(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.1'

                    Mathematica-Compatible Notebook

This notebook can be used with any Mathematica-compatible
application, such as Mathematica, MathReader or Publicon. The data
for the notebook starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do
one of the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the
  application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing
the word CacheID, otherwise Mathematica-compatible applications may
try to use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info@wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
*******************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     59554,       1234]*)
(*NotebookOutlinePosition[     60243,       1258]*)
(*  CellTagsIndexPosition[     60199,       1254]*)
(*WindowFrame->Normal*)



Notebook[{
Cell["Proof of a conjecture about class III ME", "Title"],

Cell[TextData[{
  StyleBox["Abstract",
    FontVariations->{"Underline"->True}],
  " -\nIt appears that the number 12 is exceptional also in respect to the \
notion of class III Maximally Even sets introduced by Ian Quinn in his \
dissertation: nobody could find an instance of a ",
  Cell[BoxData[
      \(TraditionalForm\`c\)]],
  " (not prime) pc-universe greater than 12 without instances of class III \
ME. We prove this is indeed impossible. The modulo 12 is yet again very \
special in this respect."
}], "Author"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: 1 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.5 0.454545 0.5 0.454545 [
[ 0 0 0 0 ]
[ 1 1 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 0 m
1 0 L
1 1 L
0 1 L
closepath
clip
newpath
.79 1 .413 r
.5 .5 m
.5 .5 .45455 0 365.73 arc
F
.209 .15 1 r
.02 w
.95012 .56326 Mdot
.93694 .62529 Mdot
.91525 .68488 Mdot
.88548 .74087 Mdot
.8482 .79218 Mdot
.80415 .83779 Mdot
.75418 .87684 Mdot
.69926 .90854 Mdot
.64046 .9323 Mdot
.57893 .94764 Mdot
.51586 .95427 Mdot
.45249 .95206 Mdot
.39004 .94104 Mdot
.32972 .92145 Mdot
.27273 .89365 Mdot
.22015 .85819 Mdot
.17303 .81575 Mdot
.13227 .76718 Mdot
.09866 .7134 Mdot
.07287 .65546 Mdot
.05539 .59451 Mdot
.04656 .53171 Mdot
.04656 .46829 Mdot
.05539 .40549 Mdot
.07287 .34454 Mdot
.09866 .2866 Mdot
.13227 .23282 Mdot
.17303 .18425 Mdot
.22015 .14181 Mdot
.27273 .10635 Mdot
.32972 .07855 Mdot
.39004 .05896 Mdot
.45249 .04794 Mdot
.51586 .04573 Mdot
.57893 .05236 Mdot
.64046 .0677 Mdot
.69926 .09146 Mdot
.75418 .12316 Mdot
.80415 .16221 Mdot
.8482 .20782 Mdot
.88548 .25913 Mdot
.91525 .31512 Mdot
.93694 .37471 Mdot
.95012 .43674 Mdot
.95455 .5 Mdot
0 g
.5 Mabswid
[ ] 0 setdash
newpath
.5 .5 .45455 0 365.73 arc
s
.05 w
.95455 .5 Mdot
.93694 .62529 Mdot
.88548 .74087 Mdot
.80415 .83779 Mdot
.69926 .90854 Mdot
.57893 .94764 Mdot
.45249 .95206 Mdot
.27273 .89365 Mdot
.17303 .81575 Mdot
.09866 .7134 Mdot
.05539 .59451 Mdot
.04656 .46829 Mdot
.07287 .34454 Mdot
.13227 .23282 Mdot
.27273 .10635 Mdot
.39004 .05896 Mdot
.51586 .04573 Mdot
.64046 .0677 Mdot
.75418 .12316 Mdot
.8482 .20782 Mdot
.91525 .31512 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  GeneratedCell->False,
  CellAutoOverwrite->False,
  ImageSize->{288, 288},
  ImageMargins->{{90.375, 0}, {0, 3.1875}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40005X0001J2000`400?l00000o`00003ooooooomY
ooooo`00oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00oooooomY
ooooo`00oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00]_ooool5o`000:gooooo002d
