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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "global_start:=time()
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "inc1:=\{W,w,X,x,Y,y,Z,
z\}:\ninc2:=\{n,m\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 204 "eq
uW:=W=(1/2+1/3)*X+Z:\nequX:=X=(1/4+1/5)*Y+Z:\nequY:=Y=(1/6+1/7)*W+Z:\n
equw:=w=(1/3+1/4)*(X+x):\nequx:=x=(1/4+1/5)*(Y+y):\nequy:=y=(1/5+1/6)*
(Z+z):\nequz:=z=(1/6+1/7)*(W+w):\nequm:=m^2=W+X:\nequn:=Y+Z=n*(n+1)/2:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "S1:=isolve(\{seq(equ.i,
i=inc1 minus \{Z\})\},k);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#S1G<*/
%\"xG,$%\"kG\"(YK*[/%\"yG,$F)\"(?e^$/%\"wG,$F)\"(gj?(/%\"XG,$F)\"(90Y(
/%\"YG,$F)\"(g!et/%\"ZG,$F)\"((Q\\T/%\"WG,$F)\")#[m.\"/%\"zG,$F)\"(8#R
a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S2:=[isolve(subs(S1,eq
um),p)];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#S2G7$<$/%\"mG,$%\"pG!)'
*p#y\"/%\"kG,$*$)F*\"\"#\"\"\"\")'*p#y\"<$/F-,(\"(\\nX%\"\"\"F*F+F/F3/
F(,&\"()\\8*)F8F*F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "for \+
i to nops(S2) do\n  kk.i:=factor(subs(S2[i],k));\n  mm.i:=factor(subs(
S2[i],m));\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$kk1G,$*$)%\"pG\"
\"#\"\"\"\")'*p#y\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mm1G,$%\"pG!
)'*p#y\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$kk2G,$*$),&%\"pG\"\"#!
\"\"\"\"\"F*\"\"\"\"(\\nX%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mm2G,
&\"()\\8*)\"\"\"%\"pG!)'*p#y\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 112 "for i to nops(S2) do\n  k.i:=factor(subs(p=(i mod 2+pp)/2,kk.
i));\n  m.i:=factor(subs(p=(i mod 2+pp)/2,mm.i));\nod;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#k1G,$*$),&\"\"\"F)%#ppGF)\"\"#\"\"\"\"(\\nX%" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m1G,&!()\\8*)\"\"\"%#ppGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k2G,$*$),&%#ppG\"\"\"!\"\"F*\"\"#\"
\"\"\"(\\nX%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m2G,&\"()\\8*)\"\"
\"%#ppG!()\\8*)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S2:=\{k=
4456749*p^2,m=-8913498*p\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G<
$/%\"kG,$*$)%\"pG\"\"#\"\"\"\"(\\nX%/%\"mG,$F+!()\\8*)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "subs(S1,S2,equn);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/,$*$)%\"pG\"\"#\"\"\"\"/.)4H!eG^,$*&%\"nG\"\"\",&F-
F.F.F.F.#F.F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fermat:=op
(1,%)-op(2,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'fermatG,&*$)%\"pG
\"\"#\"\"\"\"/.)4H!eG^*&%\"nG\"\"\",&F-F.F.F.F.#!\"\"F)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(n=N/2-1/2,fermat);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"pG\"\"#\"\"\"\"/.)4H!eG^*&,&%\"NG#\"
\"\"F'#!\"\"F'F.F.,&F,F-F-F.F.F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "fermat1:=expand(8*%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%(fermat1G,(*$)%\"pG\"\"#\"\"\"\"0C%yKU'G5%*$)%\"NGF)F*!\"\"\"
\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "d:=coeff(fermat1,
p,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"0C%yKU'G5%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(d);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*0)-%!G6#\"\"#\"\"$\"\"\"-F&6#F)\"\"\"-F&6#\"\"(F-
-F&6#\"#6F--F&6#\"#HF--F&6#\"$`$F-)-F&6#\"%dYF(F*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "df:=readlib(ifactors)(d)[2];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#dfG7)7$\"\"#\"\"$7$F(\"\"\"7$\"\"(F*7$\"#6F*7$
\"#HF*7$\"$`$F*7$\"%dYF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 
"sqrtd:=mul(i[1]^iquo(i[2],2),i=df)*sqrt(mul(i[1]^irem(i[2],2),i=df));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sqrtdG,$*$-%%sqrtG6#\"(%\\HZ\"
\"\"\"%9$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dd:=mul(i[1]^
irem(i[2],2),i=df);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ddG\"(%\\HZ
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "sqrtdd:=sqrt(dd);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'sqrtddG*$-%%sqrtG6#\"(%\\HZ\"\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 453 "algtoconfrac:=proc(x,N::
