
will be tried during the\n" }{MPLTEXT 1 0 12 "# pretest.\n" }{MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "rieselsqrt5:=proc(h
0,H,n,P) local h,L,LL;\n" }{MPLTEXT 1 0 45 "L:=pretest(h0,H,30,2,n,\{-
1\},7,P); LL:=[];\n" }{MPLTEXT 1 0 15 "for h in L do\n" }{MPLTEXT 1 0 
57 "  if riesel(h0+30*h,n,3/2,1/2,5) then LL:=[op(LL),h] fi\n" }
{MPLTEXT 1 0 19 "od;         LL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
f*6&I#h0G6\"I\"HGF%I\"nGF%I\"PGF%6%I\"hGF%I\"LGF%I#LLGF%F%F%C&>F+-I(pr
etestGF%6*F$F&\"#I\"\"#F'<#!\"\"\"\"(F(>F,7\"?&F*F+I%trueG%*protectedG
@$-I'rieselGF%6',&F$\"\"\"*&F2FAF*FAFAF'#\"\"$F3#FAF3\"\"&>F,7$-I#opGF
;6#F,F*F,F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rieselsq
rt5(3,3000,2000,1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7.\"$$>\"$8(\"
$I(\"%k7\"%P8\"%\\8\"%#R\"\"%#\\\"\"%7:\"%[@\"%HG\"%uH" }}}}{SECT 0 
{PARA 4 "" 0 "" {TEXT 206 12 "5.10. Ikerpr" }{TEXT 206 8 "\303\255" }
{TEXT 206 4 "mek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 58 "# This filter left onl
y those h's from the list L in the\n" }{MPLTEXT 1 0 66 "# resulting li
st for which (h0+c*h)*2^n+d is base-2 pseudoprime.\n" }{MPLTEXT 1 0 3 
"#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 99 "prob:=proc(h0::nonnegint,
c::posint,n::posint,d::integer,L::list(nonnegint)) local LL,h,N; LL:=[
];\n" }{MPLTEXT 1 0 15 "for h in L do\n" }{MPLTEXT 1 0 22 "  N:=(h0+c*
h)*2^n+d;\n" }{MPLTEXT 1 0 48 "  if modp(3&^(N-1),N)=1 then LL:=[op(LL
),h] fi\n" }{MPLTEXT 1 0 11 "od; LL end;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "f*6''I#h0G6\"I*nonnegintG%*protectedG'I\"cGF&I'posintGF('I\"nGF&F+
'I\"dGF&I(integerGF('I\"LGF&-I%listGF(6#F'6%I#LLGF&I\"hGF&I\"NGF&F&F&C
%>F77\"?&F8F2I%trueGF(C$>F9,&*&,&F%\"\"\"*&F*FDF8FDFDFD)\"\"#F-FDFDF/F
D@$/-I%modpGF(6$-I#&^GF&6$\"\"$,&F9FDFD!\"\"F9FD>F77$-I#opGF(6#F7F8F7F
&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 
58 "# This filter left only those h's from the list L in the\n" }
{MPLTEXT 1 0 60 "# resulting list for which (h0+c*h)*2^n+1 is prime. I
t use\n" }{MPLTEXT 1 0 19 "# the Proth test.\n" }{MPLTEXT 1 0 3 "#\n" 
}{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "exactplus:=proc(h0,c,n,L) local
 LL,h,a,S;\n" }{MPLTEXT 1 0 9 "LL:=[];\n" }{MPLTEXT 1 0 15 "for h in L
 do\n" }{MPLTEXT 1 0 45 "  if proth(h0+c*h,n) then LL:=[op(LL),h] fi\n
" }{MPLTEXT 1 0 11 "od; LL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I
#h0G6\"I\"cGF%I\"nGF%I\"LGF%6&I#LLGF%I\"hGF%I\"aGF%I\"SGF%F%F%C%>F*7\"
?&F+F(I%trueG%*protectedG@$-I&prothGF%6$,&F$\"\"\"*&F&F9F+F9F9F'>F*7$-
I#opGF36#F*F+F*F%F%F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 "5.11.
 Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 58 "# This filter left only those \+
h's from the list L in the\n" }{MPLTEXT 1 0 54 "# resulting list for w
hich (h0+30*h)*2^n-1 is prime.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 
2 "\n" }{MPLTEXT 1 0 38 "exactminus:=proc(h0,n,L) local LL,h;\n" }
{MPLTEXT 1 0 9 "LL:=[];\n" }{MPLTEXT 1 0 15 "for h in L do\n" }
{MPLTEXT 1 0 57 "  if riesel(h0+30*h,n,3/2,1/2,5) then LL:=[op(LL),h] \+
fi\n" }{MPLTEXT 1 0 11 "od; LL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f
*6%I#h0G6\"I\"nGF%I\"LGF%6$I#LLGF%I\"hGF%F%F%C%>F)7\"?&F*F'I%trueG%*pr
otectedG@$-I'rieselGF%6',&F$\"\"\"*&\"#IF6F*F6F6F&#\"\"$\"\"##F6F;\"\"
&>F)7$-I#opGF06#F)F*F)F%F%F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 
14 "5.12. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 51 "# After a pretest the
 probabilistic test for N-1,\n" }{MPLTEXT 1 0 62 "# the Riesel test wi
th D=5 and unit (1+sqrt(5))^2/4 for N-1,\n" }{MPLTEXT 1 0 62 "# the pr
obabilistic test for N+1, and the Proth test for N+1\n" }{MPLTEXT 1 0 
58 "# is used to find twin primes N+-1 with N=(h0+30*h)*2^n,\n" }
{MPLTEXT 1 0 46 "# where h runs from 0 to H-1. Only the pairs\n" }
{MPLTEXT 1 0 60 "# [3,0],[9,1],[21,3],[27,2], for [h0 mod 30,n mod 4] \+
gives\n" }{MPLTEXT 1 0 55 "# combinations for which the Riesel test ma
y used and\n" }{MPLTEXT 1 0 56 "# N+-1 is not divisible by 2,3,5, henc
e this procedure\n" }{MPLTEXT 1 0 60 "# should be used only with these
 pairs. All prime divisors\n" }{MPLTEXT 1 0 45 "# until P will be trie
d during the pretest.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 32 "twin:=proc(h0,H,n,P) local LL;\n" }{MPLTEXT 1 0 40 "L
L:=pretest(h0,H,30,2,n,\{1,-1\},7,P);\n" }{MPLTEXT 1 0 18 "print(nops(
LL));\n" }{MPLTEXT 1 0 26 "LL:=prob(h0,30,n,-1,LL);\n" }{MPLTEXT 1 0 
18 "print(nops(LL));\n" }{MPLTEXT 1 0 26 "LL:=exactminus(h0,n,LL);\n" 
}{MPLTEXT 1 0 18 "print(nops(LL));\n" }{MPLTEXT 1 0 25 "LL:=prob(h0,30
,n,1,LL);\n" }{MPLTEXT 1 0 18 "print(nops(LL));\n" }{MPLTEXT 1 0 27 "L
L:=exactplus(h0,30,n,LL)\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6&I#h0G6\"I\"HGF%I\"nGF%I\"PGF%6#I#LLGF%F%F%C+>F*-I(pre
testGF%6*F$F&\"#I\"\"#F'<$!\"\"\"\"\"\"\"(F(-I&printG%*protectedG6#-I%
nopsGF8F)>F*-I%probGF%6'F$F0F'F3F*F6>F*-I+exactminusGF%6%F$F'F*F6>F*-F
>6'F$F0F'F4F*F6>F*-I*exactplusGF%6&F$F0F'F*F%F%F%" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 23 "twin(3,20000,400,1000);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"%R<" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#s" }}{PARA 11 "
" 1 "" {XPPMATH 20 "\"#s" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7#\"&6k\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 56 "# This is a test for measuremen
t densities and times. \n" }{MPLTEXT 1 0 59 "# First a pretests is don
e for N+-1 where N=(h0+30*h)*2^n\n" }{MPLTEXT 1 0 57 "# and h runs fro
m 0 to H-1. The pretest is repeated for\n" }{MPLTEXT 1 0 58 "# prime l
imit P=2^e where e is integer from the interval\n" }{MPLTEXT 1 0 53 "#
 eint. The time and the remaining number after the\n" }{MPLTEXT 1 0 
55 "# pretest is reported, and p*ln(P)^2 also calculated,\n" }{MPLTEXT
 1 0 48 "# where p is the relative frequency of numbers\n" }{MPLTEXT 
1 0 54 "# passed the pretest. The result of the last pretest\n" }
{MPLTEXT 1 0 56 "# became the input of a probabilistic test for N-1 an
d\n" }{MPLTEXT 1 0 59 "# the result of this will be the input of the R
iesel test\n" }{MPLTEXT 1 0 58 "# with D=5 and unit (1+sqrt(5))^2/4 fo
r N-1. The time of\n" }{MPLTEXT 1 0 59 "# the probabilistic test and t
he result of booth tests is\n" }{MPLTEXT 1 0 52 "# reported. The proba
bilistic test for N+1 and the\n" }{MPLTEXT 1 0 48 "# Proth test for N+
1 also done. Only the pairs\n" }{MPLTEXT 1 0 59 "# [3,0],[9,1],[21,3],
[27,2] for [h0 mod 30,n mod 4] gives\n" }{MPLTEXT 1 0 42 "# combinatio
ns for which the Riesel test\n" }{MPLTEXT 1 0 38 "# may used and N+-1 \+
is not divisible\n" }{MPLTEXT 1 0 34 "# by 2,3,5, hence this procedure
\n" }{MPLTEXT 1 0 41 "# should be used only with these pairs.\n" }
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 48 "twintest:=
proc(h0,H,n,eint) local e,p,P,L,t,N;\n" }{MPLTEXT 1 0 42 "print(`Twint
est; h0=`,h0,`H=`,H,`n=`,n);\n" }{MPLTEXT 1 0 19 "N:=(h0+15*H)*2^n;\n"
 }{MPLTEXT 1 0 28 "print(`Pretest for N+-1`);\n" }{MPLTEXT 1 0 40 "for
 e from op(1,eint) to op(2,eint) do\n" }{MPLTEXT 1 0 11 "  P:=2^e;\n" 
}{MPLTEXT 1 0 14 "  t:=time();\n" }{MPLTEXT 1 0 41 "  L:=pretest(h0,H,
30,2,n,\{1,-1\},7,P);\n" }{MPLTEXT 1 0 16 "  t:=time()-t;\n" }{MPLTEXT
 1 0 24 "  p:=evalf(nops(L)/H);\n" }{MPLTEXT 1 0 68 "  print(`P=`,2^e,
`p=`,p,`p*ln(P)^2=`,evalf((ln(P))^2*p),`time=`,t)\n" }{MPLTEXT 1 0 5 "
od;\n" }{MPLTEXT 1 0 25 "print(`Test for N+-1`);\n" }{MPLTEXT 1 0 12 "
t:=time();\n" }{MPLTEXT 1 0 24 "L:=prob(h0,30,n,-1,L);\n" }{MPLTEXT 1 
0 24 "L:=exactminus(h0,n,L);\n" }{MPLTEXT 1 0 23 "L:=prob(h0,30,n,1,L)
;\n" }{MPLTEXT 1 0 26 "L:=exactplus(h0,30,n,L);\n" }{MPLTEXT 1 0 14 "t
:=time()-t;\n" }{MPLTEXT 1 0 22 "p:=evalf(nops(L)/H);\n" }{MPLTEXT 1 
0 58 "print(`p=`,p,`p*ln(N)^2=`,evalf((ln(N))^2*p),`time=`,t);\n" }
{MPLTEXT 1 0 3 "L\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6&I#h0G6\"I\"HGF%I\"nGF%I%eintGF%6(I\"eGF%I\"pGF%I\"PGF
%I\"LGF%I\"tGF%I\"NGF%F%F%C0-I&printG%*protectedG6(I.Twintest;~h0=GF%F
$I#H=GF%F&I#n=GF%F'>F/*&,&F$\"\"\"*&\"#:F;F&F;F;F;)\"\"#F'F;-F26#I1Pre
test~for~N+-1GF%?(F*-I#opGF36$F;F(F;-FE6$F?F(I%trueGF3C(>F,)F?F*>F.-I%