ooooo`Wo0000Zoooool00;;ooooo3Ol0002Yooooo`00/_ooool=o`000:Wooooo002aooooo`oo0000
Z?ooool008Wooooo1Ol0000Sooooo`oo0000Z?ooool008Oooooo2Ol0000Aooooo`?o=BKo3?oooolA
o`000:Oooooo0025ooooo`go00003_ooool5ocDVo`_ooooo4Ol0000=ooooo`?o=BKoUoooool008Go
oooo3Ol0000=ooooo`Go=BKo7ol0000:ooooo`Go=BKo4Oooool5o`00083ooooo0024ooooo`oo0000
2Oooool8o`0000;o=BKo2_o:ofTAo`0000;ob_mY2?l000000ooooooo=BKoocDVo`05ocDVo`kooooo
2Ol0001nooooo`00Q?ooool?o`0000?ooooo1_l00003ol[oJ@Oo=BKo2_o:ofTAo`0000[ob_mY1_l0
0002ocDVo`cooooo3Ol0001looooo`00PooooolCo`0000[ob_mY1Ole9_l<ol[oJ@oo00003?o:ofT5
ocDVo`Ko00002?ooool=o`0007cooooo0023oooooa7o00003Oo:ofT3ocDVo`gob_mY3ol0000=ol[o
J@Go=BKo1Oo:ofT4o`0000?ooooo3ol0001kooooo`00PooooolAo`0001kob_mY3Ol0000?ol[oJ@?o
=BKo2_o:ofTBo`0007_ooooo0023oooooa7o00007_o:ofT=o`0001kob_mY4Ol0001jooooo`00P_oo
oolBo`00023ob_mY2Ol0000Pol[oJA7o0000N_ooool007kooooo1?l00002ol[oJ@oo00008oo:ofT5
o`0002;ob_mY4Ol0001jooooo`00MOooool3ocDVo`Cooooo0_l00006ol[oJ@oo0000B_o:ofTAo`00
07[ooooo001dooooo`Co=BKo1?l00009ol[oJ@go0000Boo:ofTBo`0007Wooooo001cooooo`Co=BKo
00?o0000ocDVoole9_l02oo:ofT=o`0004cob_mY3ol00002ol[oJ@;o0000Moooool007?ooooo0_le
9_l2o`0000?o=BKo3Oo:ofT9o`0004kob_mY3ol00004ol[oJ@03o`000?oooooooooo07Cooooo001a
ooooo`Co00001Ole9_l?ol[oJ@Go0000DOo:ofT=o`0000Kob_mY1?l00003ooooo`?o=BKoK?ooool0
05oooooo1Ol0000;ooooo`;o00000oo:ofT5ocDVofKob_mY3Ol0000:ol[oJ@;o00001Ole9_m[oooo
o`00GOooool9o`0000Oooooo0_l00006ol[oJ@?o=BKoJOo:ofT9o`0000gob_mY00?o=BKoo`000?l0
00001?le9_mZooooo`00Foooool=o`0000?ooooo0_l0001fol[oJ@Go00003oo:ofT3ocDVo`;o0000
0_le9_mZooooo`00Foooool=o`0000;ooooo00?o0000ol[oJOo:ofT0R_o:ofT5ocDVo`03o`000?le
9_oooooo06Wooooo001Joooooa3o0000S_o:ofT5ocDVo`;o0000JOooool005[ooooo3ol0002@ol[o
J@?o=BKo0oo:ofT2o`0006Oooooo001Ioooooa7o0000Uoo:ofT2o`0000Wooooo1Ol0001Gooooo`00
FOoooolAo`0009Wob_mY0_l00005ooooo`Wo0000EOooool005Wooooo4Ol0002Kol[oJ@03o`000?oo
oooooooo00go0000Doooool005Wooooo4Ol0002Lol[oJ@oo0000Doooool005Wooooo4Ol0002Mol[o
J@oo0000D_ooool005[ooooo3ol0002Nol[oJ@oo0000D_ooool005[ooooo3ol0002Mol[oJA7o0000
DOooool005Sooooo0_l000000oo:ofWo0000o`00000;o`0009kob_mY4Ol0001Aooooo`00E_ooool2
o`0000?ob_mY3Ol0002Nol[oJA7o0000DOooool005Gooooo00?o0000ol[oJOo:ofT01Oo:ofT9o`00
0:3ob_mY4Ol0001Aooooo`00E?ooool00ol0003ob_mYol[oJ@08ol[oJ@Go0000X_o:ofTAo`00057o
oooo001=ooooo`?o=BKo0oooool00ol0003ob_mYol[oJ@2aol[oJ@oo0000D_ooool004cooooo1Ole
9_l2o`000;Cob_mY3ol0001Booooo`00Boooool5ocDVo`03o`000?le9_oob_mY0;Gob_mY3Ol00000
0oo:ofWo0000ooooo`1@ooooo`00Boooool3ocDVo`;o00000_le9_nfol[oJ@go00000_o:ofT2o`00
04oooooo001;ooooo`;o=BKo00?o0000ocDVoole9_l00_le9_nhol[oJ@Wo00001_o:ofT00ol0003o
ooooooooo`1<ooooo`00C?ooool00ole9_oo0000ocDVo`02ocDVok_ob_mY1Ol00008ol[oJ@03o`00
0?oooooooooo04cooooo001;ooooo`;o00000ole9_o:ol[oJ@;o0000C?ooool004[ooooo00?o0000
ol[oJOo:ofT0coo:ofT00ol0003oooooooooo`19ooooo`00BOooool00ol0003ob_mYol[oJ@3Aol[o
J@03o`000?oooooo=BKo00;o=BKoA_ooool004Sooooo00?o0000ol[oJOo:ofT0doo:ofT00ol0003o
=BKoocDVo`02ocDVodGooooo0016ooooo`;o0000eOo:ofT2ocDVo`;o00000ole9_m4ooooo`00AOoo
ool00ol0003ob_mYol[oJ@3Eol[oJ@Co=BKo00?o0000ocDVoole9_l0A?ooool004Cooooo00?o0000
ol[oJOo:ofT0e_o:ofT5ocDVo`03o`000?le9_oooooo04?ooooo0013ooooo`03o`000?o:ofWob_mY
0=Sob_mY1Ole9_l00ol0003oooooooooo`12ooooo`00@_ooool00ol0003ob_mYol[oJ@3Jol[oJ@?o
=BKo0_o:ofT00ol0003oooooooooo`11ooooo`00@Oooool00ol0003ob_mYol[oJ@3Qol[oJ@03o`00
0?oooooooooo043ooooo000mooooo`?o=BKo00?o0000ol[oJOo:ofT0hoo:ofT00ol0003ooooooooo
o`0oooooo`00??ooool3ocDVo`03o`000?le9_oob_mY0>Gob_mY00?o0000ooooooooool0?_ooool0
03_ooooo0ole9_l01?l0003o=BKoocDVoole9_oVol[oJ@03o`000?oooooooooo03gooooo000koooo
