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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 1 {PARA 3 "" 0 "" {TEXT 205 23 "5. Alkalmaz
\303\241sok " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 205 36 "6. Sz\303\241mo
k \303\251s polinomok" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 0 {PARA 4 "" 0 ""
 {TEXT 206 5 "6.1. " }{TEXT 206 8 "\303\226" }{TEXT 206 10 "sszehasonl
" }{TEXT 206 8 "\303\255" }{TEXT 206 1 "t" }{TEXT 206 8 "\303\241" }
{TEXT 206 3 "s, " }{TEXT 206 8 "\303\266" }{TEXT 206 6 "sszead" }{TEXT
 206 8 "\303\241" }{TEXT 206 8 "s, kivon" }{TEXT 206 8 "\303\241" }
{TEXT 206 2 "s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 37 "# We fix a base. For dem
o purposes:\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 
0 6 "B:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "`mod`:=modp;" }}{PARA 11 "" 1 "" {XPPMATH 20
 "I%modpG%*protectedG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n
" }{MPLTEXT 1 0 35 "# Carry addition to long number s\n" }{MPLTEXT 1 
0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "cadd:=proc(s,c) local
 i,r,cc,x; global B;\n" }{MPLTEXT 1 0 15 "r:=[]; cc:=c;\n" }{MPLTEXT 
1 0 28 "for i from 1 to nops(s) do\n" }{MPLTEXT 1 0 21 "  x:=s[i]+cc m
od B;\n" }{MPLTEXT 1 0 22 "  cc:=(s[i]+cc-x)/B;\n" }{MPLTEXT 1 0 16 " \+
 r:=[op(r),x]\n" }{MPLTEXT 1 0 15 "od; [r,cc] end;" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "f*6$I\"sG6\"I\"cGF%6&I\"iGF%I\"rGF%I#ccGF%I\"xGF%F%F%C&>
F)7\">F*F&?(F(\"\"\"F1-I%nopsG%*protectedG6#F$I%trueGF4C%>F+-I$modGF%6
$,&&F$6#F(F1F*F1I\"BGF%>F**&,(F=F1F*F1F+!\"\"F1F?FC>F)7$-I#opGF46#F)F+
7$F)F*F%6#F?F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }
{MPLTEXT 1 0 40 "# Carry subtraction from long number s\n" }{MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 43 "csub:=proc(s,c) loc
al i,r,cc,x; global B;\n" }{MPLTEXT 1 0 15 "r:=[]; cc:=c;\n" }{MPLTEXT
 1 0 28 "for i from 1 to nops(s) do\n" }{MPLTEXT 1 0 21 "  x:=s[i]-cc \+
mod B;\n" }{MPLTEXT 1 0 23 "  cc:=-(s[i]-cc-x)/B;\n" }{MPLTEXT 1 0 16 
"  r:=[op(r),x]\n" }{MPLTEXT 1 0 15 "od; [r,cc] end;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "f*6$I\"sG6\"I\"cGF%6&I\"iGF%I\"rGF%I#ccGF%I\"xGF%F%F%C
&>F)7\">F*F&?(F(\"\"\"F1-I%nopsG%*protectedG6#F$I%trueGF4C%>F+-I$modGF
%6$,&&F$6#F(F1F*!\"\"I\"BGF%>F*,$*&,(F=F1F*F?F+F?F1F@F?F?>F)7$-I#opGF4
6#F)F+7$F)F*F%6#F@F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "ss:
=convert(100,base,B); csub(ss,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%
\"\"!F#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7%\"\")\"\"*\"\"!F&" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 31 "# C
omparision of long numbers\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 27 "cmp:=proc(s1,s2) local i;\n" }{MPLTEXT 1 0 41 "i
f nops(s1)>nops(s2) then RETURN(1) fi;\n" }{MPLTEXT 1 0 42 "if nops(s1
)<nops(s2) then RETURN(-1) fi;\n" }{MPLTEXT 1 0 35 "for i from nops(s1