nonnegint)\n  local n,z,T,history,i,delta;\n  z:=radnormal(x,rationali
zed);\n  for n to N do\n    T[n]:=floor(z);\n    if assigned(history[z
]) then\n      i:=history[z];\n RETURN([seq(T[j],j=1..i-1)],[seq(T[j],
j=i..n-1)])\n    else\n      history[z]:=n;\n    fi;\n    delta:=radno
rmal(z-T[n]);\n    if delta=0 then RETURN([seq(T[i],i=1..n)]) \n    el
se z:=radnormal(1/delta,rationalized)\n    fi;\n  od;\n  [seq(T[n],n=1
..N)]\nend: # algtoconfrac" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
61 "start:=time():\nCFe:=algtoconfrac(sqrtdd,10000):\ntime()-start;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&'R=!\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "CFe;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$7#\"%u@7h
p\"\"\"\"\"#F&\"\"&F'\"#D\"\"$F&F&F&F&F&F&\"#:F&F'\"#;F&F'F&F&\"\")\"
\"'F&\"#@F&F&F*F&F&F&F'F'F.F&F&F(F&\"#<F&F&\"#ZF*F&F&F.F&F&F*F1F&F&F0F
&F(F&F&F.F'F'F&F&F&F*F&F&F/F&F.F-F&F&F'F&F,F'F&F+F&F&F&F&F&F&F*F)F'F(F
&F'F&\"%[V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "m1:=nops(cFe[
1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#m1G\"\"\"" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "m2:=nops(CFe[2]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#m2G\"##*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "if type(m2,even) then m3:=m2 else m3:=2*m2 fi;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#m3G\"##*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "L:=[op(CFe[1]),op(CFe[2])]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "modulus:=mul(i[1]^iquo(i[2],2),i=df);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(modulusG\"%9$*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 302 "oldA:=1:\nA:=CFe[1][1]:\noldB:=0:\nB:=1:\nfor nu to \+
m1 do\n  newA:=L[nu+1]*A+oldA;\n  newB:=L[nu+1]*B+oldB;\n  oldA:=A;\n \+
 oldB:=B;\n  A:=newA;\n  B:=newB;\nod:\nfor nu from m1+1 to m3-1 do\n \+
 nn:=nu-iquo(nu-m1,m2)*m2;\n  newA:=L[nn+1]*A+oldA;\n  newB:=L[nn+1]*B
+oldB;\n  oldA:=A;\n  oldB:=B;\n  A:=newA;\n  B:=newB;\nod:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A,B;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6$\"N\\!)3,RaL9#GBm)z\\tHGt')>$*4\"\"JSj)*3/aN(>yZuI.:VB&
[\\0&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "A^2-dd*B^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "XA:=oldA;\nYA:=A;\nXB:=oldB;\nYB:=B;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#XAG\"MMMpvuhEM_=9@:$*=*p4;'zM\")" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#YAG\"N\\!)3,RaL9#GBm)z\\tHGt')>$*4\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#XBG\"J*)[ykZpojr_9&f2`)GM$eSP" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#YBG\"JSj)*3/aN(>yZuI.:VB&[\\0&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "oldA:=1:\nA:=CFe[1][1]:\nol
dB:=0:\nB:=1:\nfor nu to m1 do\n  newA:=L[nu+1]*A+oldA mod modulus;\n \+
 newB:=L[nu+1]*B+oldB mod modulus;\n  oldA:=A;\n  oldB:=B;\n  A:=newA;
\n  B:=newB;\nod:\nfor nu from m1+1 to m3-1 do\n  nn:=nu-iquo(nu-m1,m2
)*m2;\n  newA:=L[nn+1]*A+oldA mod modulus;\n  newB:=L[nn+1]*B+oldB mod
 modulus;\n  oldA:=A;\n  oldB:=B;\n  A:=newA;\n  B:=newB;\nod:\nfor ka
ppa while B<>0 do\n  for nu from kappa*m3 to (kappa+1)*m3-1 do\n    nn
:=nu-iquo(nu-m1,m2)*m2;\n    newA:=L[nn+1]*A+oldA mod modulus;\n    ne
wB:=L[nn+1]*B+oldB mod modulus;\n    oldA:=A;\n    oldB:=B;\n    A:=ne
wA;\n    B:=newB;\n  od;\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "right_nu:=nu-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)right_nuG
\"'nU@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "A,B;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$\"%8$*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "qq:=iquo(right_nu,92,'rr');" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#qqG\"%GB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "rr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#\"*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 120 "oldC:=x:\nC:=y:\nfor nu from m3 to 2*m3-1 do
\n  nn:=nu-iquo(nu-m1,m2)*m2;\n  newC:=L[nn+1]*C+oldC;\n  oldC:=C;\n  \+
C:=newC\nod:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "M:=array