o`;o=BKo00?o0000ocDVoole9_l00_le9_oWol[oJ@03o`000?oooooooooo03cooooo000kooooo`03
ocDVool0003o=BKo00Co=BKoj?o:ofT01?l0003oooooooooooooool5o`0003Gooooo000kooooo`03
o`000?le9_oo=BKo00?o=BKoj_o:ofT:o`0003?ooooo000jooooo`03o`000?o:ofWob_mY00?o=BKo
j_o:ofT=o`00037ooooo000jooooo`03o`000?o:ofWob_mY0>gob_mY3Ol0000aooooo`00>Oooool0
0ol0003ob_mYol[oJ@3]ol[oJ@oo0000<?ooool003Sooooo00?o0000ol[oJOo:ofT0k_o:ofT?o`00
033ooooo000gooooo`03o`000?o:ofWob_mY0>kob_mY4Ol0000_ooooo`00=_ooool00ol0003ob_mY
ol[oJ@3_ol[oJA7o0000;oooool003Kooooo00?o0000ol[oJOo:ofT0koo:ofTAo`0002oooooo000]
ooooo`Go00000_ooool2o`000?;ob_mY4Ol0000_ooooo`00:oooool9o`000?Cob_mY4Ol0000_oooo
o`00:Oooool=o`000??ob_mY3ol0000`ooooo`00:Oooool=o`000??ob_mY3ol0000`ooooo`00:?oo
ool?o`000??ob_mY3Ol0000aooooo`00:?ooool?o`000??ob_mY3Ol0000aooooo`009ooooolAo`00
0?Cob_mY2Ol000000oo:ofWo0000ooooo`0`ooooo`009ooooolAo`000?Kob_mY1Ol00004ol[oJ@03
o`000?oooooooooo02kooooo000Woooooa7o0000ooo:ofT1ol[oJ@03o`000?oooooooooo02gooooo
000Woooooa7o0000ooo:ofT2ol[oJ@03o`000?oooooooooo02cooooo000Woooooa7o0000ooo:ofT2
ol[oJ@03o`000?oooooooooo02cooooo000Xooooo`oo0000ooo:ofT4ol[oJ@03o`000?oooooooooo
02_ooooo000Xooooo`oo0000ooo:ofT4ol[oJ@03o`000?oooooooooo02_ooooo000Yooooo`go0000
ooo:ofT6ol[oJ@03o`000?oooooooooo02[ooooo000Yooooo`go0000ooo:ofT7ol[oJ@03o`000?le
9_oo=BKo02Wooooo000Zooooo`[o0000ooo:ofT8ol[oJ@03ocDVool0003o=BKo00;o=BKo:?ooool0
02Wooooo00Co0000ol[oJOo:ofWob_mY1Ol0003ool[oJ@Wob_mY0ole9_l01?l0003o=BKoocDVoole
9_lWooooo`00:Oooool00ol0003ob_mYol[oJ@3ool[oJ@oob_mY0ole9_l01?l0003o=BKoocDVoole
9_lWooooo`00:?ooool00ol0003ob_mYol[oJ@3ool[oJA3ob_mY1?le9_l00ol0003o=BKoocDVo`0W
ooooo`00:?ooool00ol0003ob_mYol[oJ@3ool[oJA7ob_mY0ole9_l00ol0003o=BKoooooo`0Woooo
o`009oooool00ol0003ob_mYol[oJ@3ool[oJA?ob_mY0ole9_l00ol0003oooooooooo`0Vooooo`00
9_ooool00ol0003ob_mYol[oJ@3ool[oJASob_mY00?o0000ooooooooool09Oooool002Kooooo00?o
0000ol[oJOo:ofT0ooo:ofTHol[oJ@03o`000?oooooooooo02Gooooo000Rooooo`?o=BKo00?o0000
ol[oJOo:ofT0ooo:ofTJol[oJ@03o`000?oooooooooo02Cooooo000Qooooo`Co=BKo00?o0000ol[o
JOo:ofT0ooo:ofTJol[oJ@03o`000?oooooooooo02Cooooo000Pooooo`Co=BKo00?o0000ocDVoole
9_l0ooo:ofTLol[oJ@03o`000?oooooooooo02?ooooo000Pooooo`Co=BKo00?o0000ocDVoole9_l0
ooo:ofTLol[oJ@03o`000?oooooooooo02?ooooo000Pooooo`?o=BKo00Co0000ocDVoole9_oo=BKo
ooo:ofTMol[oJ@03o`000?oooooooooo02;ooooo000Qooooo`;o=BKo00?o0000ocDVoole9_l0ooo:
ofTNol[oJ@03o`000?oooooooooo02;ooooo000Rooooo`03o`000?le9_oo=BKo0?oob_mY8?o:ofT0
0ol0003oooooo`000004o`0001gooooo000Qooooo`03o`000?o:ofWob_mY0?oob_mY8Oo:ofT9o`00
01_ooooo000Qooooo`03o`000?o:ofWob_mY0?oob_mY7oo:ofT=o`0001Wooooo000Pooooo`03o`00
0?o:ofWob_mY0?oob_mY8?o:ofT=o`0001Wooooo000Pooooo`03o`000?o:ofWob_mY0?oob_mY7oo:
ofT?o`0001Sooooo000Oooooo`03o`000?o:ofWob_mY0?oob_mY8?o:ofT?o`0001Sooooo000Ooooo
o`03o`000?o:ofWob_mY0?oob_mY7oo:ofTAo`0001Oooooo000Nooooo`03o`000?o:ofWob_mY0?oo
b_mY8?o:ofTAo`0001Oooooo000Nooooo`03o`000?o:ofWob_mY0?oob_mY8?o:ofTAo`0001Oooooo
000Mooooo`03o`000?o:ofWob_mY0?oob_mY8Oo:ofTAo`0001Oooooo000Mooooo`03o`000?o:ofWo
b_mY0?oob_mY8Oo:ofTAo`0001Oooooo000Gooooo`Go000000?oooooo`000?o:ofT0ooo:ofTSol[o
J@oo00006?ooool001Gooooo2Ol0003ool[oJBCob_mY3ol0000Hooooo`004oooool=o`000?oob_mY
8oo:ofT=o`0001Wooooo000Cooooo`go0000ooo:ofTSol[oJ@go00006Oooool001;ooooo3ol0003o
ol[oJBCob_mY2Ol0000Kooooo`004_ooool?o`000?oob_mY9_o:ofT5o`000003ol[oJOl0003ooooo
01[ooooo000Aoooooa7o0000ooo:ofT[ol[oJ@03o`000?oooooooooo01Wooooo000Aoooooa7o0000
ooo:ofT[ol[oJ@03o`000?oooooooooo01Wooooo000Aoooooa7o0000ooo:ofT[ol[oJ@03o`000?oo
oooooooo01Wooooo000Aoooooa7o0000ooo:ofT/ol[oJ@03o`000?oooooooooo01Sooooo000Aoooo
oa7o0000ooo:ofT/ol[oJ@03o`000?oooooooooo01Sooooo000Booooo`oo0000ooo:ofT^ol[oJ@03
o`000?oooooooooo01Oooooo000Booooo`oo0000ooo:ofT^ol[oJ@03o`000?oooooooooo01Oooooo
000Cooooo`go0000ooo:ofT_ol[oJ@03o`000?oooooooooo01Oooooo000Cooooo`go0000ooo:ofT_
ol[oJ@03o`000?oooooooooo01Oooooo000Eooooo`Wo0000ooo:ofTbol[oJ@03o`000?le9_oo=BKo
01Kooooo000Gooooo`Go0000ooo:ofTcol[oJ@03ocDVool0003o=BKo00;o=BKo5Oooool001Kooooo
00?o0000ol[oJOo:ofT0ooo:ofTeol[oJ@?o=BKo00Co0000ocDVoole9_oo=BKo5?ooool001Kooooo
00?o0000ol[oJOo:ofT0ooo:ofTeol[oJ@?o=BKo00Co0000ocDVoole9_oo=BKo5?ooool001Kooooo
00?o0000ol[oJOo:ofT0ooo:ofTeol[oJ@?o=BKo00Co0000ocDVoole9_oo=BKo5?ooool001Kooooo
00?o0000ol[oJOo:ofT0ooo:ofTfol[oJ@;o=BKo00?o0000ocDVoole9_l05Oooool001Gooooo00?o
0000ol[oJOo:ofT0ooo:ofThol[oJ@;o=BKo00?o0000ooooooooool05?ooool001Gooooo00?o0000
ol[oJOo:ofT0ooo:ofTjol[oJ@03o`000?oooooooooo01Cooooo000Eooooo`03o`000?o:ofWob_mY
0?oob_mY>_o:ofT00ol0003oooooooooo`0Dooooo`005Oooool00ol0003ob_mYol[oJ@3ool[oJC[o
b_mY00?o0000ooooooooool05?ooool001Cooooo00?o0000ol[oJOo:ofT0ooo:ofTlol[oJ@03o`00
0?oooooooooo01?ooooo000Booooo`;o=BKo00?o0000ol[oJOo:ofT0ooo:ofTlol[oJ@03o`000?oo
oooooooo01?ooooo000Aooooo`?o=BKo00?o0000ocDVooo:ofT0ooo:ofTlol[oJ@03o`000?oooooo
oooo01?ooooo000@ooooo`Co=BKo00?o0000ocDVoole9_l0ooo:ofTlol[oJ@03o`000?oooooooooo
01?ooooo000@ooooo`?o=BKo00Co0000ocDVoole9_oo=BKoooo:ofTmol[oJ@03o`000?oooooooooo
01;ooooo000@ooooo`?o=BKo00Co0000ocDVoole9_oo=BKoooo:ofTmol[oJ@03o`000?oooooooooo
01;ooooo000Aooooo`;o=BKo00?o0000ocDVoole9_l0ooo:ofTnol[oJ@03o`000?oooooooooo01;o
oooo000Booooo`03ocDVool0003o=BKo0?oob_mY?oo:ofT00ol0003oooooooooo`0Booooo`004ooo
ool00ol0003ob_mYol[oJ@3ool[oJCkob_mY00?o0000ooooooooool04_ooool001?ooooo00?o0000
ol[oJOo:ofT0ooo:ofTnol[oJ@03o`000?oooooooooo01;ooooo000Booooo`03o`000?o:ofWob_mY
0?oob_mY@?o:ofT00ol0003oooooooooo`0Aooooo`004_ooool00ol0003ob_mYol[oJ@3ool[oJD3o
b_mY00?o0000ooooooooool04Oooool001;ooooo00?o0000ol[oJOo:ofT0ooo:ofU0ol[oJ@03o`00
0?le9_oo=BKo017ooooo000Booooo`03o`000?o:ofWob_mY0?oob_mY?oo:ofT00ole9_oo0000ocDV
o`02ocDVoa3ooooo000Booooo`03o`000?o:ofWob_mY0?oob_mY?_o:ofT2ocDVo`03o`000?le9_oo
=BKo00;o=BKo3oooool001;ooooo00?o0000ol[oJOo:ofT0ooo:ofTnol[oJ@;o=BKo00?o0000ocDV
oole9_l00_le9_l?ooooo`004Oooool00ol0003ob_mYol[oJ@3ool[oJCoob_mY0ole9_l01?l0003o
=BKoocDVoole9_l?ooooo`004Oooool00ol0003ob_mYol[oJ@3ool[oJD3ob_mY0_le9_l00ol0003o
=BKoocDVo`0@ooooo`004Oooool00ol0003ob_mYol[oJ@3ool[oJD7ob_mY00?o=BKoo`000?le9_l0
4Oooool000kooooo1Ol0003ool[oJD?ob_mY00?o0000ooooooooool04?ooool000cooooo2Ol0003o
ol[oJD7ob_mY00?o0000ooooooooool04?ooool000[ooooo3Ol0003ool[oJCoob_mY00?o0000oooo
ooooool04?ooool000[ooooo3Ol0003ool[oJCoob_mY00?o0000ooooooooool04?ooool000Wooooo
3ol0003ool[oJCkob_mY00?o0000ooooooooool04?ooool000Wooooo3ol0003ool[oJCoob_mY00?o
0000ooooooooool03oooool000Sooooo4Ol0003ool[oJCkob_mY00?o0000ooooooooool03oooool0
00Sooooo4Ol0003ool[oJCkob_mY00?o0000ooooooooool03oooool000Sooooo4Ol0003ool[oJCko
b_mY00?o0000ooooooooool03oooool000Sooooo4Ol0003ool[oJCkob_mY00?o0000ooooooooool0
3oooool000Sooooo4Ol0003ool[oJCkob_mY00?o0000ooooooooool03oooool000Wooooo3ol0003o
ol[oJCgob_mY1Ol0000?ooooo`002Oooool?o`000?oob_mY>oo:ofT9o`0000gooooo000:ooooo`go
0000ooo:ofTjol[oJ@go00002oooool000[ooooo3Ol0003ool[oJC[ob_mY3Ol0000;ooooo`003?oo
ool9o`000?oob_mY>oo:ofT?o`0000[ooooo000>ooooo`Go0000ooo:ofTmol[oJ@oo00002_ooool0
013ooooo00?o0000ol[oJOo:ofT0ooo:ofTlol[oJA7o00002Oooool0013ooooo00?o0000ol[oJOo:
ofT0ooo:ofTlol[oJA7o00002Oooool0013ooooo00?o0000ol[oJOo:ofT0ooo:ofTlol[oJA7o0000
2Oooool0013ooooo00?o0000ol[oJOo:ofT0ooo:ofTlol[oJA7o00002Oooool0013ooooo00?o0000
ol[oJOo:ofT0ooo:ofTlol[oJA7o00002Oooool0013ooooo00?o0000ol[oJOo:ofT0ooo:ofTmol[o
J@oo00002_ooool0013ooooo00?o0000ol[oJOo:ofT0ooo:ofTmol[oJ@oo00002_ooool0013ooooo
00?o0000ol[oJOo:ofT0ooo:ofTnol[oJ@go00002oooool0013ooooo00?o0000ol[oJOo:ofT0ooo:
ofTnol[oJ@go00002oooool0013ooooo00?o0000ol[oJOo:ofT0ooo:ofU0ol[oJ@Wo00003Oooool0
00oooooo00?o=BKoo`000?le9_l0ooo:ofU3ol[oJ@Go00003oooool000kooooo0_le9_l00ol0003o
=BKoocDVo`3ool[oJDCob_mY00?o0000ooooooooool03oooool000gooooo0ole9_l01?l0003o=BKo
ocDVoole9_oool[oJD?ob_mY00?o0000ooooooooool03oooool000gooooo0ole9_l01?l0003o=BKo
ocDVoole9_oool[oJD?ob_mY00?o0000ooooooooool03oooool000gooooo0ole9_l01?l0003o=BKo
ocDVoole9_oool[oJD?ob_mY00?o0000ooooooooool03oooool000kooooo0_le9_l00ol0003o=BKo
ocDVo`3ool[oJDCob_mY00?o0000ooooooooool03oooool000oooooo00?o=BKoo`000?le9_l0ooo:
ofU5ol[oJ@03o`000?oooooooooo00oooooo000@ooooo`03o`000?o:ofWob_mY0?oob_mYA?o:ofT0
0ol0003oooooooooo`0?ooooo`004Oooool00ol0003ob_mYol[oJ@3ool[oJD;ob_mY00?o0000oooo
ooooool04?ooool0017ooooo00?o0000ol[oJOo:ofT0ooo:ofU2ol[oJ@03o`000?oooooooooo013o
oooo000Aooooo`03o`000?o:ofWob_mY0?oob_mY@_o:ofT00ol0003oooooooooo`0@ooooo`004Ooo
ool00ol0003ob_mYol[oJ@3ool[oJD;ob_mY00?o0000ooooooooool04?ooool0017ooooo00?o0000
ol[oJOo:ofT0ooo:ofU1ol[oJ@03ocDVool0003o=BKo017ooooo000Aooooo`03o`000?o:ofWob_mY
0?oob_mY@?o:ofT2ocDVo`03o`000?le9_oo=BKo013ooooo000Aooooo`03o`000?o:ofWob_mY0?oo
b_mY?oo:ofT3ocDVo`04o`000?le9_oo=BKoocDVo`oooooo000Aooooo`03o`000?o:ofWob_mY0?oo
b_mY?oo:ofT3ocDVo`04o`000?le9_oo=BKoocDVo`oooooo000Booooo`03o`000?o:ofWob_mY0?oo
b_mY?_o:ofT2ocDVo`03o`000?le9_oo=BKo00;o=BKo3oooool001;ooooo00?o0000ol[oJOo:ofT0
ooo:ofTool[oJ@03ocDVool0003o=BKo00;o=BKo4?ooool0017ooooo1Ol0003ool[oJCoob_mY00?o
0000ocDVoole9_l04Oooool000oooooo2Ol0003ool[oJCgob_mY00?o0000ooooooooool04Oooool0
00gooooo3Ol0003ool[oJC_ob_mY00?o0000ooooooooool04Oooool000gooooo3Ol0003ool[oJC_o
b_mY00?o0000ooooooooool04Oooool000cooooo3ol0003ool[oJCWob_mY00?o0000ooooooooool0
4_ooool000cooooo3ol0003ool[oJCWob_mY00?o0000ooooooooool04_ooool000_ooooo4Ol0003o
ol[oJCSob_mY00?o0000ooooooooool04_ooool000_ooooo4Ol0003ool[oJCSob_mY00?o0000oooo
ooooool04_ooool000_ooooo4Ol0003ool[oJCSob_mY00?o0000ooooooooool04_ooool000_ooooo
4Ol0003ool[oJCSob_mY00?o0000ooooooooool04_ooool000_ooooo4Ol0003ool[oJCOob_mY00?o
0000ooooooooool04oooool000cooooo3ol0003ool[oJC?ob_mY1_l0000Eooooo`003?ooool?o`00
0?oob_mY<Oo:ofT9o`0001Cooooo000=ooooo`go0000ooo:ofT`ol[oJ@go00004_ooool000gooooo
3Ol0003ool[oJC3ob_mY3Ol0000Booooo`003oooool9o`000?oob_mY<Oo:ofT?o`00017ooooo000A
ooooo`Go0000ooo:ofTcol[oJ@oo00004Oooool001Gooooo00?o0000ol[oJOo:ofT0ooo:ofT`ol[o
JA7o00004?ooool001Kooooo00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJA7o00004?ooool001Kooooo
00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJA7o00004?ooool001Kooooo00?o0000ol[oJOo:ofT0ooo:
ofT_ol[oJA7o00004?ooool001Kooooo00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJA7o00004?ooool0
01Oooooo00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJ@oo00004Oooool001Oooooo00?o0000ol[oJOo:
ofT0ooo:ofT_ol[oJ@oo00004Oooool001Sooooo00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJ@go0000
4_ooool001Sooooo00?o0000ol[oJOo:ofT0ooo:ofT_ol[oJ@go00004_ooool001Sooooo00?o0000
ol[oJOo:ofT0ooo:ofTaol[oJ@Wo00005?ooool001Sooooo00Co0000ocDVoole9_oo=BKoooo:ofTb
ol[oJ@Go00005_ooool001Sooooo00?o=BKoo`000?le9_l00_le9_oool[oJC7ob_mY00?o0000oooo
ooooool06?ooool001Oooooo0_le9_l00ol0003o=BKoocDVo`02ocDVoooob_mY<?o:ofT00ol0003o
ooooooooo`0Hooooo`005oooool3ocDVo`04o`000?le9_oo=BKoocDVoooob_mY;oo:ofT00ol0003o
ooooooooo`0Iooooo`005oooool3ocDVo`04o`000?le9_oo=BKoocDVoooob_mY;oo:ofT00ol0003o
ooooooooo`0Iooooo`006?ooool2ocDVo`03o`000?le9_oo=BKo0?oob_mY<?o:ofT00ol0003ooooo
ooooo`0Iooooo`006Oooool00ole9_oo0000ocDVo`3ool[oJC7ob_mY00?o0000ooooooooool06Ooo
ool001_ooooo00?o0000ol[oJOo:ofT0ooo:ofT^ol[oJ@03o`000?oooooooooo01[ooooo000Loooo
o`03o`000?o:ofWob_mY0?oob_mY;Oo:ofT00ol0003oooooooooo`0Jooooo`007?ooool00ol0003o
b_mYol[oJ@3ool[oJBcob_mY00?o0000ooooooooool06oooool001gooooo00?o0000ol[oJOo:ofT0
ooo:ofTXol[oJ@;o=BKo00?o0000ooooooooool07?ooool001gooooo00?o0000ol[oJOo:ofT0ooo:
ofTWol[oJ@?o=BKo00?o0000ocDVooooool07?ooool001gooooo00?o0000ol[oJOo:ofT0ooo:ofTV
ol[oJ@Co=BKo00?o0000ocDVoole9_l07?ooool001gooooo00?o0000ol[oJOo:ofT0ooo:ofTVol[o
J@Co=BKo00?o0000ocDVoole9_l07?ooool001kooooo00?o0000ol[oJOo:ofT0ooo:ofTUol[oJ@?o
=BKo00Co0000ocDVoole9_oo=BKo7?ooool001kooooo00?o0000ol[oJOo:ofT0ooo:ofTVol[oJ@;o
=BKo00?o0000ocDVoole9_l07Oooool001oooooo1_l0003ool[oJB?ob_mY00?o0000ocDVoole9_l0
7_ooool001kooooo2Ol0003ool[oJB7ob_mY00?o0000ooooooooool07_ooool001cooooo3Ol0003o
ol[oJAkob_mY00?o0000ooooooooool07oooool001cooooo3Ol0003ool[oJAkob_mY00?o0000oooo
ooooool07oooool001_ooooo3ol0003ool[oJAcob_mY00?o0000ooooooooool08?ooool001_ooooo
3ol0003ool[oJA_ob_mY00?o0000ooooooooool08Oooool001[ooooo4Ol0003ool[oJA[ob_mY00?o
0000ooooooooool08Oooool001[ooooo4Ol0003ool[oJAWob_mY00?o0000ooooooooool08_ooool0
01[ooooo4Ol0003ool[oJAWob_mY00?o0000ooooooooool08_ooool001[ooooo4Ol0003ool[oJASo
b_mY00?o0000ooooooooool08oooool001[ooooo4Ol0003ool[oJA7ob_mY1Ol00002ol[oJ@03o`00
0?oooooooooo02?ooooo000Kooooo`oo0000ooo:ofT@ol[oJ@Wo00009_ooool001_ooooo3ol0003o
ol[oJ@kob_mY3Ol0000Tooooo`007?ooool=o`000?oob_mY3oo:ofT=o`0002Cooooo000Looooo`go
0000ooo:ofT>ol[oJ@oo00008oooool001kooooo2_l0003ool[oJ@oob_mY3ol0000Sooooo`008?oo
ool5o`0000?ooooo00?o0000ol[oJOo:ofT0ooo:ofT;ol[oJA7o00008_ooool002Sooooo00?o0000
ol[oJOo:ofT0ooo:ofT;ol[oJA7o00008_ooool002Wooooo00?o0000ol[oJOo:ofT0ooo:ofT:ol[o
JA7o00008_ooool002Wooooo00?o0000ol[oJOo:ofT0ooo:ofT:ol[oJA7o00008_ooool002[ooooo
00?o0000ol[oJOo:ofT0ooo:ofT9ol[oJA7o00008_ooool002_ooooo00?o0000ol[oJOo:ofT0ooo:
ofT9ol[oJ@oo00008oooool002_ooooo00?o0000ol[oJOo:ofT0ooo:ofT9ol[oJ@oo00008oooool0
02cooooo00?o0000ol[oJOo:ofT0ooo:ofT9ol[oJ@go00009?ooool002cooooo00?o0000ol[oJOle
9_l00_le9_oool[oJ@Oob_mY3Ol0000Tooooo`00;Oooool00ol0003o=BKoocDVo`02ocDVoooob_mY
1oo:ofT:o`0002Kooooo000/ooooo`;o=BKo00?o0000ocDVoole9_l00_le9_oool[oJ@Gob_mY00Co
0000oooooooooooooooo1Ol0000Xooooo`00;?ooool3ocDVo`04o`000?le9_oo=BKoocDVoooob_mY
1Oo:ofT00ol0003oooooooooo`0^ooooo`00;?ooool4ocDVo`03o`000?le9_oo=BKo0?oob_mY1?o:
ofT00ol0003oooooooooo`0_ooooo`00;Oooool3ocDVo`03o`000?le9_oob_mY0?oob_mY1?o:ofT0
0ol0003oooooooooo`0_ooooo`00;_ooool3ocDVo`03o`000?o:ofWob_mY0?oob_mY0_o:ofT00ol0
003oooooooooo`0`ooooo`00<_ooool00ol0003ob_mYol[oJ@3ool[oJ@03o`000?oooooooooo037o
oooo000booooo`03o`000?o:ofWob_mY0?oob_mY00?o0000ooooooooool0<Oooool003?ooooo0_l0
003jol[oJ@?o=BKo0_l0000dooooo`00=Oooool00ol0003ob_mYol[oJ@3fol[oJ@?o=BKo00?o0000
ocDVooooool0=?ooool003Gooooo00?o0000ol[oJOo:ofT0mOo:ofT4ocDVo`03o`000?le9_oo=BKo
03Cooooo000fooooo`03o`000?o:ofWob_mY0?Cob_mY0ole9_l01?l0003o=BKoocDVoole9_ldoooo
o`00=oooool01?l0003ob_mYol[oJOo:ofT5o`000>gob_mY0_le9_l00ol0003o=BKoocDVo`02ocDV
ocCooooo000hooooo`[o0000k?o:ofT00ol0003o=BKoocDVo`02ocDVocGooooo000gooooo`go0000
jOo:ofT00ol0003oooooocDVo`02ocDVocKooooo000gooooo`go0000jOo:ofT00ol0003ooooooooo
o`0hooooo`00=_ooool?o`000>Oob_mY00?o0000ooooooooool0>Oooool003Kooooo3ol0003Vol[o
J@03o`000?oooooooooo03[ooooo000eoooooa7o0000i?o:ofT00ol0003oooooooooo`0kooooo`00