timeGF3F%>F--I(pretestGF%6*F$F&\"#IF?F'<$!\"\"F;\"\"(F,>F.,&FNF;F.FV>F
+-I&evalfGF36#*&-I%nopsGF36#F-F;F&FV-F26*I#P=GF%FLI#p=GF%F+I+p*ln(P)^2
=GF%-Ffn6#*&)-I#lnG6$F3I(_syslibGF%6#F,F?F;F+F;I&time=GF%F.-F26#I.Test
~for~N+-1GF%FM>F--I%probGF%6'F$FTF'FVF->F--I+exactminusGF%6%F$F'F->F--
F`p6'F$FTF'F;F->F--I*exactplusGF%6&F$FTF'F-FXFZ-F26(F_oF+I+p*ln(N)^2=G
F%-Ffn6#*&)-Ffo6#F/F?F;F+F;FjoF.F-F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "twintest(3,20000,400,3..15);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6(I.Twintest;~h0=G6\"\"\"$I#H=GF$\"&++#I#n=GF$\"$+%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I1Pretest~for~N+-1G6\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6*I#P=G6\"\"\")I#p=GF$$\"+++]Ur!#5I+p*ln(P)^2=GF$$\"+(
3s%)3$!\"*I&time=GF$$\"%Z[!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=
G6\"\"#;I#p=GF$$\"+++]W\\!#5I+p*ln(P)^2=GF$$\"+%))f4!Q!\"*I&time=GF$$
\"%2C!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"#KI#p=GF$$\"+++
])4$!#5I+p*ln(P)^2=GF$$\"+g\"4<s$!\"*I&time=GF$$\"%k5!\"$" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6*I#P=G6\"\"#kI#p=GF$$\"++++zA!#5I+p*ln(P)^2=GF$$
\"+3(G=%R!\"*I&time=GF$$\"$%o!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I
#P=G6\"\"$G\"I#p=GF$$\"+++]0<!#5I+p*ln(P)^2=GF$$\"+:=7:S!\"*I&time=GF$
$\"$#\\!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"$c#I#p=GF$$\"
++++K8!#5I+p*ln(P)^2=GF$$\"+aew&4%!\"*I&time=GF$$\"$[%!\"$" }}{PARA 11
 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"$7&I#p=GF$$\"+++]i5!#5I+p*ln(P)^2=GF
$$\"+_()*[8%!\"*I&time=GF$$\"$$R!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6*I#P=G6\"\"%C5I#p=GF$$\"++++X')!#6I+p*ln(P)^2=GF$$\"+1j^`T!\"*I&time=
GF$$\"$(Q!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"%[?I#p=GF$$
\"++++&=(!#6I+p*ln(P)^2=GF$$\"+Ok)p<%!\"*I&time=GF$$\"$H%!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"%'4%I#p=GF$$\"++++Xf!#6I+p*l
n(P)^2=GF$$\"+i@18T!\"*I&time=GF$$\"$#\\!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6*I#P=G6\"\"%#>)I#p=GF$$\"++++l^!#6I+p*ln(P)^2=GF$$\"+\"H
-Q>%!\"*I&time=GF$$\"$N'!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6
\"\"&%Q;I#p=GF$$\"++++IW!#6I+p*ln(P)^2=GF$$\"+IunrT!\"*I&time=GF$$\"$$
))!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"&oF$I#p=GF$$\"++++
gQ!#6I+p*ln(P)^2=GF$$\"+FWtsT!\"*I&time=GF$$\"%/9!\"$" }}{PARA 11 "" 1
 "" {XPPMATH 20 "I.Test~for~N+-1G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6(I#p=G6\"$\"+++++]!#9I+p*ln(N)^2=GF$$\"+?IC,U!\"*I&time=GF$$\"%J8!\"$
" }}{PARA 11 "" 1 "" {XPPMATH 20 "7#\"&6k\"" }}}}{SECT 0 {PARA 4 "" 0 
"" {TEXT 206 23 "5.13. Sophie Germain pr" }{TEXT 206 8 "\303\255" }
{TEXT 206 4 "mek." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 64 "# A pretest will be do
ne for Sophie Germain prime candidates, \n" }{MPLTEXT 1 0 68 "# to be \+
more exact, for (h0+c*h)*b^n-1 and for (h0+c*h)*b^(n+1)-1,\n" }
{MPLTEXT 1 0 64 "# where h runs from 0 to H-1. All prime divisors from
 P0 to P1\n" }{MPLTEXT 1 0 66 "# will be tried. P0 must be odd, and b,
c must have prime factors\n" }{MPLTEXT 1 0 68 "# smaller then P0. The \+
values h0+c*h, for which (h0+c*h)*b^n-1 and\n" }{MPLTEXT 1 0 66 "# (h0
+c*h)*b^(n+1)-1 are not divisible with primes from P0 to P1\n" }
{MPLTEXT 1 0 19 "# are given back.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 
1 0 2 "\n" }{MPLTEXT 1 0 62 "presophie:=proc(h0::nonnegint,H::posint,c
::posint,b::posint,\n" }{MPLTEXT 1 0 34 "n::posint,P0::posint,P1::posi
nt)\n" }{MPLTEXT 1 0 42 "local T,h,p,hh,L,nn,bb,bbb,cc,bT,cT,i,j;\n" }
{MPLTEXT 1 0 19 "T:=array(0..H-1);\n" }{MPLTEXT 1 0 36 "for h from 0 t
o H-1 do T[h]:=1 od;\n" }{MPLTEXT 1 0 69 "bT:=array(0..b-1);          \+
              # table of inverses for b\n" }{MPLTEXT 1 0 37 "for i fro
m 0 to b-1 do bT[i]:=0 od;\n" }{MPLTEXT 1 0 24 "for i from 0 to b-1 do
\n" }{MPLTEXT 1 0 65 "  for j from 0 to b-1 do if irem(i*j+1,b)=0 then
 bT[i]:=j fi od\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 69 "cT:=array
(0..c-1);                        # table of inverses for c\n" }
{MPLTEXT 1 0 37 "for i from 0 to c-1 do cT[i]:=0 od;\n" }{MPLTEXT 1 0 
24 "for i from 0 to c-1 do\n" }{MPLTEXT 1 0 65 "  for j from 0 to c-1 \+
do if irem(i*j+1,c)=0 then cT[i]:=j fi od\n" }{MPLTEXT 1 0 5 "od;\n" }
{MPLTEXT 1 0 29 "for p from P0 to P1 by 2 do\n" }{MPLTEXT 1 0 22 "  if
 isprime(p) then\n" }{MPLTEXT 1 0 30 "    cc:=(cT[p mod c]*p+1)/c;\n" 
}{MPLTEXT 1 0 22 "    nn:=modp(n,p-1);\n" }{MPLTEXT 1 0 32 "    bb:=(b
T[modp(p,b)]*p+1)/b;\n" }{MPLTEXT 1 0 26 "    bbb:=modp(bb&^nn,p);\n" 
}{MPLTEXT 1 0 30 "    hh:=modp((bbb-h0)*cc,p);\n" }{MPLTEXT 1 0 47 "  \+
  for h from hh to H-1 by p do T[h]:=0; od;\n" }{MPLTEXT 1 0 26 "    b
bb:=modp(bbb*bb,p);\n" }{MPLTEXT 1 0 30 "    hh:=modp((bbb-h0)*cc,p);
\n" }{MPLTEXT 1 0 45 "    for h from hh to H-1 by p do T[h]:=0 od\n" }
{MPLTEXT 1 0 6 "  fi\n" }{MPLTEXT 1 0 12 "od; L:=[];\n" }{MPLTEXT 1 0 
25 "for h from 0 to H-1 do \n" }{MPLTEXT 1 0 34 "  if T[h]=1 then L:=[
op(L),h] fi\n" }{MPLTEXT 1 0 18 "od;         L end;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "f*6)'I#h0G6\"I*nonnegintG%*protectedG'I\"HGF&I'posintGF
('I\"cGF&F+'I\"bGF&F+'I\"nGF&F+'I#P0GF&F+'I#P1GF&F+6/I\"TGF&I\"hGF&I\"
pGF&I#hhGF&I\"LGF&I#nnGF&I#bbGF&I$bbbGF&I#ccGF&I#bTGF&I#cTGF&I\"iGF&I
\"jGF&F&F&C.>F7-I&arrayGF(6#;\"\"!,&F*\"\"\"FL!\"\"?(F8FJFLFKI%trueGF(
>&F76#F8FL>F@-FG6#;FJ,&F/FLFLFM?(FBFJFLFWFO>&F@6#FBFJ?(FBFJFLFWFO?(FCF
JFLFWFO@$/-I%iremGF(6$,&*&FBFLFCFLFLFLFLF/FJ>FZFC>FA-FG6#;FJ,&F-FLFLFM
?(FBFJFLFdoFO>&FAFenFJ?(FBFJFLFdoFO?(FCFJFLFdoFO@$/-F[o6$F]oF-FJ>FgoFC
?(F9F3\"\"#F5FO@$-I(isprimeG6$F(I(_syslibGF&6#F9C+>F?*&,&*&&FA6#-I$mod
GF&6$F9F-FLF9FLFLFLFLFLF-FM>F<-I%modpGF(6$F1,&F9FLFLFM>F=*&,&*&&F@6#-F
cq6$F9F/FLF9FLFLFLFLFLF/FM>F>-Fcq6$-I#&^GF&6$F=F<F9>F:-Fcq6$*&,&F>FLF%
FMFLF?FLF9?(F8F:F9FKFO>FQFJ>F>-Fcq6$*&F>FLF=FLF9FdrFir>F;7\"?(F8FJFLFK
FO@$/FQFL>F;7$-I#opGF(6#F;F8F;F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 62 "# After a pretest the probabili
stic test for N-1, the Riesel\n" }{MPLTEXT 1 0 55 "# test with D=5 and
 unit (1+sqrt(5))^2/4 for N-1, the\n" }{MPLTEXT 1 0 63 "# probabilisti
c test for 2*N-1, and the Riesel test for 2*N-1\n" }{MPLTEXT 1 0 53 "#
 with D=5 and unit (1+sqrt(5))^2/4 is used to find\n" }{MPLTEXT 1 0 
60 "# Sophie Germain primes with N=(h0+30*h)*2^n, where h runs\n" }
{MPLTEXT 1 0 61 "# from 0 to H-1. Only h0=9 and n mod 4=0 used; in thi
s case\n" }{MPLTEXT 1 0 63 "# the Riesel test can be used and N-1, 2*N
-1 is not divisible\n" }{MPLTEXT 1 0 63 "# by 2,3,5 hence this procedu
re should be used only with this\n" }{MPLTEXT 1 0 61 "# pair. All prim
e divisors until P will be tried during the\n" }{MPLTEXT 1 0 12 "# pre
test.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 41 "
sophiegermain:=proc(h0,H,n,P) local LL;\n" }{MPLTEXT 1 0 33 "LL:=preso
phie(h0,H,30,2,n,7,P);\n" }{MPLTEXT 1 0 18 "print(nops(LL));\n" }
{MPLTEXT 1 0 26 "LL:=prob(h0,30,n,-1,LL);\n" }{MPLTEXT 1 0 18 "print(n
ops(LL));\n" }{MPLTEXT 1 0 26 "LL:=exactminus(h0,n,LL);\n" }{MPLTEXT 
1 0 18 "print(nops(LL));\n" }{MPLTEXT 1 0 28 "LL:=prob(h0,30,n+1,-1,LL
);\n" }{MPLTEXT 1 0 18 "print(nops(LL));\n" }{MPLTEXT 1 0 28 "LL:=exac
tminus(h0,n+1,LL);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6&I#h0G6\"I\"HGF%I\"nGF%I\"PGF%6#I#LLGF%F%F%C+>F*-I*pre
sophieGF%6)F$F&\"#I\"\"#F'\"\"(F(-I&printG%*protectedG6#-I%nopsGF5F)>F
*-I%probGF%6'F$F0F'!\"\"F*F3>F*-I+exactminusGF%6%F$F'F*F3>F*-F;6'F$F0,
&F'\"\"\"FFFFF=F*F3>F*-F@6%F$FEF*F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "sophiegermain(9,20000,400,1000);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"%P<" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#p" }}{PARA 11 "
" 1 "" {XPPMATH 20 "\"#p" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7$\"%.G\"%*G'" }}}}{SECT 0 {PARA 4 "" 0 
"" {TEXT 206 12 "5.14. Ikerpr" }{TEXT 206 8 "\303\255" }{TEXT 206 26 "
m, amely Sophie Germain pr" }{TEXT 206 8 "\303\255" }{TEXT 206 5 "m is
." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 48 "# This part contains some speci
al calculations\n" }{MPLTEXT 1 0 43 "# which are useful for search of \+
primes N\n" }{MPLTEXT 1 0 42 "# for which N+2 and 2*N+1 are prime too.
\n" }{MPLTEXT 1 0 43 "# We try to find p in the form N=h*2^e-1.\n" }
{MPLTEXT 1 0 39 "# The test given by Riesel works with\n" }{MPLTEXT 1 
0 45 "# discriminnant D if the there are integers\n" }{MPLTEXT 1 0 47 
"# a,b,r such that (a+b*sqrt(D))^2/r is a unit\n" }{MPLTEXT 1 0 38 "# \+
such that r=|a^2-b^2*D|, (D|N)=-1,\n" }{MPLTEXT 1 0 49 "# (r|D)(a^2-b^
2*D)/r=-1. We are here interested\n" }{MPLTEXT 1 0 39 "# for the choos
e D=13, a=3, b=1, r=4.\n" }{MPLTEXT 1 0 41 "# The test is then applica
ble for those\n" }{MPLTEXT 1 0 38 "# N, for which (13|N)=(N|13)=-1 and
 \n" }{MPLTEXT 1 0 40 "# (13|2*N+1)=(2*N+1|13)=-1. This gives\n" }
{MPLTEXT 1 0 47 "# the condition h*7^e mod 13 is in \{3,6,8\}.\n" }
{MPLTEXT 1 0 41 "# In these cases N+2 mod 13 also not 0.\n" }{MPLTEXT 
1 0 51 "# We plan to use the modulus 30030=2*3*5*7*11*13.\n" }{MPLTEXT
 1 0 42 "# Because N, N+2 and 2*N+1 are certainly\n" }{MPLTEXT 1 0 41 
"# not divisible by 2, 3, 5, 7 and 11 if\n" }{MPLTEXT 1 0 48 "# h divi
sible by 3, 5, 7 and 11, independently\n" }{MPLTEXT 1 0 48 "# of the c
hoose of n, we obtain a lot of cases\n" }{MPLTEXT 1 0 47 "# in which t
he discriminant D=13 may be used,\n" }{MPLTEXT 1 0 15 "# as follows:\n
" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 49 "# e mod 12 =0  : h mod 30030
 = 5775,26565,10395\n" }{MPLTEXT 1 0 49 "# e mod 12 =1  : h mod 30030 \+
= 10395,5775,12705\n" }{MPLTEXT 1 0 50 "# e mod 12 =2  : h mod 30030 =
 12705,10395,28875\n" }{MPLTEXT 1 0 50 "# e mod 12 =3  : h mod 30030 =
 28875,12705,21945\n" }{MPLTEXT 1 0 49 "# e mod 12 =4  : h mod 30030 =
 21945,28875,3465\n" }{MPLTEXT 1 0 49 "# e mod 12 =5  : h mod 30030 = \+
3465,21945,24255\n" }{MPLTEXT 1 0 49 "# e mod 12 =6  : h mod 30030 = 2
4255,3465,19635\n" }{MPLTEXT 1 0 50 "# e mod 12 =7  : h mod 30030 = 19
635,24255,17325\n" }{MPLTEXT 1 0 49 "# e mod 12 =8  : h mod 30030 = 17
325,19635,1155\n" }{MPLTEXT 1 0 48 "# e mod 12 =9  : h mod 30030 = 115
5,17325,8085\n" }{MPLTEXT 1 0 48 "# e mod 12 =10 : h mod 30030 = 8085,
1155,26565\n" }{MPLTEXT 1 0 49 "# e mod 12 =11 : h mod 30030 = 265775,
8085,5775\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 
0 43 "# These where calculated by the following\n" }{MPLTEXT 1 0 20 "#
 simple program: \n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 49 "calcsophietwinclasses:=proc() local n,L; L:=[];\n" }
{MPLTEXT 1 0 23 "for n from 0 to 11 do\n" }{MPLTEXT 1 0 55 "  [n,chrem
([1,0,0,0,0,3*7^n mod 13],[2,3,5,7,11,13]),\n" }{MPLTEXT 1 0 52 "  chr
em([1,0,0,0,0,6*7^n mod 13],[2,3,5,7,11,13]),\n" }{MPLTEXT 1 0 53 "  c
hrem([1,0,0,0,0,8*7^n mod 13],[2,3,5,7,11,13])];\n" }{MPLTEXT 1 0 17 "
  L:=[op(L),%];\n" }{MPLTEXT 1 0 20 "od;         L end;\n" }}{PARA 11 
"" 1 "" {XPPMATH 20 "f*6\"6$I\"nGF#I\"LGF#F#F#C%>F&7\"?(F%\"\"!\"\"\"
\"#6I%trueG%*protectedGC$7&F%-I&chremGF#6$7(F,F+F+F+F+-I$modGF#6$,$*&
\"\"$F,)\"\"(F%F,F,\"#87(\"\"#F;\"\"&F=F-F>-F36$7(F,F+F+F+F+-F76$,$*&
\"\"'F,F<F,F,F>F?-F36$7(F,F+F+F+F+-F76$,$*&\"\")F,F<F,F,F>F?>F&7$-I#op
GF/6#F&I\"%GF#F&F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#
\n" }{MPLTEXT 1 0 49 "# More generally, such calculations can be done
\n" }{MPLTEXT 1 0 63 "# by the following program, which find all congr
uence classes\n" }{MPLTEXT 1 0 66 "# for h for a given discriminant D=
5,13,17,29,53 and primes from\n" }{MPLTEXT 1 0 60 "# the list P, for w
hich h*2^(n+e)+d<>0 mod p and all cases\n" }{MPLTEXT 1 0 56 "# where d
=-1 are appropriate for the Riesel test by D.\n" }{MPLTEXT 1 0 36 "# L
ist L contains the pairs [e,d].\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 
0 2 "\n" }{MPLTEXT 1 0 61 "classes:=proc(D::posint,n::integer,P::list(
posint),L::list)\n" }{MPLTEXT 1 0 26 "local l,i,ll,r,p,LL,q,x;\n" }
{MPLTEXT 1 0 37 "ll:=ilcm(phi(D),op(map(i->i-1,P)));\n" }{MPLTEXT 1 0 
16 "r:=[i$i=1..D];\n" }{MPLTEXT 1 0 15 "for l in L do\n" }{MPLTEXT 1 
0 19 "  if l[2]=-1 then\n" }{MPLTEXT 1 0 54 "    r:=remove(x->jacobi(D
,x*2^(n+l[1])+l[2])<>-1,r);\n" }{MPLTEXT 1 0 8 "  else\n" }{MPLTEXT 1 
0 53 "    r:=select(x->jacobi(D,x*2^(n+l[1])+l[2])<>0,r);\n" }{MPLTEXT
 1 0 7 "  fi;\n" }{MPLTEXT 1 0 18 "od; LL:=[[D,r]];\n" }{MPLTEXT 1 0 
15 "for p in P do\n" }{MPLTEXT 1 0 20 "  r:=[i$i=0..p-1];\n" }{MPLTEXT
 1 0 17 "  for l in L do\n" }{MPLTEXT 1 0 50 "    r:=select(x->modp(x*
2^(n+l[1])+l[2],p)<>0,r)\n" }{MPLTEXT 1 0 27 "  od; LL:=[op(LL),[p,r]]
;\n" }{MPLTEXT 1 0 14 "od; ll,LL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"f*6&'I\"DG6\"I'posintG%*protectedG'I\"nGF&I(integerGF('I\"PGF&-I%list
GF(6#F''I\"LGF&F/6*I\"lGF&I\"iGF&I#llGF&I\"rGF&I\"pGF&I#LLGF&I\"qGF&I
\"xGF&F&F&C(>F6-I%ilcmGF&6$-_I*numtheoryG6$F(I(_syslibGF&I$phiGF&6#F%-
I#opGF(6#-I$mapGF(6$f*6#F5F&6$I)operatorGF&I&arrowGF&F&,&F5\"\"\"FT!\"
\"F&F&F&F->F77#-I\"$GF(6$F5/F5;FTF%?&F4F2I%trueGF(@%/&F46#\"\"#FU>F7-I
'removeGF(6$f*6#F;F&FPF&0-_FCI'jacobiGF&6$F%,&*&F;FT)F]o,&F*FT&F46#FTF
TFTFTF[oFTFUF&F&6(F%9$F*9%F48$F7>F7-I'selectGF(6$f*FcoF&FPF&0Feo\"\"!F
&F&F_pF7>F97#7$F%F7?&F8F-FhnC%>F77#-FY6$F5/F5;Fip,&F8FTFTFU?&F4F2Fhn>F
7-Fep6$f*FcoF&FPF&0-I%modpGF(6$FioF8FipF&F&6(F*FapF4FbpF88(F7>F97$-FI6
#F97$F8F76$F6F9F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "cl
asses(5,60000,[2,3,7,11,13],[[0,-1],[1,-1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#g7(7$\"\"&7#\"\"%7$\"\"#7$\"\"!\"\"\"7$\"\"$7#F,7$\"
\"(7'F,F*F/F&\"\"'7$\"#67+F,F*F/F(F&F2\"\")\"\"*\"#57$\"#87-F,F*F/F(F&
F4F8F9F:F6\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "chrem([1,
0,4],[2,3,5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "classes(13,1983*128,[2,3,5,7,11],[[
0,-1],[1,-1],[0,+1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g7(7$\"#8
7%\"\"$\"\"'\"\")7$\"\"#7$\"\"!\"\"\"7$F(7#F.7$\"\"&7$F.F,7$\"\"(7&F.F
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 1 0 37 "chrem([1,0,0,0,0,3],[2,3,5,7,11,13]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"%vd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "class
es(13,1983*128,[2,3,5,7,11],[[0,-1],[1,+1],[0,+1]]);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$\"#g7(7$\"#87&\"\"$\"\"(\"\")\"\"*7$\"\"#7$\"\"!\"\"
\"7$F(7#F/7$\"\"&7$F/F(7$F)7&F/F-\"\"%F47$\"#67*F/F(F8F4\"\"'F)F*\"#5"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "chrem([1,0,0,0,0,9],[2,3
,5,7,11,13]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&Dt\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "classes(13,21*2^14-128,[2,3,5,7,11]
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\"#87%\"\"\"\"\"#\"\"(7$F)7$\"\"!F(7$\"\"$7#F-7$\"\"&7$F-F)7$F*7&F-F(F
2\"\"'7$\"#67*F-F(F)F/\"\"%F*\"\"*\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "chrem([0,0,0,0,0,1],[2,3,5,7,11,13]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "\"%Ip" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "
classes(13,247*128,[2,3,5,7,11,13],[[0,-1],[1,-1],[2,-1]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"#g7)7$\"#87$\"\"*\"#67$\"\"#7$\"\"!\"\"\"7
$\"\"$7#F-7$\"\"&7$F-F+7$\"\"(7&F-F0F3\"\"'7$F)7*F-F.F+F0F8F6F(\"#57$F
&7,F-F.F+F3F8F6F(F;F)\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "classes(13,273*128,[2,3,5,7,11,13],[[0,-1],[1,-1],[2,-1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g7)7$\"#87$\"\"$\"\")7$\"\"#7$\"\"!
\"\"\"7$F(7#F-7$\"\"&7$F-F+7$\"\"(7&F-F(F2\"\"'7$\"#67*F-F.F+F(\"\"%F7
F5F)7$F&7,F-F+F(F;F2F7F)\"\"*F9\"#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "273*128;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&W\\$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "chrem([1,0,0,0,0,3],[2,3,5,7
,11,13]);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%vd" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "ee:=273; log[10.](2.)*128*
ee; (ee/247)^4.;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$t#" }}{PARA 11 ""
 1 "" {XPPMATH 20 "$\"+<#>>0\"!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"+.GK#\\\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "a:=expan
d((3+1*sqrt(13))^2/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&#\"#6\"\"#\"
\"\"*&#\"\"$F%F&)\"#8#F&F%F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "riesel(5110664609396115,34944,11/2,3/2,13);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "riesel(5110664609396115,34945,11/2,3/2,13);" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%trueG%*protectedG"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "riesel(5110664609396115,
34946,11/2,3/2,13);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20
 "I%trueG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "2;
 log[2.](%); 2*3; log[2.](%); 2*3*5;log[2.](%); 2*3*5*7; log[2.](%);\n
" }{MPLTEXT 1 0 52 "2*3*5*7*11; log[2.](%); 2*3*5*7*11*13; log[2.](%);
\n" }{MPLTEXT 1 0 64 "2*3*5*7*11*13*17; log[2.](%); 2*3*5*7*11*13*17*1
9; log[2.](%);\n" }{MPLTEXT 1 0 76 "2*3*5*7*11*13*17*19*23; log[2.](%)
; 2*3*5*7*11*13*17*19*23*29; log[2.](%);\n" }{MPLTEXT 1 0 43 "2*3*5*7*
11*13*17*19*23*29*31; log[2.](%);\n" }{MPLTEXT 1 0 46 "2*3*5*7*11*13*1
7*19*23*29*31*37; log[2.](%);\n" }{MPLTEXT 1 0 49 "2*3*5*7*11*13*17*19
*23*29*31*37*41; log[2.](%);\n" }{MPLTEXT 1 0 52 "2*3*5*7*11*13*17*19*
23*29*31*37*41*47; log[2.](%);\n" }{MPLTEXT 1 0 55 "2*3*5*7*11*13*17*1
9*23*29*31*37*41*47*53; log[2.](%);\n" }{MPLTEXT 1 0 58 "2*3*5*7*11*13
*17*19*23*29*31*37*41*47*53*59; log[2.](%);\n" }{MPLTEXT 1 0 61 "2*3*5
*7*11*13*17*19*23*29*31*37*41*47*53*59*61; log[2.](%);\n" }{MPLTEXT 1 
0 62 "2*3*5*7*11*13*17*19*23*29*31*37*41*47*53*59*61*67; log[2.](%);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 
"$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 
"" {XPPMATH 20 "$\"+,D'\\e#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#I"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+'f!*o!\\!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"$5#" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+=bC9x!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"%5B" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"+9xO<6!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&I+$" }}{PARA 11 "" 1 
"" {XPPMATH 20 "$\"+&o6u[\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"'50
^" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+qz:'*=!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"(!p*p*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+@2&4K#!\")"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "\"*qG4B#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+<pItF!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"+IKppk" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+;]5fK!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"-I,\\g0?" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+ZY_aP!\")
" }}{PARA 11 "" 1 "" {XPPMATH 20 "\".5[8Q2U(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+%)*paF%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"05s_j-
D/$" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+%=D7\"[!\")" }}{PARA 11 "" 1 
"" {XPPMATH 20 "\"2q)ydQi(*H9" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+pSo
m`!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"35,GYkS()yv" }}{PARA 11 "" 1
 "" {XPPMATH 20 "$\"+:hZRf!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"5!
\\E0L!)pN:Z%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+?/uFl!\")" }}{PARA 
11 "" 1 "" {XPPMATH 20 "\"7!*e6i,!exOws#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "$\"+`T\"37(!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"9IYwh3hyRm^F=" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+sIUFx!\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "chrem([1,3],[2,13]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "class
es(13,61*128,[2,3,5,7,11,17],[[0,-1],[1,-1],[2,-1],[3,-1]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"$S#7)7$\"#87#\"#67$\"\"#7$\"\"!\"\"\"7$\"
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{MPLTEXT 1 0 29 "chrem([1,0,0,11],[2,3,5,13]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"$v$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "class
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}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$S#7)7$\"#87\"7$\"\"#7$\"\"!\"\"\"7
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{MPLTEXT 1 0 65 "classes(13,30*128,[2,3,5,7,11,17],[[0,-1],[1,-1],[2,-
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "classes(17,30*128,[2,3,5,7,11,13],[
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "classes(13,29*64,[2,3,5,7,11,17,19,
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{XPPMATH 20 "7-7$\"#87#\"#67$\"\"#7$\"\"!\"\"\"7$\"\"$7#F*7$\"\"&F.7$
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{MPLTEXT 1 0 74 "classes(17,29*64,[2,3,5,7,11,13,19,23,29,31],[[4,-1],
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{XPPMATH 20 "7-7$\"#<7&\"\"#\"\"&\"#6\"#87$F&7$\"\"!\"\"\"7$\"\"$7#F,7
$F'7%F,F&\"\"%7$\"\"(7'F,F&F/F'\"\"'7$F(7+F,F&F/F3F'F5\"\")\"\"*\"#57$
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F$\"#=7$\"#B77F,F-F&F/F3F'F5F:F;F<F(F)FC\"#:FDF$FEFA\"#?\"#@\"#A7$\"#H
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "\{op(cl13[6][2])\}; \{op(cl1
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<)\"\"!\"\"\"\"\"$\"\"'\"\"(\"\"*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 
"<+\"\"!\"\"#\"\"$\"\"%\"\"&\"\"(\"\")\"\"*\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<'\"\"!\"\"$\"\"(\"\"*\"#5" }}{PARA 11 "" 1 "" {XPPMATH 
20 "\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "\{op(cl13[8][2
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 {XPPMATH 20 "<1\"\"!\"\"\"\"\"#\"\"%\"\"(\"\")\"\"*\"#5\"#6\"#8\"#9\"
#:\"#;\"#<\"#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "<3\"\"!\"\"\"\"\"#\"\"
$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#7\"#8\"#9\"#;\"#<\"#=" }}{PARA 
11 "" 1 "" {XPPMATH 20 "</\"\"!\"\"\"\"\"#\"\"%\"\"(\"\")\"\"*\"#5\"#8
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "\{op(cl13[9][2])\}; \{op(cl17[9][2]
)\}; % intersect %%; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<5\"\"!
\"\"$\"\"&\"\"'\"\"(\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?