) by -1 to 1 do\n" }{MPLTEXT 1 0 37 "  if s1[i]>s2[i] then RETURN(1) f
i;\n" }{MPLTEXT 1 0 38 "  if s1[i]<s2[i] then RETURN(-1) fi;\n" }
{MPLTEXT 1 0 10 "od; 0 end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I#s1G
6\"I#s2GF%6#I\"iGF%F%F%C&@$2-I%nopsG%*protectedG6#F&-F-6#F$-I'RETURNGF
.6#\"\"\"@$2F0F,-F36#!\"\"?(F(F0F:F5I%trueGF.C$@$2&F&F'&F$F'F2@$2FAF@F
8\"\"!F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "s1:=convert
(123,base,B); s2:=convert(321,base,B); cmp(s1,s2);" }}{PARA 11 "" 1 ""
 {XPPMATH 20 "7%\"\"$\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"
\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 32 "# Addition wi
th carry, n digit\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "\n" }
{MPLTEXT 1 0 49 "addc:=proc(s1,s2,c,n) local i,r,cc,x; global B;\n" }
{MPLTEXT 1 0 15 "r:=[]; cc:=c;\n" }{MPLTEXT 1 0 22 "for i from 1 to n \+
do\n" }{MPLTEXT 1 0 28 "  x:=s1[i]+s2[i]+cc mod B;\n" }{MPLTEXT 1 0 
29 "  cc:=(s1[i]+s2[i]+cc-x)/B;\n" }{MPLTEXT 1 0 16 "  r:=[op(r),x]\n"
 }{MPLTEXT 1 0 16 "od; [r,cc]; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f
*6&I#s1G6\"I#s2GF%I\"cGF%I\"nGF%6&I\"iGF%I\"rGF%I#ccGF%I\"xGF%F%F%C&>F
+7\">F,F'?(F*\"\"\"F3F(I%trueG%*protectedGC%>F--I$modGF%6$,(&F$6#F*F3&
F&F=F3F,F3I\"BGF%>F,*&,*F<F3F>F3F,F3F-!\"\"F3F?FC>F+7$-I#opGF56#F+F-7$
F+F,F%6#F?F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "addc(s,s1,2
,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$7%,&&I\"sG6\"6#\"\"\"F)\"\"&F)
,&&F&6#\"\"#F)F.F),&&F&6#\"\"$F)F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 35 "# Subtraction with carry, n di
git\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 3 " \n" }{MPLTEXT 1 0 39 "s
ubc:=proc(s1,s2,c,n) local i,r,cc,x;\n" }{MPLTEXT 1 0 15 "r:=[]; cc:=c
;\n" }{MPLTEXT 1 0 22 "for i from 1 to n do\n" }{MPLTEXT 1 0 28 "  x:=
s1[i]-s2[i]-cc mod B;\n" }{MPLTEXT 1 0 30 "  cc:=-(s1[i]-s2[i]-cc-x)/B
;\n" }{MPLTEXT 1 0 16 "  r:=[op(r),x]\n" }{MPLTEXT 1 0 15 "od; [r,cc] \+
end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I#s1G6\"I#s2GF%I\"cGF%I\"nGF
%6&I\"iGF%I\"rGF%I#ccGF%I\"xGF%F%F%C&>F+7\">F,F'?(F*\"\"\"F3F(I%trueG%
*protectedGC%>F--I$modGF%6$,(&F$6#F*F3&F&F=!\"\"F,F?I\"BGF%>F,,$*&,*F<
F3F>F?F,F?F-F?F3F@F?F?>F+7$-I#opGF56#F+F-7$F+F,F%F%F%" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "subc(s1,s2,2,3);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "7$7%\"\"!F$\"\")\"\"\"" }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT 206 10 "6.2. Szorz" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "s " }
{TEXT 206 8 "\303\251" }{TEXT 206 14 "s polinomszorz" }{TEXT 206 8 "\3
03\241" }{TEXT 206 2 "s." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x:='x'; i:='i';\n" }{MPLTEXT 1 0 
64 "s1:=convert(123,base,B); p1:=add(s1[i]*x^(i-1),i=1..nops(s1));\n" 
}{MPLTEXT 1 0 64 "s2:=convert(321,base,B); p2:=add(s2[i]*x^(i-1),i=1..