=OoooolAo`000>?ob_mY00?o0000ooooooooool0??ooool003Gooooo4Ol0003Hol[oJ@Go00001Oo:
ofT00ol0003oooooooooo`0mooooo`00=OoooolAo`000=Kob_mY2Ol00002ol[oJ@03o`000?oooooo
oooo03kooooo000eoooooa7o0000e?o:ofT=o`00047ooooo000fooooo`oo0000eOo:ofT=o`00047o
oooo000fooooo`oo0000e?o:ofT?o`00043ooooo000gooooo`go0000eOo:ofT?o`00043ooooo000g
ooooo`ko0000doo:ofTAo`0003oooooo000iooooo`Wo00000oooool2o`000=7ob_mY4Ol0000ooooo
o`00>oooool5o`0000Oooooo00?o0000ol[oJOo:ofT0c_o:ofTAo`0003oooooo0018ooooo`03o`00
0?o:ofWob_mY0<gob_mY4Ol0000oooooo`00BOooool00ol0003ob_mYol[oJ@3<ol[oJA7o0000?ooo
ool004[ooooo0_l0003=ol[oJ@oo0000@?ooool004cooooo00Co0000ocDVoole9_oo=BKobOo:ofT?
o`00043ooooo001<ooooo`03o`000?le9_oo=BKo00;o=BKobOo:ofT=o`00047ooooo001;ooooo`;o
=BKo0_l00003ocDVolKob_mY3ol00011ooooo`00Boooool4ocDVo`03o`000?le9_oo=BKo0<Gob_mY
00?o0000ooooooooool00_ooool9o`0004?ooooo001;ooooo`Go=BKo0_l00033ol[oJ@;o00001ooo
ool5o`0004Gooooo001<ooooo`Go=BKo00?oooooo`000?l00000_oo:ofT2o`0005?ooooo001=oooo
o`?o=BKo1?ooool00ol0003ob_mYol[oJ@2fol[oJ@?o=BKo0_o:ofT00ol0003oooooooooo`1Coooo
o`00E?ooool00ol0003ob_mYol[oJ@2eol[oJ@Go=BKo00?ob_mYo`000?ooool0E?ooool005Gooooo
0_l00008ol[oJ@Go0000Yoo:ofT5ocDVo`;o0000E_ooool005Oooooo0_l00004ol[oJ@Wo0000YOo:
ofT3ocDVo`;o00000_le9_mFooooo`00FOooool00ol0003ob_mYo`00000<o`000:?ob_mY0_le9_l0
0ol0003o=BKoocDVo`02ocDVoeKooooo001Jooooo`ko0000Xoo:ofT2o`0000Co=BKoEoooool005[o
oooo3ol0002Pol[oJ@;o00000_ooool3ocDVoeSooooo001Jooooo`oo0000Soo:ofT5o`0000[ob_mY
0_l0001Oooooo`00FOoooolAo`0008cob_mY2Ol00007ol[oJ@03o`000?oooooooooo05oooooo001I
oooooa7o0000R_o:ofT=o`0000?ob_mY0_l0001Rooooo`00FOoooolAo`0008[ob_mY3Ol000000oo:
ofWo0000o`00001Tooooo`00FOoooolAo`0008Wob_mY3ol0001Vooooo`00FOoooolAo`0008Wob_mY
3ol0001Vooooo`00F_ooool@o`0008Sob_mY4Ol0001Uooooo`00F_ooool?o`000003ooooool0003o
000008Kob_mY4Ol0001Uooooo`00Foooool=o`0000Cooooo0_l00024ol[oJA7o0000IOooool005_o
oooo3Ol00006ooooo`;o00001Oo:ofT3ocDVog[ob_mY4Ol0001Uooooo`00GOooool9o`0000[ooooo
0_l00002ol[oJ@Go=BKoNOo:ofTAo`0006Gooooo001Oooooo`Go00003_ooool3o`0000Go=BKoN?o:
ofT@o`0006Kooooo001cooooo`;o=BKo0_l00003ocDVogKob_mY0_l000000ooooooo0000o`00000=
o`0006Kooooo001cooooo`Co=BKo0_l000000ole9_oob_mYol[oJ@1Xol[oJ@?o=BKo1oo:ofT2o`00
00Cooooo3Ol0001Wooooo`00M?ooool5ocDVo`Co0000D?o:ofT5o`00017ob_mY1Ole9_l2ol[oJ@Co
00001_ooool=o`0006Oooooo001eooooo`?o=BKo1Oooool2o`0004cob_mY2Ol0000>ol[oJ@Ko=BKo
0_l0000<ooooo`Wo0000JOooool007oooooo1?l00007ol[oJ@?o=BKo4oo:ofT5o`0002Cob_mY3Ol0
000<ol[oJ@;o=BKo1?l000000ole9_ooooooooooo`0=ooooo`Go0000Joooool008?ooooo0_l00004
ol[oJ@Go=BKo4?o:ofT9o`0002;ob_mY3Ol0000<ol[oJ@;o00001Ole9_moooooo`00QOooool4o`00
00Ko=BKo3Oo:ofT=o`0001oob_mY3ol00007ol[oJ@Co000000?oooooocDVoole9_l00ole9_n0oooo
o`00R?ooool00ole9_oo0000o`000002o`0000;o=BKo3Oo:ofT=o`0001oob_mY3ol00003ol[oJ@Co
00001_ooool3ocDVoh7ooooo0028ooooo`Go=BKo1?l0000:ol[oJ@oo00007Oo:ofTCo`0008kooooo
0029ooooo`Go=BKo0oooool6o`0000Cob_mY3ol0000>ol[oJ@?o=BKo3?o:ofTAo`00093ooooo002:
ooooo`?o=BKo2_oooolDo`0000cob_mY1Ole9_l;ol[oJA7o0000T?ooool009[ooooo4Ol0000;ol[o
J@Oo=BKo1Oo:ofTFo`00093ooooo002JooooobSo00001OoooolAo`00093ooooo002Joooooa7o0000
2oooool7ocDVo`_ooooo3ol0002Aooooo`00V_oooolAo`0000cooooo1Ole9_l<ooooo`oo0000TOoo
ool009_ooooo3ol0000>ooooo`?o=BKo3_ooool=o`0009;ooooo002Kooooo`oo00007oooool=o`00
09;ooooo002Looooo`go00008_ooool9o`0009Cooooo002Looooo`go00009?ooool5o`0009Kooooo
002Nooooo`Wo0000`Oooool00:3ooooo1Ol00033ooooo`00oooooomYooooo`00oooooomYooooo`00
oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00oooooomYooooo`00
oooooomYooooo`00\
\>"],
  ImageRangeCache->{{{0, 359}, {359, 0}} -> {-1.10001, -1.10001, 0.00766025, \
0.00766025}}],