\"#@\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "<7\"\"!\"\"\"\"\"#\"\"$\"\"%
\"\"&\"\"(\"\")\"\"*\"#5\"#6\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "<3\"\"!\"\"$\"\"&\"\"(\"\"*\"#5\"#6\"#
8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A" }}{PARA 11 "" 1 "" {XPPMATH 20 
"\"#<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "\{op(cl13[10][2])
\}; \{op(cl17[10][2])\}; % intersect %%; nops(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<;\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5
\"#6\"#7\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#A\"#C\"#D\"#F\"#G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "<=\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"
\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#E\"#
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "<9\"\"!\"\"\"\"\"#\"\"$\"\"%\"\"&\"
\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#9\"#:\"#;\"#<\"#=\"#>\"#?\"#A\"#C\"#G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#B" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "\{op(cl13[11][2])\}; \{op(cl17[11][2])\}; % intersect
 %%; nops(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "<=\"\"!\"\"\"\"\"$\"\"&
\"\"'\"\"(\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#
C\"#D\"#E\"#F\"#G\"#H\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "<?\"\"!\"\"
#\"\"$\"\"%\"\"&\"\"'\"\"(\"\")\"\"*\"#5\"#6\"#7\"#8\"#9\"#:\"#<\"#=\"
#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "<<\"\"!\"\"$\"\"&\"\"'\"\"(\"\"*\"#5\"#6\"#7\"#8\"#9\"#:
\"#<\"#=\"#>\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#G\"#H\"#I" }}{PARA 11 "
" 1 "" {XPPMATH 20 "\"#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 
"classes(17,1983*128,[2,3,5,7,11,13],[[4,-1],[5,-1],[6,-1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$S#7)7$\"#<7#\"#67$\"\"#7$\"\"!\"\"
\"7$\"\"$7#F,7$\"\"&7$F,F*7$\"\"(7&F,F/F2\"\"'7$F(7*F,F/F2F7F5\"\")\"
\"*\"#57$\"#87,F,F-F*F/\"\"%F2F7F5F:F<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "classes(5,1983*128,[2,3,7,11,13],[[0,-1],[0,+1]]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g7(7$\"\"&7#\"\"$7$\"\"#7$\"\"!\"
\"\"7$F(7#F,7$\"\"(7'F,F*F(\"\"%F&7$\"#67+F,F-F(F3F&\"\"'F1\"\")\"#57$
\"#87-F,F*F(F3F&F7F1F8\"\"*F9F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 57 "classes(5,1983*128,[2,3,7,11,13],[[0,-1],[0,+1],[1,+1]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#g7(7$\"\"&7#\"\"$7$\"\"#7$\"\"!\"\"
\"7$F(7#F,7$\"\"(7&F,F*\"\"%F&7$\"#67*F,F(F3F&\"\"'F1\"\")\"#57$\"#87,
F,F*F(F3F&F1F8\"\"*F9F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
chrem([1,0,3,0,0,0],[2,3,5,7,11,13]);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"\"%.I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "classes(5,247*128
,[2,3,7,11,13],[[0,+1],[1,+1],[2,+1]]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$\"#g7(7$\"\"&7$\"\"$F&7$\"\"#7$\"\"!\"\"\"7$F(7#F,7$\"\"(7&F,F-F
*\"\"%7$\"#67*F,F-F*F3F&\"\")\"\"*\"#57$\"#87,F,F-F*F(F3\"\"'F1F7F5\"#
7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "classes(5,61*128,[2,3,
7,11,13,17],[[0,+1],[1,+1],[2,+1],[3,+1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"$S#7)7$\"\"&7#F&7$\"\"#7$\"\"!\"\"\"7$\"\"$7#F+7$\"\"
(7&F+F,F)\"\"%7$\"#67)F+F,F)F.F3\"\"'\"\")7$\"#87+F+F,F)F.F3F7F1F8\"#7
7$\"#<7/F+F,F.F&F7F1\"\"*\"#5F5F<F:\"#9\"#:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 23 "chrem([1,0,0],[2,3,5]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "classe
s(5,30*128,[2,3,7,11,13,17,19],[[0,+1],[1,+1],[2,+1],[3,+1],[4,+1]]);"
 }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$?(7*7$\"\"&7#F&7$\"\"#7$\"\"!\"
\"\"7$\"\"$7#F+7$\"\"(7&F+F,F)\"\"%7$\"#67(F+F,F.\"\"'F1\"\"*7$\"#87*F
+F,F)F&F1F8\"#5F57$\"#<7.F+F.F&F7F1F8F<F5\"#7F:\"#9\"#:7$\"#>70F+F.F&F
7F1F8F5F@F:FAFB\"#;F>\"#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
90 "classes(5,29*64,[2,3,7,11,13,17,19,23,29,31],[[0,+1],[1,+1],[2,+1]
,[3,+1],[4,+1],[5,+1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&Sa&7-7$
\"\"&7#F&7$\"\"#7$\"\"!\"\"\"7$\"\"$7#F+7$\"\"(7&F+F,F)\"\"%7$\"#67'F+
F,F)F3\"\")7$\"#87)F+F,F)F.F3F1F77$\"#<7-F+F.F&\"\"'F1\"#5F5\"#7F9\"#9
\"#:7$\"#>7/F+F,F)F.F&F>\"\"*F?F5F@FBF<\"#=7$\"#B73F+F,F)F.F3F&F>F1F7F
FF?F@F9FA\"#;FG\"#?7$\"#H79F+F,F&F1FFF?F5F@F9FAFBF<FGFDFL\"#@\"#AFI\"#
C\"#D\"#E\"#F\"#G7$\"#J7<F+F,F)F.F3F&F>F1F7FFF?F5F@F9FAFKF<FGFDFLFPFQF
RFSFTFV" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "classes(17,0,[2,
3,5,7,11],[[0,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$S#7(7$\"#<7
*\"\"%\"\"'\"\"(\"\")\"#6\"#7\"#8\"#:7$\"\"#7#\"\"!7$\"\"$7$F3F17$\"\"
&7&F3F1F5F(7$F*7(F3F1F5F(F8F)7$F,7,F3F1F5F(F8F)F*F+\"\"*\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "classes(17,0,[2,3,5,7,11],[[
0,-1],[1,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$S#7(7$\"#<7&\"\"
%\"\"'\"#7\"#:7$\"\"#7#\"\"!7$\"\"$F.7$\"\"&7%F/F-F(7$\"\"(7'F/F-F1F3F
)7$\"#67+F/F-F1F(F3F6\"\")\"\"*\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "classes(17,0,[2,3,5,7,11],[[0,-1],[1,-1],[2,-1]]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$S#7(7$\"#<7#\"\"'7$\"\"#7#\"\"!7$
\"\"$F+7$\"\"&7$F,F*7$\"\"(7&F,F.F0F(7$\"#67*F,F*\"\"%F0F3\"\")\"\"*\"
#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "classes(17,0,[2,3,5,7
,11],[[0,-1],[1,-1],[2,-1],[3,-1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$\"$S#7(7$\"#<7\"7$\"\"#7#\"\"!7$\"\"$F*7$\"\"&F*7$\"\"(7&F+F-F/\"\"'
7$\"#67)F+F)\"\"%F/\"\")\"\"*\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "classes(29,0,[2,3,5,7,11],[[0,-1],[1,-1],[2,-1],[3,-1
]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$?%7(7$\"#H7#\"#?7$\"\"#7#\"
\"!7$\"\"$F+7$\"\"&F+7$\"\"(7&F,F.F0\"\"'7$\"#67)F,F*\"\"%F0\"\")\"\"*
\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "classes(29,0,[2,3,5
,7,11],[[0,-1],[1,-1],[2,-1],[3,-1],[4,-1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"$?%7(7$\"#H7\"7$\"\"#7#\"\"!7$\"\"$F*7$\"\"&F*7$\"\"(
7&F+F-F/\"\"'7$\"#67(F+F)\"\"%F/\"\")\"#5" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 57 "classes(53,0,[2,3,5,7,11],[[0,-1],[1,-1],[2,-1],[3,-
1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$!y7(7$\"#`7#\"#@7$\"\"#7#
\"\"!7$\"\"$F+7$\"\"&F+7$\"\"(7&F,F.F0\"\"'7$\"#67)F,F*\"\"%F0\"\")\"
\"*\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "classes(53,0,[2,
3,5,7,11],[[0,-1],[1,-1],[2,-1],[3,-1],[4,-1]]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"$!y7(7$\"#`7\"7$\"\"#7#\"\"!7$\"\"$F*7$\"\"&F*7$\"\"(
7&F+F-F/\"\"'7$\"#67(F+F)\"\"%F/\"\")\"#5" }}}}{SECT 0 {PARA 4 "" 0 ""
 {TEXT 206 25 "5.15. n^2+1 es n^4+1 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 3 "#
\n" }{MPLTEXT 1 0 58 "# A pretest will be done for prime candidates ha
ving the\n" }{MPLTEXT 1 0 59 "# form n^2+1, to be more exact, for (h0+
c*h)^2*b^(2*n)+1,\n" }{MPLTEXT 1 0 58 "# where h runs from 0 to H-1. A
ll prime divisors from P0\n" }{MPLTEXT 1 0 53 "# to P1 will be tried. \+
P0 must be odd, and b,c must\n" }{MPLTEXT 1 0 57 "# have prime factors
 smaller then P0. The values h, for\n" }{MPLTEXT 1 0 59 "# which (h0+c
*h)^2*b^(2*n)+1 is not divisible with primes\n" }{MPLTEXT 1 0 33 "# fr
om P0 to P1 are given back.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 39 "presquareplus:=proc(h0,H,c,b,n,P0,P1)\n" }
{MPLTEXT 1 0 46 "local T,h,p,hh,L,nn,bb,bbb,cc,bT,cT,i,j,R,r;\n" }
{MPLTEXT 1 0 19 "T:=array(0..H-1);\n" }{MPLTEXT 1 0 36 "for h from 0 t
o H-1 do T[h]:=1 od;\n" }{MPLTEXT 1 0 69 "bT:=array(0..b-1);          \+
              # table of inverses for b\n" }{MPLTEXT 1 0 37 "for i fro
m 0 to b-1 do bT[i]:=0 od;\n" }{MPLTEXT 1 0 24 "for i from 0 to b-1 do
\n" }{MPLTEXT 1 0 26 "  for j from 0 to b-1 do\n" }{MPLTEXT 1 0 41 "  \+
  if irem(i*j+1,b)=0 then bT[i]:=j fi\n" }{MPLTEXT 1 0 6 "  od\n" }
{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 69 "cT:=array(0..c-1);           \+
             # table of inverses for c\n" }{MPLTEXT 1 0 37 "for i from
 0 to c-1 do cT[i]:=0 od;\n" }{MPLTEXT 1 0 24 "for i from 0 to c-1 do
\n" }{MPLTEXT 1 0 26 "  for j from 0 to c-1 do\n" }{MPLTEXT 1 0 41 "  \+
  if irem(i*j+1,c)=0 then cT[i]:=j fi\n" }{MPLTEXT 1 0 6 "  od\n" }
{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 29 "for p from P0 to P1 by 2 do\n
" }{MPLTEXT 1 0 22 "  if isprime(p) then\n" }{MPLTEXT 1 0 22 "    r:=m
sqrt(p-1,p);\n" }{MPLTEXT 1 0 21 "    if r<>FAIL then\n" }{MPLTEXT 1 
0 19 "      R:=[r,p-r];\n" }{MPLTEXT 1 0 32 "      cc:=(cT[p mod c]*p+
1)/c;\n" }{MPLTEXT 1 0 24 "      nn:=n mod (p-1);\n" }{MPLTEXT 1 0 32 
"      bb:=(bT[p mod b]*p+1)/b;\n" }{MPLTEXT 1 0 26 "      bbb:=bb&^nn
 mod p;\n" }{MPLTEXT 1 0 21 "      for r in R do\n" }{MPLTEXT 1 0 34 "
        hh:=(r*bbb-h0)*cc mod p;\n" }{MPLTEXT 1 0 38 "        for h fr
om hh to H-1 by p do\n" }{MPLTEXT 1 0 20 "          T[h]:=0;\n" }
{MPLTEXT 1 0 13 "        od;\n" }{MPLTEXT 1 0 11 "      od;\n" }
{MPLTEXT 1 0 9 "    fi;\n" }{MPLTEXT 1 0 7 "  fi;\n" }{MPLTEXT 1 0 5 "
od;\n" }{MPLTEXT 1 0 8 "L:=[];\n" }{MPLTEXT 1 0 25 "for h from 0 to H-
1 do \n" }{MPLTEXT 1 0 39 "  if T[h]=1 then L:=[op(L),h0+c*h] fi\n" }
{MPLTEXT 1 0 10 "od; L end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6)I#h0G
6\"I\"HGF%I\"cGF%I\"bGF%I\"nGF%I#P0GF%I#P1GF%61I\"TGF%I\"hGF%I\"pGF%I#
hhGF%I\"LGF%I#nnGF%I#bbGF%I$bbbGF%I#ccGF%I#bTGF%I#cTGF%I\"iGF%I\"jGF%I
\"RGF%I\"rGF%F%F%C.>F--I&arrayG%*protectedG6#;\"\"!,&F&\"\"\"FE!\"\"?(
F.FCFEFDI%trueGF@>&F-6#F.FE>F6-F?6#;FC,&F(FEFEFF?(F8FCFEFPFH>&F66#F8FC
?(F8FCFEFPFH?(F9FCFEFPFH@$/-I%iremGF@6$,&*&F8FEF9FEFEFEFEF(FC>FSF9>F7-
F?6#;FC,&F'FEFEFF?(F8FCFEF]oFH>&F7FTFC?(F8FCFEF]oFH?(F9FCFEF]oFH@$/-FZ
6$FfnF'FC>F`oF9?(F/F*\"\"#F+FH@$-I(isprimeG6$F@I(_syslibGF%6#F/C$>F;-_
I*numtheoryGF]pI&msqrtGF%6$,&F/FEFEFFF/@$0F;I%FAILGF@C(>F:7$F;,&F/FEF;
FF>F5*&,&*&&F76#-I$modGF%6$F/F'FEF/FEFEFEFEFEF'FF>F2-Ffq6$F)Fgp>F3*&,&
*&&F66#-Ffq6$F/F(FEF/FEFEFEFEFEF(FF>F4-Ffq6$-I#&^GF%6$F3F2F/?&F;F:FHC$
>F0-Ffq6$*&,&*&F;FEF4FEFEF$FFFEF5FEF/?(F.F0F/FDFH>FJFC>F17\"?(F.FCFEFD
FH@$/FJFE>F17$-I#opGF@6#F1,&F$FE*&F'FEF.FEFEF1F%F%F%" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 58 "# This filter left \+
only those h's from the list L in the\n" }{MPLTEXT 1 0 52 "# resulting
 list for which (h0+c*h)^2*2^(2*n)+1 is\n" }{MPLTEXT 1 0 19 "# probabl
y prime.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 
46 "squareplusprob:=proc(h0,c,n,L) local LL,h,N;\n" }{MPLTEXT 1 0 9 "L
L:=[];\n" }{MPLTEXT 1 0 15 "for h in L do\n" }{MPLTEXT 1 0 28 "  N:=(h
0+c*h)^2*2^(2*n)+1;\n" }{MPLTEXT 1 0 48 "  if modp(2&^(N-1),N)=1 then \+
LL:=[op(LL),h] fi\n" }{MPLTEXT 1 0 19 "od;         LL end;" }}{PARA 11
 "" 1 "" {XPPMATH 20 "f*6&I#h0G6\"I\"cGF%I\"nGF%I\"LGF%6%I#LLGF%I\"hGF
%I\"NGF%F%F%C%>F*7\"?&F+F(I%trueG%*protectedGC$>F,,&*&),&F$\"\"\"*&F&F
9F+F9F9\"\"#F9)F;,$*&F;F9F'F9F9F9F9F9F9@$/-I%modpGF26$-I#&^GF%6$F;,&F,
F9F9!\"\"F,F9>F*7$-I#opGF26#F*F+F*F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 58 "# This filter left only those h
's from the list L in the\n" }{MPLTEXT 1 0 52 "# resulting list for wh
ich (h0+c*h)^2*2^(2*n)+1 is\n" }{MPLTEXT 1 0 10 "# prime.\n" }{MPLTEXT
 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 47 "squareplusproth:=p
roc(h0,c,n,L) local LL,h,N;\n" }{MPLTEXT 1 0 9 "LL:=[];\n" }{MPLTEXT 
1 0 15 "for h in L do\n" }{MPLTEXT 1 0 28 "  N:=(h0+c*h)^2*2^(2*n)+1;
\n" }{MPLTEXT 1 0 51 "  if proth((h0+c*h)^2,2*n) then LL:=[op(LL),h] f
i\n" }{MPLTEXT 1 0 19 "od;         LL end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6&I#h0G6\"I\"cGF%I\"nGF%I\"LGF%6%I#LLGF%I\"hGF%I\"NGF%F
%F%C%>F*7\"?&F+F(I%trueG%*protectedGC$>F,,&*&),&F$\"\"\"*&F&F9F+F9F9\"
\"#F9)F;,$*&F;F9F'F9F9F9F9F9F9@$-I&prothGF%6$*$F7F9F=>F*7$-I#opGF26#F*
F+F*F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT
 1 0 51 "# This is a test to measure densities and times. \n" }
{MPLTEXT 1 0 39 "# First pretests are done for N where\n" }{MPLTEXT 1 
0 52 "# N=(h0+2*h)^2*2^(2*n)+1 and h runs from 0 to H-1.\n" }{MPLTEXT 
1 0 55 "# The pretest is repeated for prime limit P=2^e where\n" }
{MPLTEXT 1 0 49 "# e is integer from the interval eint. The time\n" }
{MPLTEXT 1 0 60 "# and the number of remaining candidates after the pr
etest\n" }{MPLTEXT 1 0 60 "# is reported, and p*ln(P) also calculated,
 where p is the\n" }{MPLTEXT 1 0 57 "# relative frequency of numbers p
assed the pretest. The\n" }{MPLTEXT 1 0 55 "# result of the last prete
st became the input of the \n" }{MPLTEXT 1 0 53 "# probabilistic test \+
for N. The time and the result\n" }{MPLTEXT 1 0 34 "# of this test is \+
also reported.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT
 1 0 54 "squareplustest:=proc(h0,H,n,eint) local e,p,P,L,t,N;\n" }
{MPLTEXT 1 0 48 "print(`Squareplustest; h0=`,h0,`H=`,H,`n=`,n);\n" }
{MPLTEXT 1 0 22 "N:=(h0+H)^2*2^(2*n);\n" }{MPLTEXT 1 0 25 "print(`Pret
est for N`);\n" }{MPLTEXT 1 0 40 "for e from op(1,eint) to op(2,eint) \+
do\n" }{MPLTEXT 1 0 11 "  P:=2^e;\n" }{MPLTEXT 1 0 14 "  t:=time();\n"
 }{MPLTEXT 1 0 37 "  L:=presquareplus(h0,H,2,2,n,3,P);\n" }{MPLTEXT 1 
0 16 "  t:=time()-t;\n" }{MPLTEXT 1 0 24 "  p:=evalf(nops(L)/H);\n" }
{MPLTEXT 1 0 64 "  print(`P=`,2^e,`p=`,p,`p*ln(P)=`,evalf((ln(P))*p),`
time=`,t)\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 36 "print(`Probabil
istic test for N`);\n" }{MPLTEXT 1 0 12 "t:=time();\n" }{MPLTEXT 1 0 
30 "L:=squareplusprob(h0,2,n,L);\n" }{MPLTEXT 1 0 14 "t:=time()-t;\n" 
}{MPLTEXT 1 0 22 "p:=evalf(nops(L)/H);\n" }{MPLTEXT 1 0 54 "print(`p=`
,p,`p*ln(N)=`,evalf((ln(N))*p),`time=`,t);\n" }{MPLTEXT 1 0 28 "print(
`Proth test for N`);\n" }{MPLTEXT 1 0 12 "t:=time();\n" }{MPLTEXT 1 0 
31 "L:=squareplusproth(h0,2,n,L);\n" }{MPLTEXT 1 0 14 "t:=time()-t;\n"
 }{MPLTEXT 1 0 22 "p:=evalf(nops(L)/H);\n" }{MPLTEXT 1 0 54 "print(`p=
`,p,`p*ln(N)=`,evalf((ln(N))*p),`time=`,t);\n" }{MPLTEXT 1 0 6 "L end;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I#h0G6\"I\"HGF%I\"nGF%I%eintGF%6
(I\"eGF%I\"pGF%I\"PGF%I\"LGF%I\"tGF%I\"NGF%F%F%C3-I&printG%*protectedG
6(I4Squareplustest;~h0=GF%F$I#H=GF%F&I#n=GF%F'>F/*&),&F$\"\"\"F&F<\"\"
#F<)F=,$*&F=F<F'F<F<F<-F26#I.Pretest~for~NGF%?(F*-I#opGF36$F<F(F<-FF6$
F=F(I%trueGF3C(>F,)F=F*>F.-I%timeGF3F%>F--I.presquareplusGF%6)F$F&F=F=
F'\"\"$F,>F.,&FOF<F.!\"\">F+-I&evalfGF36#*&-I%nopsGF36#F-F<F&FX-F26*I#
P=GF%FMI#p=GF%F+I)p*ln(P)=GF%-Fen6#*&-I#lnG6$F3I(_syslibGF%6#F,F<F+F<I
&time=GF%F.-F26#I9Probabilistic~test~for~NGF%FN>F--I/squareplusprobGF%
6&F$F=F'F-FVFY-F26(F^oF+I)p*ln(N)=GF%-Fen6#*&-Fdo6#F/F<F+F<FhoF.-F26#I
1Proth~test~for~NGF%FN>F--I0squareplusprothGF%F_pFVFYF`pF-F%F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "squareplustest(1,20000,1000,
2..15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(I4Squareplustest;~h0=G6\"\"
\"\"I#H=GF$\"&++#I#n=GF$\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "I.Prete
st~for~NG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"\"%I#p=GF$$
\"\"\"\"\"!I)p*ln(P)=GF$$\"+hVH'Q\"!\"*I&time=GF$$\"%t)*!\"$" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"\")I#p=GF$$\"+++++g!#5I)p*ln(P)=GF
$$\"+D\\mZ7!\"*I&time=GF$$\"%*y$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6*I#P=G6\"\"#;I#p=GF$$\"+++]w]!#5I)p*ln(P)=GF$$\"+lY]29!\"*I&time=GF$$
\"%RF!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"#KI#p=GF$$\"+++
+rT!#5I)p*ln(P)=GF$$\"+X%ebW\"!\"*I&time=GF$$\"%N>!\"$" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6*I#P=G6\"\"#kI#p=GF$$\"+++]!\\$!#5I)p*ln(P)=GF$$\"+
S\"e;X\"!\"*I&time=GF$$\"%U9!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#
P=G6\"\"$G\"I#p=GF$$\"+++]pI!#5I)p*ln(P)=GF$$\"+!pI$*[\"!\"*I&time=GF$
$\"%!=\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"$c#I#p=GF$$
\"+++]dF!#5I)p*ln(P)=GF$$\"+!o#3H:!\"*I&time=GF$$\"%=5!\"$" }}{PARA 11
 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"$7&I#p=GF$$\"++++]C!#5I)p*ln(P)=GF$$
\"+L&*QG:!\"*I&time=GF$$\"$r)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I
#P=G6\"\"%C5I#p=GF$$\"+++]8A!#5I)p*ln(P)=GF$$\"+%G\"GM:!\"*I&time=GF$$
\"$9)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"%[?I#p=GF$$\"++
+]6?!#5I)p*ln(P)=GF$$\"+4@pL:!\"*I&time=GF$$\"$l(!\"$" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6*I#P=G6\"\"%'4%I#p=GF$$\"++++c=!#5I)p*ln(P)=GF$$\"+,
uxV:!\"*I&time=GF$$\"$l(!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6
\"\"%#>)I#p=GF$$\"++++5<!#5I)p*ln(P)=GF$$\"+#=m3a\"!\"*I&time=GF$$\"$A
)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*I#P=G6\"\"&%Q;I#p=GF$$\"+++]&
f\"!#5I)p*ln(P)=GF$$\"+dGG[:!\"*I&time=GF$$\"$%)*!\"$" }}{PARA 11 "" 1
 "" {XPPMATH 20 "6*I#P=G6\"\"&oF$I#p=GF$$\"+++]%[\"!#5I)p*ln(P)=GF$$\"
+%[lMa\"!\"*I&time=GF$$\"%&Q\"!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I9
Probabilistic~test~for~NG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(I#p=G6
\"$\"+++++S!#8I)p*ln(N)=GF$$\"+XdSCc!#5I&time=GF$$\"']\"3\"!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "I1Proth~test~for~NG6\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6(I#p=G6\"$\"+++++S!#8I)p*ln(N)=GF$$\"+XdSCc!#5I&time=
GF$$\"$(H!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7*\"&4b\"\"&*p:\"&\\O#
\"&**=$\"&&zL\"&4g$\"&*=Q\"&*>Q" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 54 "# Estimate of the powplusone density Cpo
wplusone for\n" }{MPLTEXT 1 0 69 "# polynomial (X*2^n)^(2^e)+1 (n fix)
 by calculating the product for\n" }{MPLTEXT 1 0 19 "# primes below x.
\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 51 "Cpowp
lusone:=proc(e::posint,x::posint) local P,p;\n" }{MPLTEXT 1 0 25 "P:=2
.; p:=nextprime(2);\n" }{MPLTEXT 1 0 14 "while p<x do\n" }{MPLTEXT 1 
0 29 "  if modp(p,2^(e+1))=1 then\n" }{MPLTEXT 1 0 28 "    P:=P*(1-2^e
/p)/(1-1/p)\n" }{MPLTEXT 1 0 8 "  else\n" }{MPLTEXT 1 0 18 "    P:=P/(
1-1/p)\n" }{MPLTEXT 1 0 7 "  fi;\n" }{MPLTEXT 1 0 19 "  p:=nextprime(p
)\n" }{MPLTEXT 1 0 10 "od; P end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6
$'I\"eG6\"I'posintG%*protectedG'I\"xGF&F'6$I\"PGF&I\"pGF&F&F&C&>F,$\"
\"#\"\"!>F--I*nextprimeG6$F(I(_syslibGF&6#F1?(F&\"\"\"F:F&2F-F*C$@%/-I
%modpGF(6$F-)F1,&F%F:F:F:F:>F,*(F,F:,&F:F:*&)F1F%F:F-!\"\"FIF:,&F:F:*$
F-FIFIFI>F,*&F,F:FJFI>F--F56#F-F,F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "Cpowplusone(1,10); Cpowplusone(1,100); Cpowplusone(1,
1000);\n" }{MPLTEXT 1 0 68 "Cpowplusone(1,10000); Cpowplusone(1,100000
); Cpowplusone(1,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"++++DE!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+_=4.F!\"*" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "$\"+*o24u#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+J]/U
F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+:5qWF!\"*" }}{PARA 11 "" 1 
"" {XPPMATH 20 "$\"+<5iXF!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 61 "Cpowplusone(2,10); Cpowplusone(2,100); Cpowplusone(2,1000);\n" }
{MPLTEXT 1 0 68 "Cpowplusone(2,10000); Cpowplusone(2,100000); Cpowplus
one(2,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"++++vV!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"+?a@k\\!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+Ccg2`!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+\"p3aM
&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+'[y;N&!\"*" }}{PARA 11 "" 1
 "" {XPPMATH 20 "$\"+/E,d`!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 45 "# This procedure calculate the factor qe
AB.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 54 "qe
AB:=proc(e::posint,A::posint,B::posint) local P,p;\n" }{MPLTEXT 1 0 
27 "P:=1.; p:=nextprime(A-1);\n" }{MPLTEXT 1 0 14 "while p<B do\n" }
{MPLTEXT 1 0 48 "  if modp(p,2^(e+1))=1 then P:=P*(1-2^e/p) fi;\n" }
{MPLTEXT 1 0 19 "  p:=nextprime(p)\n" }{MPLTEXT 1 0 10 "od; P end;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"eG6\"I'posintG%*protectedG'I\"AG
F&F''I\"BGF&F'6$I\"PGF&I\"pGF&F&F&C&>F.$\"\"\"\"\"!>F/-I*nextprimeG6$F
(I(_syslibGF&6#,&F*F3F3!\"\"?(F&F3F3F&2F/F,C$@$/-I%modpGF(6$F/)\"\"#,&
F%F3F3F3F3>F.*&F.F3,&F3F3*&)FFF%F3F/F<F<F3>F/-F76#F/F.F&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# square plus one prime" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f:=h->((2*h+1)*2.^(340000.*4
))^2+1;\n" }{MPLTEXT 1 0 17 "g:=h->1/ln(f(h));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6#I\"hG6\"F%6$I)operatorGF%I&arrowGF%F%,&*&),&*&\"\"#\"
\"\"F$F/F/F/F/F.F/))$F.\"\"!*&$\"'++MF3F/\"\"%F/F.F/F/F/F/F%F%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"hG6\"F%6$I)operatorGF%I&arrowGF%F
%*$-I#lnG6$%*protectedGI(_syslibGF%6#-I\"fGF%F#!\"\"F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "H:=2.^21; evalf(H/6*(g(0)+4*g(H/2)+
g(H)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"(_r4#\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "$\"++/K76!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "%*2.745621017; # expected primes" }}{PARA 11 "" 1 "" {XPPMATH 
20 "$\"+o-,aI!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "qeAB(1
,3,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+:7x:6!#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "%*(ln(1000000.)/(40*ln(2.)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"+Awwfb!#6" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 41 "%*H*160*20/3.054010268; # SPU time in sec" }}{PARA 
11 "" 1 "" {XPPMATH 20 "$\"+4Tq@7!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 47 "%/24/3600/16; # time for one Cell blade in days" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"+f'ev$))!\")" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "# quad plus one prime" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "f:=h->((2*h+1)*2.^(340000.*2))^4+1;\n" }{MPLTEXT 
1 0 17 "g:=h->1/ln(f(h));" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"hG6
\"F%6$I)operatorGF%I&arrowGF%F%,&*&),&*&\"\"#\"\"\"F$F/F/F/F/\"\"%F/))
$F.\"\"!*&$\"'++MF4F/F.F/F0F/F/F/F/F%F%F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "f*6#I\"hG6\"F%6$I)operatorGF%I&arrowGF%F%*$-I#lnG6$%*prot
ectedGI(_syslibGF%6#-I\"fGF%F#!\"\"F%F%F%" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 42 "H:=2.^15; evalf(H/6*(g(0)+4*g(H/2)+g(H)));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "$\"&oF$\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
$\"+G3*zt\"!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "%*5.35701
2604; # expected primes" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+r(Q/J*!#6
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "qeAB(2,3,1000000);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"+AT*p<#!#5" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "%*(ln(1000000.)/(40*ln(2.)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+!psZ3\"!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "%*H*160*20/0.09310438771; # SPU time in sec" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+n5r@7!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "%/24/3600/16; # time for one Cell blade in days" }}{PARA 11 "" 1 "
" {XPPMATH 20 "$\"+#**3w$))!\")" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
206 9 "5.16. Egy" }{TEXT 206 8 "\303\251" }{TEXT 206 7 "b speci" }
{TEXT 206 8 "\303\241" }{TEXT 206 8 "lis alak" }{TEXT 206 8 "\303\272"
 }{TEXT 206 3 " pr" }{TEXT 206 8 "\303\255" }{TEXT 206 4 "mek." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "#\n" }{MPLTEXT 1 0 69 "# Cullen and Woodall numbers: n*2^n+-1; al
l known below n<=8000000.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n
" }{MPLTEXT 1 0 60 "x:=20000000; evalf(int(y->2/y/ln(2.)/ln(ln(y)),800
0000..x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\")+++?" }}{PARA 11 "" 1 "
" {XPPMATH 20 ",&$\")v'>Y*!\")\"\"\"*&$\"\"!F)F&^#F&F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 45 "# This proced
ure calculate the factor qsAB.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 
2 "\n" }{MPLTEXT 1 0 54 "qsAB:=proc(s::posint,A::posint,B::posint) loc
al P,p;\n" }{MPLTEXT 1 0 27 "P:=1.; p:=nextprime(A-1);\n" }{MPLTEXT 1 
0 48 "while p<B do P:=P*(1-s/p); p:=nextprime(p) od;\n" }{MPLTEXT 1 0 
6 "P end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%'I\"sG6\"I'posintG%*pro
tectedG'I\"AGF&F''I\"BGF&F'6$I\"PGF&I\"pGF&F&F&C&>F.$\"\"\"\"\"!>F/-I*
nextprimeG6$F(I(_syslibGF&6#,&F*F3F3!\"\"?(F&F3F3F&2F/F,C$>F.*&F.F3,&F
3F3*&F%F3F/F<F<F3>F/-F76#F/F.F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "qsAB(1,3,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"+;;kF\")!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "%*(ln(1000
000.)/(50*ln(2.))); # decreasing factor of sieve" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+BH$*RK!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "%*12000000*160*16*20; # SPU time in sec" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+)y91*>\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "%/24/3600/16; # time for one Cell blade in days" }}{PARA 11 "" 1
 "" {XPPMATH 20 "$\"+)=q*R9!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 70 "# Primorial numbers: p#+-1; p#><exp(p); \+
record (02.2011): 843301#-1.\n" }{MPLTEXT 1 0 63 "# Expected number of
 such primes for +1 and -1, respectively,\n" }{MPLTEXT 1 0 59 "# up to
 x is exp(gamma)*ln(x), hence the next expected by\n" }{MPLTEXT 1 0 4 
"# \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 57 "x:=evalf(exp(exp(-gamma)
+ln(843301.))); # around this p\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 
44 "log[10.](exp(x)); # so many decimal digits\n" }{MPLTEXT 1 0 2 "\n"
 }{MPLTEXT 1 0 62 "x/ln(x)-843301/ln(843301.); # so many to test, with
out sieve\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+O,]y9!\"$" }}{PARA 11
 "" 1 "" {XPPMATH 20 "$\"+1X/@k!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"+3L#pA%!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "%*300*128*
3; # SPU time in sec" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+^cTp[\"\"!" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "%/24/3600/420; # time for
 all PS3 in days" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+:/)=M\"!\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 74 "# Fact
orial numbers: n!+-1; n!><n^n/exp(n); record (02.2011): 103040!-1.\n" 
}{MPLTEXT 1 0 63 "# Expected number of such primes for +1 and -1, resp
ectively,\n" }{MPLTEXT 1 0 59 "# up to x is exp(gamma)*ln(x), hence th
e next expected by\n" }{MPLTEXT 1 0 4 "# \n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 57 "x:=evalf(exp(exp(-gamma)+ln(102030.))); # around this
 n\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 48 "log[10.](x^x/exp(x)); # s
o many decimal digits\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 44 "x-1030
40; # so many to test, without sieve\n" }}{PARA 11 "" 1 "" {XPPMATH 20
 "$\"+,+#))y\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+5;->')!\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"*,+Ue(!\"%" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 28 "%*300*200; # SPU time in sec" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+1+_]X\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "%/24/3600/420; # time for all PS3 in days" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+C8+a7!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "#\n" }{MPLTEXT 1 0 63 "# Primes in arithmetic sequences, 3 primes; \+
record (02.2011):\n" }{MPLTEXT 1 0 64 "# 1185*2^529445-1539*2^263359+1
, d=1185*2^529444-1539*2^263359\n" }{MPLTEXT 1 0 28 "# (159382 decimal
 digits).\n" }{MPLTEXT 1 0 4 "# \n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 
0 64 "evalf(2^64/3/5/7/11/13/17/19/23/29/31); # so many to sieve out\n
" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 117 "(2/1)*(3/2)*(5/4)*(7/6)*(11/
10)*(13/12)*(17/16)*(19/18)*(23/22)*(29/28)*(31/30); evalf(%); # so ma
ny times more dense" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+X#>&R=!\"\"" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "#\")BF#o)\")S5F8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+4(pAa'!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "qsAB(1,32,1000000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"++8meE!#5"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "%*(ln(1000000.)/(50*ln(2
.))); # decreasing factor of sieve" }}{PARA 11 "" 1 "" {XPPMATH 20 "$
\"++e#)f5!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "1/200000/ln
(10.); # natural density of primes having 200000 digits" }}{PARA 11 ""
 1 "" {XPPMATH 20 "$\"+5CZr@!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 71 "%*6.542269709/0.1059825800; 1/%; # increased density and expec
ted tests" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+dFWS8!#8" }}{PARA 11 ""
 1 "" {XPPMATH 20 "$\"+$f@-Y(!\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 73 "# after 30000000 test we have 400 primes, 80000 pairs
, seems to be enough" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "300
0000*160*5; # SPU time in sec" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"+++++
C" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "%/24/3600/16; evalf(%)
; # time for one Cell blade in days" }}{PARA 11 "" 1 "" {XPPMATH 20 "#
\"&Dc\"\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+666O<!\"'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "1/160000/ln(10.); # natural \+
density of primes having 160000 digits" }}{PARA 11 "" 1 "" {XPPMATH 20
 "$\"+70M9F!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "%*6.54226
9709/0.1059825800; 1/%; # increased density and expected tests" }}
{PARA 11 "" 1 "" {XPPMATH 20 "$\"+YMbv;!#8" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+vs<of!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "# after 1800000 test we have 300 primes, 45000 pairs, seems to be \+
enough" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "1800000*300*12; #
 SPU time in sec" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"++++!['" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "%/24/3600/420; evalf(%); # t
ime for all PS3 in days" }}{PARA 11 "" 1 "" {XPPMATH 20 "#\"%]7\"\"(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+'G9dy\"!\"(" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "# A ten-year jump:\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 66 "1/310000/ln(10.); # natural density of primes having \+
310000 digits" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+U*\\4S\"!#:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "%*6.542269709/0.1059825800; \+
1/%; # increased density and expected tests" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+(y<![')!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+ZVLc6
!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "# after 5800000 tes
t we have 500 primes, 125000 pairs, seems to be enough\n" }{MPLTEXT 1 
0 2 "\n" }{MPLTEXT 1 0 33 "5800000*300*24; # SPU time in sec" }}{PARA 
11 "" 1 "" {XPPMATH 20 "\",+++g<%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "%/24/3600/420; evalf(%); # time for all PS3 in days" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "#\"&+D(\"#j" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"+^Oz]6!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "repunit:=proc(n) local i;\n" }{MPLTEXT 1 0 59 "for i to n do if is
prime((10^i-1)/9) then print(i) fi od;\n" }{MPLTEXT 1 0 4 "end;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I\"nG6\"6#I\"iGF%F%F%?(F'\"\"\"F)F$I
%trueG%*protectedG@$-I(isprimeG6$F+I(_syslibGF%6#,&*&#F)\"\"*F))\"#5F'
F)F)F4!\"\"-I&printGF+F&F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "repunit(1000);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$<$" }}}}{SECT 0 {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 3 "#\n" }{MPLTEXT 1 0 64 "# This is a simple p
rocedure to get all the prime pairs (p,q) \n" }{MPLTEXT 1 0 55 "# for \+
which the quotient (p+1)/(q+1)=n is an integer,\n" }{MPLTEXT 1 0 58 "#
 1<n<=N and 2<q<=Q. The limit Q is supposed to be small\n" }{MPLTEXT 
1 0 67 "# and N is supposed to be large, hence primality testing is us
ed.\n" }{MPLTEXT 1 0 58 "# The result is a list of pairs; each pair co
nsist of n \n" }{MPLTEXT 1 0 58 "# and a list of all pairs [p,q] corre
sponding to this n.\n" }{MPLTEXT 1 0 38 "# The pairs are in increasing
 order.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 
36 "prime_plus_one_quotient:=proc(Q,N)\n" }{MPLTEXT 1 0 32 "local q,n,
p,QQ,LL,LLL; QQ:=[];\n" }{MPLTEXT 1 0 33 "for q from 3 by 2 while q<=Q
 do\n" }{MPLTEXT 1 0 41 "  if isprime(q) then QQ:=[op(QQ),q] fi;\n" }
{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 1 0 9 "LL:=[];\n" }{MPLTEXT 1 0 28 "f
or n from 2 while n<=N do\n" }{MPLTEXT 1 0 12 "  LLL:=[];\n" }{MPLTEXT
 1 0 18 "  for q in QQ do\n" }{MPLTEXT 1 0 17 "    p:=n*q+n-1;\n" }
{MPLTEXT 1 0 49 "    if isprime(p) then LLL:=[op(LLL),[p,q]] fi;\n" }
{MPLTEXT 1 0 7 "  od;\n" }{MPLTEXT 1 0 25 "  LL:=[op(LL),[n,LLL]];\n" 
}{MPLTEXT 1 0 11 "od; LL end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"
QG6\"I\"NGF%6(I\"qGF%I\"nGF%I\"pGF%I#QQGF%I#LLGF%I$LLLGF%F%F%C'>F+7\"?
(F(\"\"$\"\"#F%1F(F$@$-I(isprimeGF%6#F(>F+7$-I#opG%*protectedG6#F+F(>F
,F0?(F)F3\"\"\"F%1F)F&C%>F-F0?&F(F+I%trueGF=C$>F*,(*&F)FAF(FAFAF)FAFA!
\"\"@$-F76#F*>F-7$-F<6#F-7$F*F(>F,7$-F<6#F,7$F)F-F,F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L:=prime_plus_one_quotient(10,100);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "7_q7$\"\"#7$7$\"\"(\"\"$7$\"#6\"\"&7
$F(7%7$F*F(7$\"#<F+7$\"#BF'7$\"\"%7$7$F2F+7$\"#JF'7$F+7$7$\"#>F(7$\"#H
F+7$\"\"'7$7$F2F(7$\"#ZF'7$F'7#7$\"#TF+7$\"\")7$7$F8F(7$FDF+7$\"\"*7$7
$\"#`F+7$\"#rF'7$\"#57$7$\"#fF+7$\"#zF'7$F*7#7$\"#VF(7$\"#77$7$FDF(7$F
TF+7$\"#87#7$\"$.\"F'7$\"#97#7$\"#$)F+7$\"#:7$7$FYF(7$\"#*)F+7$\"#;7#7
$\"$F\"F'7$F07$7$\"#nF(7$\"$,\"F+7$\"#=7$7$FTF(7$\"$2\"F+7$F<7$7$\"$8
\"F+7$\"$^\"F'7$\"#?7#7$FenF(7$\"#@7$7$FhoF(7$\"$n\"F'7$\"#A7#7$\"$J\"
F+7$F27#7$\"$P\"F+7$\"#C7#7$\"$\">F'7$\"#D7$7$\"$\\\"F+7$\"$*>F'7$\"#E
7#7$FcoF(7$\"#F7#7$F_qF(7$\"#G7$7$F_rF+7$\"$B#F'7$F>7#7$\"$t\"F+7$\"#I
7$7$\"$z\"F+7$\"$R#F'7$F87\"7$\"#K7$7$FcpF(7$F]sF+7$\"#L7%7$FdrF(7$\"$
(>F+7$\"$j#F'7$\"#M7#7$\"$r#F'7$\"#N7#7$\"$R\"F(7$\"#OF_u7$\"#PF_u7$\"
#Q7$7$FeqF(7$\"$F#F+7$\"#R7$7$\"$L#F+7$\"$6$F'7$\"#S7#7$F]uF+7$FH7#7$
\"$j\"F(7$\"#U7$7$F_rF(7$\"$^#F+7$Fin7#7$\"$d#F+7$\"#W7#7$F\\vF+7$\"#X
7%7$F[uF(7$\"$p#F+7$\"$f$F'7$\"#Y7#7$\"$n$F'7$FD7#7$\"$\"GF+7$\"#[7$7$
F]sF(7$\"$$QF'7$\"#\\7#7$\"$$HF+7$\"#]7#7$FdsF(7$\"#^F_u7$\"#_7#7$FgwF
+7$FR7$7$\"$6#F(7$\"$<$F+7$\"#a7#7$\"$J%F'7$\"#b7#7$\"$R%F'7$\"#c7#7$F
btF(7$\"#d7#7$F`wF(7$\"#e7$7$\"$Z$F+7$\"$j%F'7$FY7#7$\"$`$F+7$\"#g7%7$
F]uF(7$FeyF+7$\"$z%F'7$\"#h7#7$\"$([F'7$\"#iF_u7$\"#j7$7$FexF(7$\"$.&F
'7$\"#k7#7$FdzF+7$\"#l7#7$\"$*QF+7$\"#m7#7$F\\vF(7$Fgp7#7$\"$,%F+7$\"#
o7#7$FavF(7$\"#pF_u7$\"#q7#7$\"$>%F+7$FT7#7$\"$$GF(7$\"#s7#7$F^\\lF+7$
\"#tF_u7$\"#u7#7$\"$V%F+7$\"#v7$7$\"$\\%F+7$\"$*fF'7$\"#w7#7$\"$2'F'7$
\"#x7$7$\"$2$F(7$\"$h%F+7$\"#y7$7$FgwF(7$\"$n%F+7$Fen7#7$\"$J'F'7$\"#!
)7#7$F]^lF+7$\"#\")7#7$\"$Z'F'7$\"##)7#7$\"$\"\\F+7$Fho7#7$\"$J$F(7$\"
#%)7#7$Fj^lF+7$\"#&)7#7$\"$4&F+7$\"#')F_u7$\"#()7$7$F`]lF(7$\"$@&F+7$
\"#))F_u7$F^pF_u7$\"#!*7$7$FeyF(7$\"$>(F'7$\"#\"*7#7$\"$F(F'7$\"##*7#7
$FjyF(7$\"#$*7$7$\"$d&F+7$\"$V(F'7$\"#%*7$7$\"$j&F+7$\"$^(F'7$\"#&*7$7
$\"$z$F(7$\"$p&F+7$\"#'*7#7$FdzF(7$\"#(*F_u7$\"#)*7#7$\"$(eF+7$\"#**7#
7$\"$$fF+7$\"$+\"7#7$F\\blF+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "#\n" }{MPLTEXT 1 0 59 "# This routine collect statistics from the
 results of the\n" }{MPLTEXT 1 0 53 "# above routine. For all number 0
<=m<=M, where M is\n" }{MPLTEXT 1 0 60 "# the number of odd prime less
 then Q, the number of those\n" }{MPLTEXT 1 0 61 "# numbers 1<n<=N giv
en back, for which exactly m solutions \n" }{MPLTEXT 1 0 62 "# of n=(p
+1)/(q+1) is found. L is the list calculated by the\n" }{MPLTEXT 1 0 
18 "# above program.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 51 "stat_prime_plus:=proc(Q,L) local q,M,MM,LL; M:=0;\n" 
}{MPLTEXT 1 0 32 "for q from 3 by 2 while q<Q do\n" }{MPLTEXT 1 0 33 "
  if isprime(q) then M:=M+1 fi;\n" }{MPLTEXT 1 0 5 "od;\n" }{MPLTEXT 
1 0 18 "MM:=array(0..M);\n" }{MPLTEXT 1 0 36 "for q from 0 to M do MM[
q]:=0; od;\n" }{MPLTEXT 1 0 56 "for LL in L do MM[nops(LL[2])]:=MM[nop
s(LL[2])]+1; od;\n" }{MPLTEXT 1 0 25 "convert(MM,listlist) end;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"QG6\"I\"LGF%6&I\"qGF%I\"MGF%I#MMG
F%I#LLGF%F%F%C(>F)\"\"!?(F(\"\"$\"\"#F%2F(F$@$-I(isprimeG6$%*protected
GI(_syslibGF%6#F(>F),&F)\"\"\"F<F<>F*-I&arrayGF76#;F.F)?(F(F.F<F)I%tru
eGF7>&F*F9F.?&F+F&FC>&F*6#-I%nopsGF76#&F+6#F1,&FHF<F<F<-I(convertGF76$
F*I)listlistGF%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "st
at_prime_plus(10,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"#6\"#_\"#K\"
\"%" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 206 7 "5.18. P" }{TEXT 206 8 "
\303\251" }{TEXT 206 4 "lda." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 
"5.19. Feladat." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 34 "# This is a simple facto
rization\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 "# factors
 and the e's are the exponents.\n" }{MPLTEXT 1 0 47 "# The factors are
 anyway in increasing order.\n" }{MPLTEXT 1 0 41 "# Only primes <= P a
re tried, hence the\n" }{MPLTEXT 1 0 38 "# last \"factor\" may composi
te, 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::pos
int,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 m
od 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@F<F@F0F@F@>F,7#7$F<F.@$3/-I$modGF&6$F0\"\"$F?1FNF*C$
?(F.F?F@F&FJ>F0,$*&#F@FNF@F0F@F@>F,7$-I#opGF(6#F,7$FNF.>F/F<>F-\"\"&?(
F&F@F@F&31F-F*1*$)F-F<F@F0C%@$/-FL6$F0F-F?C$?(F.F?F@F&Fao>F0*&F0F@F-!
\"\">F,7$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<vJ_p-g&zTOV" }}}{EXCHG {PARA 0 "> " 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$*RM#>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "modp(3&^
(n6-1),n6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"<'Q>0@M];Yv*eq\"=" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT 206 8 "5.20. Pr" }{TEXT 206 8 "\303\255"
 }{TEXT 206 3 "msz" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "mk" }{TEXT 
206 8 "\303\263" }{TEXT 206 3 "dol" }{TEXT 206 8 "\303\241" }{TEXT 
206 2 "s." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 14 "5.21. Feladat." }}
{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 183 "p:=safeprime(15634567888142561788867656615552613429876453213456
6543212345678877209127512109876123212233332123343412343212334445432123
4320948725467845467788859812342365); log[2.](p);\n" }{MPLTEXT 1 0 190 
"q:=safeprime(29841524475159001676561453467890987651234254321456490998
7676267881825143256789099872365142341542323965878747789933777220049883
76667882767156363888377626677728888); log[2.](q);\n" }{MPLTEXT 1 0 9 "
n:=p*q;\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 52 "e:=28763541324536789
09987653432123409887635423125;\n" }{MPLTEXT 1 0 50 "igcdex(e,(p-1)*(q-
1),'d'); d; d*e mod (p-1)*(q-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"et
zJO7)f))yna%yYD([4KM7KaWMB@VBTVL7KLB7K7w)4@^F\"4s()ycM7KamX8KXw)HMh_bh
cw'))yhD9))ycMc\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+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$F+F/F%F%F<F5F-\"#*
*F'F)F;F(F/\"$:\"F'F$F?F6F5F-F)F8F?F5F6F7F7F6F$F6F/F&F'\"$-\"F/F-F,F6F
,F?F+F&F,F&F&F)F8F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"b^l$*)=cl\"z&)o
h7Il*zqJn\")\\`[RxVx[pfF`(4y\\&*[!fw*[54)oDJ:!px#yr-Dz$o@[b3<JS*Q\"*>A
u5G6n*f%)>]DNDJQ\"fIWf4Y(>ioa*GEOt]53G)Hya\"[x%RJc,aq`N3CyVuh!p6At7EB6
@TY%y*Q#eM#Rg4*=6uS061CY+oG&>" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I%modp
G%*protectedG6$-I#&^G6\"6$\"b^l$*)=cl\"z&)oh7Il*zqJn\")\\`[RxVx[pfF`(4
y\\&*[!fw*[54)oDJ:!px#yr-Dz$o@[b3<JS*Q\"*>Au5G6n*f%)>]DNDJQ\"fIWf4Y(>i
oa*GEOt]53G)Hya\"[x%RJc,aq`N3CyVuh!p6At7EB6@TY%y*Q#eM#Rg4*=6uS061CY+oG
&>I\"eGF(I\"nGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "c&^d mo
d n; convert(%,base,256); convert(%,'bytes');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"b^l$*)=cl\"z&)oh7Il*zqJn\")\\`[RxVx[pfF`(4y\\&*[!fw*[54
)oDJ:!px#yr-Dz$o@[b3<JS*Q\"*>Au5G6n*f%)>]DNDJQ\"fIWf4Y(>ioa*GEOt]53G)H
ya\"[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\"#XF/F'F3
F,\"$1\"F%F,F-F5F%F6F)\"$p\"F%F&F'F)F8F4\"$@\"F,\"$+\"F%F)F8F-F7F7F,F'
F&F$F+F/F%F%F<F5F-\"#**F'F)F;F(F/\"$:\"F'F$F?F6F5F-F)F8F?F5F6F7F7F6F$F
6F/F&F'\"$-\"F/F-F,F6F,F?F+F&F,F&F&F)F8F-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "Q\\tMint~`v|^w|huz`~alatti,~`elmer|^w|gvlt`~harangok|+|+`
hint|^w|\\uznak`-e~`hajnalonk|^w|dunt`~`|^w|\\ugyadn|^w|\\ul`|+|+a~tiz
ennyolc~`|^w|duves`~`iskol|^w|\\usok`|+|+kiket~`felakasztatt|^w|\\ul`6
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 65 
"# This simple procedure generates a pair of safe primes p and q\n" }
{MPLTEXT 1 0 56 "# congruent 3 mod 4, and calculates the product n=p*q
,\n" }{MPLTEXT 1 0 61 "# a Blum integer. The output is a sequence of t
hese numbers\n" }{MPLTEXT 1 0 64 "# in this order. The input is two la
rge numbers, these are the\n" }{MPLTEXT 1 0 62 "# starting points for \+
the search for p and q, respectively. \n" }{MPLTEXT 1 0 63 "# For exam
ple, 100 and 104 digit starting points may be used.\n" }{MPLTEXT 1 0 
55 "# The best to choose them randomly and independently.\n" }{MPLTEXT
 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 41 "setBlum:=proc(p0,q
0) local p,q,n,e,d,k;\n" }{MPLTEXT 1 0 19 "p:=safeprime(p0);\n" }
{MPLTEXT 1 0 43 "while p mod 4 <> 3 do p:=safeprime(p) od;\n" }
{MPLTEXT 1 0 19 "q:=safeprime(q0);\n" }{MPLTEXT 1 0 43 "while q mod 4 \+
<> 3 do q:=safeprime(q) od;\n" }{MPLTEXT 1 0 27 "n:=p*q;         p,q,n
; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#p0G6\"I#q0GF%6(I\"pGF%I
\"qGF%I\"nGF%I\"eGF%I\"dGF%I\"kGF%F%F%C(>F(-_I*numtheoryG6$%*protected
GI(_syslibGF%I*safeprimeGF%6#F$?(F%\"\"\"F9F%0-I$modGF%6$F(\"\"%\"\"$>
F(-F16#F(>F)-F16#F&?(F%F9F9F%0-F<6$F)F>F?>F)-F16#F)>F**&F(F9F)F96%F(F)
F*F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 
1 0 57 "# This inner routine make the encryption and decryption\n" }
{MPLTEXT 1 0 55 "# for an arbitrary texts. Here n is the Blum integer,
\n" }{MPLTEXT 1 0 53 "# L is the text, x0 is the first quadratic resid
ue,\n" }{MPLTEXT 1 0 45 "# k is the number of bits used in one step.\n
" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 42 "Blumi:=
proc(n,L,x0,k) local xi,b,i,j,nL;\n" }{MPLTEXT 1 0 17 "nL:=[]; xi:=x0;
\n" }{MPLTEXT 1 0 32 "for i from 0 to nops(L)/k-1 do\n" }{MPLTEXT 1 0 
34 "  b:=convert(xi mod 2^k,base,2);\n" }{MPLTEXT 1 0 17 "  for j to k
 do\n" }{MPLTEXT 1 0 56 "    if nops(b)>j then nL:=[op(nL),L[i*k+j]+b[
j] mod 2]\n" }{MPLTEXT 1 0 35 "    else nL:=[op(nL),L[i*k+j]] fi\n" }
{MPLTEXT 1 0 7 "  od;\n" }{MPLTEXT 1 0 22 "  xi:=xi &^ 2 mod n;\n" }
{MPLTEXT 1 0 22 "od;         xi,nL end;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "f*6&I\"nG6\"I\"LGF%I#x0GF%I\"kGF%6'I#xiGF%I\"bGF%I\"iGF%I\"jGF%I#n
LGF%F%F%C&>F.7\">F*F'?(F,\"\"!\"\"\",&*&-I%nopsG%*protectedG6#F&F5F(!
\"\"F5F5F<I%trueGF:C%>F+-I(convertGF:6%-I$modGF%6$F*)\"\"#F(I%baseGF%F
G?(F-F5F5F(F=@%2F--F96#F+>F.7$-I#opGF:6#F.-FD6$,&&F&6#,&*&F,F5F(F5F5F-
F5F5&F+6#F-F5FG>F.7$FPFV>F*-FD6$-I#&^GF%6$F*FGF$6$F*F.F%F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 49 "# This
 make the encryption for arbitrary texts.\n" }{MPLTEXT 1 0 55 "# Here \+
n is the Blum integer, L is the text, x is the\n" }{MPLTEXT 1 0 58 "# \+
random number to generate the first quadratic residue,\n" }{MPLTEXT 1 
0 45 "# k is the number of bits used in one step.\n" }{MPLTEXT 1 0 3 "
#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 39 "Blume:=proc(n,x,L,k) local
 pt,i,j,nL;\n" }{MPLTEXT 1 0 63 "if igcd(x,n)>1 then ERROR(`x is not r
elative prime to n`) fi;\n" }{MPLTEXT 1 0 21 "convert(L,'bytes');\n" }
{MPLTEXT 1 0 39 "pt:=sum(%[i]*256^(i-1),i=1..nops(%));\n" }{MPLTEXT 1 
0 25 "pt:=convert(pt,base,2);\n" }{MPLTEXT 1 0 31 "if k>1 then pmod(no
ps(pt),k);\n" }{MPLTEXT 1 0 44 "  if %<>0 then pt:=[op(pt),0$i=1..k-%]
 fi;\n" }{MPLTEXT 1 0 5 "fi;\n" }{MPLTEXT 1 0 31 "Blumi(n,pt,x &^ 2 mo
d n,k) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"nG6\"I\"xGF%I\"LGF
%I\"kGF%6&I#ptGF%I\"iGF%I\"jGF%I#nLGF%F%F%C(@$2\"\"\"-I%igcdG%*protect
edG6$F&F$-I&ERRORGF46#I=x~is~not~relative~prime~to~nGF%-I(convertGF46$
F'.I&bytesGF%>F*-I$sumG6$F4I(_syslibGF%6$*&&I\"%GF%6#F+F1)\"$c#,&F+F1F
1!\"\"F1/F+;F1-I%nopsGF46#FG>F*-F;6%F*I%baseGF%\"\"#@$2F1F(C$-I%pmodGF
%6$-FP6#F*F(@$0FG\"\"!>F*7$-I#opGF4Fhn-I\"$GF46$F[o/F+;F1,&F(F1FGFL-I&
BlumiGF%6&F$F*-I$modGF%6$-I#&^GF%6$F&FVF$F(F%F%F%" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 49 "# This make the decryp
tion for arbitrary texts.\n" }{MPLTEXT 1 0 65 "# Here p and q are the \+
primes, x is the last quadratic residue,\n" }{MPLTEXT 1 0 66 "# L is t
he string, and k is the number of bits used in one step.\n" }{MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 48 "Blumd:=proc(p,q,x,L
,k) local a,b,u,v,i,n,t,nL;\n" }{MPLTEXT 1 0 34 "igcdex(p,q,'a','b'); \+
t:=nops(L);\n" }{MPLTEXT 1 0 43 "u:=x &^ (((p+1)/4) &^ t mod (p-1)) mo
d p;\n" }{MPLTEXT 1 0 43 "v:=x &^ (((q+1)/4) &^ t mod (q-1)) mod q;\n"
 }{MPLTEXT 1 0 46 "nL:=Blumi(p*q,L,a*u+b*v mod p*q,k)[2]; n:=0;\n" }
{MPLTEXT 1 0 46 "for i from t to 1 by -1 do n:=n+n+nL[i]; od;\n" }
{MPLTEXT 1 0 50 "convert        (convert(n,base,256),'bytes'); end;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"pG6\"I\"qGF%I\"xGF%I\"LGF%I\"kGF
%6*I\"aGF%I\"bGF%I\"uGF%I\"vGF%I\"iGF%I\"nGF%I\"tGF%I#nLGF%F%F%C*-I'ig
cdexGF%6&F$F&.F+.F,>F1-I%nopsG%*protectedG6#F(>F--I$modGF%6$-I#&^GF%6$
F'-F@6$-FC6$,&*&#\"\"\"\"\"%FLF$FLFLFKFLF1,&F$FLFL!\"\"F$>F.-F@6$-FC6$
F'-F@6$-FC6$,&*&FKFLF&FLFLFKFLF1,&F&FLFLFOF&>F2&-I&BlumiGF%6&*&F$FLF&F
LF(-F@6$,&*&F+FLF-FLFL*&F,FLF.FLFLF[oF)6#\"\"#>F0\"\"!?(F/F1FOFLI%true
GF<>F0,&*&FboFLF0FLFL&F26#F/FL-I(convertGF<6$-F]p6%F0I%baseGF%\"$c#.I&
bytesGF%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p,q,n:=se
tBlum(10^100,10^104);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"`qBP/++++++
+++++++++++++++++++++++++++++++++++++++++\"\"dq$yS++++++++++++++++++++
+++++++++++++++++++++++++++++5\"hw4^:$y,++++++++++++++++++++++++++++++
++++++++++++++IyqsV+++++++++++++++++++++++++++++++++++++++++++++++5" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x,L:=Blume(n,55555,M,1);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"gwC&Qy\\@(y(Gi'RRe\"y.gbw\"*\\l]9$
ywh='\\<_kbg+!eJ\"GIi\">j6_)z)yMqTHJJ\\VmLH5vu\\gs*G(>P2d(eu,#3k(pCGJn
m&R!)HSLjEtrW;hQIgV=7c[o\"\"\"\"\"!F%F%F&F&F%F&F%F&F&F%F&F%F%F&F&F%F%F
%F&F%F%F&F&F&F%F&F%F%F%F&F&F&F&F&F&F%F&F&F&F%F%F&F%F%F%F&F%F%F&F&F&F&F
%F%F%F&F%F%F&F%F&F%F&F%F&F%F%F%F%F&F&F&F&F&F&F%F&F&F%F&F&F&F&F%F%F&F&F
&F%F%F&F%F%F&F%F&F&F&F&F%F%F&F&F&F%F&F%F%F%F&F&F&F%F&F%F%F%F&F%F&F&F%F
&F%F%F&F&F&F%F%F&F%F&F&F&F&F&F&F&F%F&F&F%F&F%F&F&F%F%F&F&F&F%F%F&F%F%F
&F%F&F%F%F&F%F%F&F%F&F%F&F&F%F%F&F&F%F&F&F%F%F%F&F%F%F&F&F&F&F%F%F&F&F
%F%F%F%F&F%F&F&F%F%F&F%F%F&F&F&F%F&F%F%F%F&F&F&F&F&F&F%F&F&F&F&F&F%F&F
%F%F&F%F&F&F&F&F%F%F&F&F%F&F&F%F%F%F&F%F&F&F&F&F%F%F&F&F%F%F%F&F%F%F&F
%F%F%F&F&F%F%F&F%F%F%F%F&F%F%F&F%F%F&F%F&F%F%F&F&F%F&F%F&F&F&F&F&F%F&F
%F&F&F&F&F&F&F&F%F&F%F%F&F%F&F&F%F&F%F%F&F&F%F%F%F&F%F%F&F&F&F%F&F%F%F
%F&F%F%F&F&F&F&F%F%F%F&F&F&F&F%F&F%F&F%F&F%F%F%F%F&F&F%F%F%F&F%F%F&F%F
&F&F&F&F%F%F&F%F%F&F%F&F%F%F&F%F&F%F%F&F%F&F&F%F&F%F&F&F%F%F&F&F&F&F&F
&F%F&F&F&F&F&F%F&F%F%F&F%F&F&F&F&F%F%F&F&F%F&F%F&F%F%F&F&F%F%F%F&F%F%F
&F%F&F&F&F&F%F%F&F&F&F%F%F&F%F%F&F%F%F%F%F&F%F%F&F&F%F%F%F&F%F%F&F%F%F
&F%F&F%F%F&F%F%F&F&F&F&F%F%F%F&F&F%F&F%F&F%F&F%F%F%F&F%F%F&F&F&F%F&F%F
%F%F&F&F&F&F&F&F%F&F&F%F%F&F&F&F&F%F%F%F&F&F&F&F%F&F%F%F%F%F&F&F%F%F&F
%F&F&F%F%F%F%F&F%F&F&F&F&F%F%F&F&F&F%F&F&F%F%F&F&F%F%F%F&F%F%F&F%F%F&F
&F&F&F%F%F%F&F&F&F&F%F&F%F&F&F%F%F&F%F%F&F&F%F&F%F&F&F&F&F&F%F&F%F&F&F
&F&F%F&F&F&F&F%F%F&F&F&F&F&F&F%F&F&F&F&F%F&F%F%F%F&F%F&F&F%F&F%F%F&F&F
%F&F%F%F%F%F&F%F&F%F&F&F%F%F&F&F%F%F%F&F%F%F&F&F%F%F%F&F%F%F&F%F&F&F%F
%F%F%F&F%F%F%F%F&F%F%F&F&F&F%F%F&F%F%F&F%F%F&F&F&F%F%F&F&F&F&F&F&F%F&F
&F%F%F&F&F&F&F%F%F%F&F&F%F&F%F&F%F&F%F%F&F%F%F%F&F%F&F%F&F&F%F%F&F%F%F
&F&F%F%F%F&F&F&F&F&F&F%F&F&F%F&F&F%F&F%F%F&F%F%F&F&F%F%F%F&F%F%F&F%F&F
%F%F&F%F%F%F%F&F%F%F&F&F&F%F%F&F%F%F&F%F%F&F&F&F&F%F%F%F&F&F&F&F%F&F%F
%F%F&F&F%F%F%F&F%F%F%F%F&F%F%F&F%F%F&F%F&F%F%F&F&F%F&F%F&F&F&F&F&F%F&F
%F&F&F&F&F%F%F&F%F&F%F%F&F%F&F&F%F&F%F%F&F%F%F&F%F&F%F%F&F%F&F%F&F&F%F
%F&F&F&F%F&F%F%F%F&F&F&F&F&F&F%F&F&F&F%F%F&F&F%F%F&F%F&F%F&F&F%F%F&F&F
&F%F%F&F%F%F&F%F&F&F&F&F%F%F&F%F%F&F%F&F%F%F&F%F&F&F&F&F%F%F&F%F%F&F&F
%F%F%F&F&F%F&F%F%F%F%F&F&F&F%F&F%F%F%F&F%F&F&F&F&F%F%F&F&F&F%F&F%F%F%F
&F&F&F%F&F%F%F%F&F%F%F&F&F&F&F%F%F%F&F&F&F&F%F&F%F&F&F%F%F&F%F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Blumd(p,q,x,L,1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "Q\\tMint~`v|^w|huz`~alatti,~`elmer|^w|gvlt`~ha
rangok|+|+`hint|^w|\\uznak`-e~`hajnalonk|^w|dunt`~`|^w|\\ugyadn|^w|\\u
l`|+|+a~tizennyolc~`|^w|duves`~`iskol|^w|\\usok`|+|+kiket~`felakasztat
t|^w|\\ul`6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }
{MPLTEXT 1 0 67 "# This simple procedure generates a safe prime q, and
 a generator\n" }{MPLTEXT 1 0 63 "# g of the reduced residue classes. \+
For the given exponent d,\n" }{MPLTEXT 1 0 53 "# calculates e=g^d mod \+
q. The output is a sequence \n" }{MPLTEXT 1 0 57 "# of the numbers [q,
g,e,d] in this order. The input is \n" }{MPLTEXT 1 0 70 "# three large
 numbers, these are the starting points for the search \n" }{MPLTEXT 
1 0 50 "# for q and e, respectively, and the exponent d.\n" }{MPLTEXT 
1 0 55 "# For example, 200 digit starting points may be used.\n" }
{MPLTEXT 1 0 55 "# The best to choose them randomly and independently.
\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 46 "setEl
Gamal:=proc(q0,g0,d) local f,p,q,g,e,P;\n" }{MPLTEXT 1 0 55 "q:=safepr
ime(q0); P:=factorset(q-1); f:=false; g:=g0;\n" }{MPLTEXT 1 0 25 "whil
e not f do f:=true;\n" }{MPLTEXT 1 0 30 "  for p in P while f=true do
\n" }{MPLTEXT 1 0 60 "    if modp(g &^ ((q-1)/p),q) = 1 then f:=false;
 g:=g+1 fi\n" }{MPLTEXT 1 0 7 "  od;\n" }{MPLTEXT 1 0 46 "od; e:=modp(
g &^ d,q);         [q,g,e,d]; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*
6%I#q0G6\"I#g0GF%I\"dGF%6(I\"fGF%I\"pGF%I\"qGF%I\"gGF%I\"eGF%I\"PGF%F%
F%C)>F+-_I*numtheoryG6$%*protectedGI(_syslibGF%I*safeprimeGF%6#F$>F.-_
F3I*factorsetGF%6#,&F+\"\"\"F?!\"\">F)I&falseGF5>F,F&?(F%F?F?F%4F)C$>F
)I%trueGF5?&F*F./F)FH@$/-I%modpGF56$-I#&^GF%6$F,*&F>F?F*F@F+F?C$FA>F,,
&F,F?F?F?>F--FN6$-FQ6$F,F'F+7&F+F,F-F'F%F%F%" }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 68 "# This make the encryption \+
for short texts. Here q is the modulus,\n" }{MPLTEXT 1 0 63 "# g is th
e generator, e the encryption exponent, k the random\n" }{MPLTEXT 1 0 
25 "# exponent, L the text.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 47 "ElGamale:=proc(q,g,e,k,L) convert(L,'bytes');\n"
 }{MPLTEXT 1 0 69 "modp(g &^ k,q),modp(sum(%[i]*256^(i-1),i=1..nops(%)
)*(e &^ k),q) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"qG6\"I\"gGF
%I\"eGF%I\"kGF%I\"LGF%F%F%F%C$-I(convertG%*protectedG6$F).I&bytesGF%6$
-I%modpGF-6$-I#&^GF%6$F&F(F$-F36$*&-I$sumG6$F-I(_syslibGF%6$*&&I\"%GF%
6#I\"iGF%\"\"\")\"$c#,&FDFEFE!\"\"FE/FD;FE-I%nopsGF-6#FBFE-F66$F'F(FEF
$F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 
0 68 "# This make the decryption for short texts. Here q is the modulu
s,\n" }{MPLTEXT 1 0 44 "# d the decryption exponent, x the number.\n" 
}{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 27 "ElGamald:
=proc(q,g,d,x,y)\n" }{MPLTEXT 1 0 65 "convert(convert(modp((x &^ (q-d-
1))*y,q),base,256),'bytes'); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6
'I\"qG6\"I\"gGF%I\"dGF%I\"xGF%I\"yGF%F%F%F%-I(convertG%*protectedG6$-F
+6%-I%modpGF,6$*&-I#&^GF%6$F(,(F$\"\"\"F'!\"\"F8F9F8F)F8F$I%baseGF%\"$
c#.I&bytesGF%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "L:=s
etElGamal(10^200,2*10^199,3*10^199); q:=L[1]; g:=L[2]; e:=L[3]; d:=L[4
];" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&\"dwZBj+++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
++\"\"cw++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++?\"cwH8fRVzqIGb`^9$z]rCTm`QmuRwJ)
=qEp_.4,4&3nOy\"z>w+5\"ymv,p`=wsHQ3H%GB#po9#Qx-x_T>i%[Xp83J=&>?G6n?!fn
P$z:&y@Y2W&\"cw+++++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++I" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"dwZBj++++++++++++++++++++++++++++++++++++++++++++++++++
+++++++++++++++++++++++++++++++++++++++++++++++\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"cw+++++++++++++++++++++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++?" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"cwH8fRVzqIGb`^9$z]rCTm`QmuRwJ)=qEp_.4,4&3nOy\"z>w+5\"ym
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2m`([\"[$pl^'\"cwu(=j)fKr\\.eYK7\")4Bz)3i\\YbyF%G`_(*fMk$>B3ZrNd[/)[SQ
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gvlt`~harangok6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "debug(
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03\241mok \303\251s polinomok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}
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\241ci\303\263" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 ""
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3\255mteszt elliptikus g\303\266rb\303\251kkel" }}{PARA 0 "" 0 "" 
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3\241mtest szita" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}}{SECT 1 {PARA 3 
"" 0 "" {TEXT 205 16 "16. Vegyes probl" }{TEXT 205 8 "\303\251" }{TEXT
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205 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 201 0 
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