nops(s2));\n" }{MPLTEXT 1 0 41 "convert(123*321,base,B); expand(p1*p2)
;\n" }{MPLTEXT 1 0 34 "convert(1002003*3002001,base,B^3);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"iG6
\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"$\"\"#\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 ",(\"\"$\"\"\"*&\"\"#F$I\"xG6\"F$F$*$)F'F&F$F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"\"\"\"#\"\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ",(\"\"\"F#*&\"\"#F#I\"xG6\"F#F#*&\"\"$F#)F&F%F#F#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "7'\"\"$\"\")\"\"%\"\"*F#" }}{PARA 11 "" 
1 "" {XPPMATH 20 ",,\"\"$\"\"\"*&\"\")F$I\"xG6\"F$F$*&\"#9F$)F'\"\"#F$
F$*&F&F$)F'F#F$F$*&F#F$)F'\"\"%F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "7
'\"\"$\"\")\"#9F$F#" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 36 "6.3. Kl
asszikus algoritmusok a szorz" }{TEXT 206 8 "\303\241" }{TEXT 206 4 "s
ra " }{TEXT 206 8 "\303\251" }{TEXT 206 9 "s az oszt" }{TEXT 206 8 "\3
03\241" }{TEXT 206 4 "sra." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
206 19 "6.4. Karacuba szorz" }{TEXT 206 8 "\303\241" }{TEXT 206 2 "s."
 }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 3 "#\n" }{MPLTEXT 1 0 31 "# Single digit multiplication\n" }
{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 28 "mul11:=proc(x,y) global B;\n" }
{MPLTEXT 1 0 28 "convert(x[1]*y[1],base,B);\n" }{MPLTEXT 1 0 16 "[%[1]
,%[2]] end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6$I\"xG6\"I\"yGF%F%F%F%
C$-I(convertG%*protectedG6%*&&F$6#\"\"\"F/&F&F.F/I%baseGF%I\"BGF%7$&I
\"%GF%F.&F56#\"\"#F%6#F2F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "mul11([2],[7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"%\"\"\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 56 "# Kar
atsuba's method; x and y are lists of nonnegative\n" }{MPLTEXT 1 0 27 
"# digits with lenght 2^n.\n" }{MPLTEXT 1 0 3 "#\n" }{MPLTEXT 1 0 2 "
\n" }{MPLTEXT 1 0 19 "kara:=proc(x,y,n)\n" }{MPLTEXT 1 0 67 "local m0,
m00,m01,m1,m10,m11,m2,m20,m21,f,x0,x1,y0,y1,z0,z1,c0,c1;\n" }{MPLTEXT 
1 0 36 "if n=0 then RETURN(mul11(x,y)) fi;\n" }{MPLTEXT 1 0 22 "x0:=[x
[1..2^(n-1)]];\n" }{MPLTEXT 1 0 26 "x1:=[x[1+2^(n-1)..2^n]];\n" }
{MPLTEXT 1 0 22 "y0:=[y[1..2^(n-1)]];\n" }{MPLTEXT 1 0 26 "y1:=[y[1+2^
(n-1)..2^n]];\n" }{MPLTEXT 1 0 22 "m0:=kara(x0,y0,n-1);\n" }{MPLTEXT 
1 0 24 "m00:=[m0[1..2^(n-1)]];\n" }{MPLTEXT 1 0 28 "m01:=[m0[1+2^(n-1)
..2^n]];\n" }{MPLTEXT 1 0 22 "m2:=kara(x1,y1,n-1);\n" }{MPLTEXT 1 0 
24 "m20:=[m2[1..2^(n-1)]];\n" }{MPLTEXT 1 0 28 "m21:=[m2[1+2^(n-1)..2^
n]];\n" }{MPLTEXT 1 0 10 "f:=true;\n" }{MPLTEXT 1 0 32 "if cmp(x1,x0,2
^(n-1))>=0 then \n" }{MPLTEXT 1 0 32 "  z0:=subc(x1,x0,0,2^(n-1))[1]\n
" }{MPLTEXT 1 0 6 "else\n" }{MPLTEXT 1 0 33 "  z0:=subc(x0,x1,0,2^(n-1
))[1];\n" }{MPLTEXT 1 0 13 "  f:=not f;\n" }{MPLTEXT 1 0 5 "fi;\n" }
{MPLTEXT 1 0 32 "if cmp(y1,y0,2^(n-1))>=0 then \n" }{MPLTEXT 1 0 33 " \+
 z1:=subc(y1,y0,0,2^(n-1))[1];\n" }{MPLTEXT 1 0 6 "else\n" }{MPLTEXT 
1 0 33 "  z1:=subc(y0,y1,0,2^(n-1))[1];\n" }{MPLTEXT 1 0 13 "  f:=not \+
f;\n" }{MPLTEXT 1 0 5 "fi;\n" }{MPLTEXT 1 0 24 "  m1:=kara(z0,z1,n-1);
\n" }{MPLTEXT 1 0 26 "  m10:=[m1[1..2^(n-1)]];\n" }{MPLTEXT 1 0 30 "  \+
m11:=[m1[1+2^(n-1)..2^n]];\n" }{MPLTEXT 1 0 32 "  z0:=addc(m01,m20,0,2
^(n-1));\n" }{MPLTEXT 1 0 14 "  c0:=z0[2];\n" }{MPLTEXT 1 0 34 "  z1:=
addc(z0[1],m00,0,2^(n-1));\n" }{MPLTEXT 1 0 17 "  c1:=c0+z1[2];\n" }
{MPLTEXT 1 0 35 "  z0:=addc(z0[1],m21,c1,2^(n-1));\n" }{MPLTEXT 1 0 
17 "  c0:=c0+z2[2];\n" }{MPLTEXT 1 0 36 "  m21:=cadd(z0[1],m21,c0,2^(n
-1));\n" }{MPLTEXT 1 0 11 "if f then\n" }{MPLTEXT 1 0 34 "  z1:=subc(z
1[1],m10,0,2^(n-1));\n" }{MPLTEXT 1 0 14 "  c1:=z1[2];\n" }{MPLTEXT 1 
0 35 "  z0:=subc(z0[1],m11,c1,2^(n-1));\n" }{MPLTEXT 1 0 14 "  c0:=z0[
2];\n" }{MPLTEXT 1 0 25 "  m21:=csub(m21[2],c2);\n" }{MPLTEXT 1 0 6 "e
lse\n" }{MPLTEXT 1 0 34 "  z1:=addc(z1[1],m10,0,2^(n-1));\n" }{MPLTEXT
 1 0 14 "  c1:=z1[2];\n" }{MPLTEXT 1 0 35 "  z0:=addc(z0[1],m11,c1,2^(
n-1));\n" }{MPLTEXT 1 0 14 "  c0:=z2[2];\n" }{MPLTEXT 1 0 20 "  cadd(m
21[1],c0);\n" }{MPLTEXT 1 0 49 "fi; [op(m00),op(z1[1]),op(z0[1]),op(m2
1[1])] end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"xG6\"I\"yGF%I\"nGF
%64I#m0GF%I$m00GF%I$m01GF%I#m1GF%I$m10GF%I$m11GF%I#m2GF%I$m20GF%I$m21G
F%I\"fGF%I#x0GF%I#x1GF%I#y0GF%I#y1GF%I#z0GF%I#z1GF%I#c0GF%I#c1GF%F%F%C
<@$/F'\"\"!-I'RETURNG%*protectedG6#-I&mul11GF%6$F$F&>F37#&F$6#;\"\"\")
\"\"#,&F'FKFK!\"\">F47#&F$6#;,&FKFKFLFK)FMF'>F57#&F&FI>F67#&F&FS>F)-I%
karaGF%6%F3F5FN>F*7#&F)FI>F+7#&F)FS>F/-Fin6%F4F6FN>F07#&F/FI>F17#&F/FS
>F2I%trueGFA@%1F>-I$cmpGF%6%F4F3FL>F7&-I%subcGF%6&F4F3F>FL6#FKC$>F7&-F
dp6&F3F4F>FLFfp>F24F2@%1F>-F_p6%F6F5FL>F8&-Fdp6&F6F5F>FLFfpC$>F8&-Fdp6
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