(1..2,1..2,\n   [[subs(x=1,y=0,oldC),subs(x=0,y=1,oldC)],\n    [subs(x
=1,y=0,C),subs(x=0,y=1,C)]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"M
G-%'matrixG6#7$7$\"J*)[ykZpojr_9&f2`)GM$eSP\"N?@/=h.p*3RipOVwd-%[V#oE;
7$\"JSj)*3/aN(>yZuI.:VB&[\\0&\"N47RPIR)\\EH,\"y$)ohIAjnl#)>#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Mnorm:=max(M[1,1]+M[1,1],M[2
,1]+M[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&MnormG\"N\\vPYM$RBY
2\\bo\">$\\Y<r6x)>#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eval
f(Mnorm^qq*max(XA,XB,YA,YB));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(
zkvK%\"'jK5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "L:=convert(q
q,base,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG7.\"\"!F&F&\"\"\"F
'F&F&F&F'F&F&F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "start:=
time():\nMM:=M:\nP:=&*():\nfor i from 2 to nops(L) do\n  MM:=evalm(MM&
*MM);\n  if L[i]=1 then \n    P:=evalm(P&*MM) \n  fi;\n  print(i,time(
)-start);\nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#$\"#]!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$$\"#(*!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\"%$\"$Q#!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"
&$\"$i$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"'$\"$;%!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"($\"$\"[!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"\")$\"$(f!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"
*$\"%s5!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#5$\"%B<!\"$" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"#6$\"%kN!\"$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"#7$\"&=Z\"!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 65 "A:=evalm(P&*vector([XA,YA]))[2]:\nB:=evalm(P&*vector([XB,YB]))
[2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "length(A),length(B)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"'tK5\"'qK5" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 14 "A mod modulus;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#\"%8$*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "B mod modul
us;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "evalb(A^2-dd*B^2=1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "S
OL:=subs(remove(has,S2,m),\{N=A,p=iquo(B,modulus)\},\{n=N/2-1/2\} unio
n S1 union S2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "for i in
 inc1 union inc2 do\n  sol.i:=subs(SOL,i);\n  print(i,length(sol.i))\n
od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"nG\"'tK5" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$%\"xG\"'Wl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"yG
\"'Wl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"wG\"'Xl?" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$%\"XG\"'Xl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"
YG\"'Xl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"ZG\"'Wl?" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$%\"WG\"'Xl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%
\"zG\"'Wl?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"mG\"'tK5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "time()-global_start;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"'7@D!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(B,50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S9
_VBZb\\lQh#*pzTHk0r<&*e)*RJ<\"'?K5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "B mod 10^50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"Sge
u\"\\$)G@\"*GSQN#*)Q$[6&fOjp4#>)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "writeto(\"archimedes.html\"):\nB;\nwriteto(terminal):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "S:=subs(\{seq(i=sol.i,i
=inc1 union inc2)\},x+y+z+w+X+Y+Z+W):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "length(S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"'Xl?
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(S);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#$\"+19Fgx\"'Nl?" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{MARK "50 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }