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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# Sz\341melm\351let \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "with(numtheory): # Sz
\341melm\351let csomag" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "#
 Oszthat\363s\341g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "irem(
23,4,'q'); q; # marad\351kos oszt\341s marad\351ka \351s q=h\341nyados
a" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "iqu
o(23,4,'r'); r; # marad\351kos oszt\341s h\341nyadosa \351s r=marad
\351ka" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "igc
d(38,78); # legnagyobb k\366z\366s oszt\363 (lnko)" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "i
gcd(5,27,32,14);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ilcm(35,67); ilcm(12,27,38); # legk
isebb k\366z\366s t\366bbsz\366r\366s (lkkt)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"%XB" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%_?" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "igcdex(1165,275,'x','y'); x
, y; # lnko az euklideszi algoritmussal \351s x,y megad\341sa \372gy, \+
hogy (a,b)=ax+by" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$\"#<!#s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 43 "ifactor(123456700); # pr\355mt\351nyez\365kre bont\341s" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#**)-%!G6#\"\"#F(\"\"\")-F&6#\"\"&F(F)-
F&6#\"$F\"\"\"\"-F&6#\"%@(*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "5^4 mod 7; # marad\351k mod n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "2^(2^5)+1 mod 641
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 32 "p_i:=ithprime(600); #i-edik prim" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$p_iG\"%4W" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "primek:=seq(ithprime(n),n=1..10);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'primekG6,\"\"#\"\"$\"\"&\"\"(\"#6\"#8\"#<\"#>
\"#B\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "primeksorozata:
=seq([n,ithprime(n)],n=1..10);   " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%/primeksorozataG6,7$\"\"\"\"\"#7$F(\"\"$7$F*\"\"&7$\"\"%\"\"(7$F,\"#6
7$\"\"'\"#87$F/\"#<7$\"\")\"#>7$\"\"*\"#B7$\"#5\"#H" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "isprime(145), isprime(173); # pr\355m-e?
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&falseG%%trueG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "primteszt:=seq([n,isprime(n)],n=1..10); \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*primtesztG6,7$\"\"\"%&falseG7$
\"\"#%%trueG7$\"\"$F+7$\"\"%F(7$\"\"&F+7$\"\"'F(7$\"\"(F+7$\"\")F(7$\"
\"*F(7$\"#5F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "prevprime(
1000), nextprime(1000); # n el\365tti, ill. ut\341ni pr\355m" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"$(**\"%45" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "issqrfree(30), issqrfree(50); # negyzetmentes-e?" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$%%trueG%&falseG" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "a:=seq([n,issqrfree(n)],n=1..10);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%\"aG6,7$\"\"\"%%trueG7$\"\"#F(7$\"\"$F(7$\"\"
%%&falseG7$\"\"&F(7$\"\"'F(7$\"\"(F(7$\"\")F/7$\"\"*F/7$\"#5F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "divisors(12); # oszt\363k ha
lmaza " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(\"\"\"\"\"#\"\"$\"\"%\"\"'
\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "seq([n,divisors(n)],n=1..10);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6,7$\"\"\"<#F$7$\"\"#<$F$F'7$\"\"$<$F$F*7$\"\"%<%F
$F'F-7$\"\"&<$F$F07$\"\"'<&F$F'F*F37$\"\"(<$F$F67$\"\")<&F$F'F-F97$\"
\"*<%F$F*F<7$\"#5<&F$F'F0F?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "tau(1000); # osztok szama" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq([n,tau(n)],n=1..10)
; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6,7$\"\"\"F$7$\"\"#F&7$\"\"$F&7$\"
\"%F(7$\"\"&F&7$\"\"'F*7$\"\"(F&7$\"\")F*7$\"\"*F(7$\"#5F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq(tau(n),n=1..20); " }}{PARA 11 "
" 1 "" {XPPMATH 20 "66\"\"\"\"\"#F$\"\"$F$\"\"%F$F&F%F&F$\"\"'F$F&F&\"
\"&F$F'F$F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sigma(1000);
 # oszt\363k \366sszege" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%SB" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq([n,sigma(n)],n=1..10);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6,7$\"\"\"F$7$\"\"#\"\"$7$F'\"\"%7$F)
\"\"(7$\"\"&\"\"'7$F.\"#77$F+\"\")7$F2\"#:7$\"\"*\"#87$\"#5\"#=" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "seq(sigma(n),n=1..20); " }}
{PARA 11 "" 1 "" {XPPMATH 20 "66\"\"\"\"\"$\"\"%\"\"(\"\"'\"#7\"\")\"#
:\"#8\"#=F(\"#G\"#9\"#CF/\"#JF,\"#R\"#?\"#U" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 27 "phi(1000); # Euler-f\374ggv\351ny" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#\"$+%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "seq([n,phi(n)],n=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,7$\"\"
\"F$7$\"\"#F$7$\"\"$F&7$\"\"%F&7$\"\"&F*7$\"\"'F&7$\"\"(F.7$\"\")F*7$
\"\"*F.7$\"#5F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "seq(phi(
n),n=1..10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,\"\"\"F#\"\"#F$\"\"%F$
\"\"'F%F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "seq([n,mobiu
s(n)],n=1..10); # M\366bius-f\374ggv\351ny" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6,7$\"\"\"F$7$\"\"#!\"\"7$\"\"$F'7$\"\"%\"\"!7$\"\"&F'7$
\"\"'F$7$\"\"(F'7$\"\")F,7$\"\"*F,7$\"#5F$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "seq(mobius(n),n=1..10); " }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6,\"\"\"!\"\"F$\"\"!F$F#F$F%F%F#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "fermat(n); # Fermat-sz\341mok" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&)\"\"#)F%%\"nG\"\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 11 "fermat(5); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"+
(Hn\\H%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "fermat(5,'w'); \+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"+(Hn\\H%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "w; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;it~is~
completely~factored~G*&-%!G6#,&*&)-F&6#\"\"#\"\"(\"\"\"-F&6#\"\"&\"\"
\"F3F3F3F3-F&6#,&*(F*F/-F&6#\"\"$F3-F&6#\"&\\u\"F3F3F3F3F3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "mersenne(5); mersenne(11); # a 2^n-
1 \351rt\351ke, ha az pr\355m, Mersenne-sz\341m" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"#J" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "mersenne([4]); # az n-edik M
ersenne-pr\355m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$F\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "legendre(5,13), legendre(4,13); # L
egendre-szimb\363lum" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\"\"\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "lambda(13), lambda(200);  \+
# Carmichael-f\374ggv\351ny" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#7\"#
?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "bigomega(40); # Omega-
f\374ggv\351ny (pr\355mhatv\341nyoszt\363k sz\341ma)" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 
"mcombine(7,4,11,5); # K\355nai marad\351kt\351tel, az mcombine(m,a,n,
b) megad egy olyan x eg\351sz sz\341mot, amelyre x=a (mod m), x=b (mod
 n) " }{TEXT -1 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#g" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "mcombine(5,4,15,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "order(3,10); # order(a,m) az a rendje (mod m)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "mlog(11,3,17); # index: mlog(a,g,m) megadja az ind_g(
a) mod n sz\341mot " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "mlog(11,4,17);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 82 "primroot(4,17);  # primroot(a,m) a legkisebb a-n\\'al negyobb pr
imit\355v gy\366k (mod m)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "49 0 0" 0 }
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