Cell[CellGroupData[{

Cell[TextData[{
  "Lemma 1\nFor all composite ",
  Cell[BoxData[
      \(TraditionalForm\`c > 12\)]],
  " there exists a divisor ",
  Cell[BoxData[
      \(TraditionalForm\`k | c\)]],
  " and a prime number ",
  Cell[BoxData[
      \(TraditionalForm\`p, \ p < \(\(k\)\(-\)\(1\)\(\ \)\)\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`p\)]],
  " is not a factor of ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  "."
}], "Section"],

Cell[TextData[{
  "Note this is ",
  StyleBox["false",
    FontWeight->"Bold"],
  " for ",
  Cell[BoxData[
      \(TraditionalForm\`c = 12\)]],
  " : even for ",
  Cell[BoxData[
      \(TraditionalForm\`k = 6\)]],
  " all numbers < 5 have a common factor with ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  "."
}], "Text"],

Cell[TextData[{
  "Proof : it is known since Bertrand and Tchebitchiev (who proved this using \
17 lemmas) that there is always a prime number between ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`2  n\)]],
  " for ",
  Cell[BoxData[
      \(TraditionalForm\`n \[GreaterEqual] 2\)]],
  ".\nLet ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  " be the greatest divisor of ",
  Cell[BoxData[
      \(TraditionalForm\`c\ \((k \[NotEqual] c)\)\)]],
  ": we have either\n\[Bullet] ",
  Cell[BoxData[
      \(TraditionalForm\`k = 2  n + 1\)]],
  " or \n\[Bullet] ",
  Cell[BoxData[
      \(TraditionalForm\`k = 2  n + 2\)]],
  ". \nFor ",
  Cell[BoxData[
      \(TraditionalForm\`c > 24\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`c\)]],
  " composite we can claim ",
  Cell[BoxData[
      \(TraditionalForm\`k \[GreaterEqual] 5\)]],
  " and hence ",
  Cell[BoxData[
      \(TraditionalForm\`n \[GreaterEqual] 2\)]],
  " (see by hand for ",
  Cell[BoxData[
      \(TraditionalForm\`c = 14, \ 15, \ 16\)]],
  "\[Ellipsis]24). We take our prime ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(\(p\)\(\[Element]\)\)\(]\)\) n, 
      2 \( \(n\)\([\)\)\)]],
  " or more precisely ",
  Cell[BoxData[
      \(TraditionalForm\`p \[Element] \([n + 1, 2  n - 1]\)\)]],
  ": then ",
  Cell[BoxData[
      \(TraditionalForm\`k/2 < p < k - 1\)]],
  "."
}], "Exercise"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Lemma 2 : there is a ME of cardinality ",
  Cell[BoxData[
      \(TraditionalForm\`p\)]],
  " in ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalZ]\_k\)]],
  "."
}], "Section"],

Cell[TextData[{
  "Proof: this is well known, a way to produce it is to take ",
  Cell[BoxData[
      \(TraditionalForm\`{0, p, 2  p, \[Ellipsis]\ \((p - 1)\) p}\)]],
  " modulo ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  ". It is generated (G) and its cardinality ",
  Cell[BoxData[
      \(TraditionalForm\`p\)]],
  " being coprime with the modulo ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  " it is not a LTM (Limited Transposition Mode): hence it is class I."
}], "Exercise"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[{
  "Theorem :\nFor ",
  Cell[BoxData[
      \(TraditionalForm\`c > 12\)]],
  ", there exists a ME of class III in ",
  Cell[BoxData[
      \(TraditionalForm\`\[DoubleStruckCapitalZ]\_c\)]],
  ", with cardinality ",
  Cell[BoxData[
      \(TraditionalForm\`d = \ p\ c\/k\)]],
  "with notations as above. It is the set ",
  Cell[BoxData[
      \(TraditionalForm\`{\[LeftFloor]\(k\ n\)\/p\[RightFloor]\  | \ k = 
          0\ \[Ellipsis]\ d - 1}\)]],
  "."
}], "Section"],

Cell[CellGroupData[{

Cell[TextData[{
  "Proof: We produce this ME by double generation, concatenating ",
  Cell[BoxData[
      \(TraditionalForm\`c\/k\)]],
  " copies of the ME provided by the last Lemma. Another way to put it is to \
exhibit the generating function"
}], "Exercise"],

Cell[BoxData[
    \(TraditionalForm\`J : 
          n \[Rule] \[LeftFloor]\(n\ c\)\/\(\(c\ p\)\/k\)\[RightFloor] = \
\[LeftFloor]\(k\ n\)\/p\[RightFloor]\)], "DisplayFormula"]
}, Open  ]],

Cell[TextData[{
  "Clearly, the first ",
  Cell[BoxData[
      \(TraditionalForm\`p\)]],
  " images are spread between 0 and ",
  Cell[BoxData[
      \(TraditionalForm\`k - 1\)]],
  "; equally clearly, the set is ",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  " periodic as ",
  Cell[BoxData[
      \(TraditionalForm\`J(n + p)\  = \ J(n) + k\)]],
  ".\nThese generating functions provide ME (Clough & , thm 1.5).\nIt only \
remains to verify this is a class III set.\n\[MathematicaIcon] it has \
transpositional symmetry (",
  Cell[BoxData[
      \(TraditionalForm\`k\)]],
  "-periodic).\n\[MathematicaIcon] we must check that ",
  Cell[BoxData[
      \(TraditionalForm\`Gcd(c, d) \[NotEqual] c - d\)]],
  ". Notice ",
  Cell[BoxData[
      \(TraditionalForm\`\(\(d\)\(=\)\)\)]],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`\(c\/k\) \(\(p\)\(.\)\)\)]],
  " There are two cases to consider, as ",
  Cell[BoxData[
      \(TraditionalForm\`c\/k\)]],
  " obviously divides ",
  Cell[BoxData[
      \(TraditionalForm\`c\)]],
  ":\n\t\[SixPointedStar] the gcd is ",
  Cell[BoxData[
      \(TraditionalForm\`c\/k\)]],
  ". But \n\t",
  Cell[BoxData[
      \(TraditionalForm\`c\/k = \(\(c\)\(-\)\)\)]],
  Cell[BoxData[
      \(TraditionalForm\`\(c\/k\) p\)]],
  " means ",
  Cell[BoxData[
      \(TraditionalForm\`p = k - 1\)]],
  ", which is forbidden by construction.\n\t\[SixPointedStar] the gcd is ",
  Cell[BoxData[
      \(TraditionalForm\`\(c\/k\) p\)]],
  ". But \n\t",
  Cell[BoxData[
      \(TraditionalForm\`\(c\/k\) p = \(\(c\)\(-\)\)\)]],
  Cell[BoxData[
      \(TraditionalForm\`\(c\/k\) p\)]],
  " means ",
  Cell[BoxData[
      \(TraditionalForm\`p = k/2\)]],
  ", which is forbidden by construction too.\nThis ends the proof."
}], "Exercise"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Implementations", "Section"],

Cell[CellGroupData[{

Cell["Class 3", "Subsubsection"],

Cell[BoxData[{
    \(findPK[
        c_] := \[IndentingNewLine]Module[{k, p, 
          i = 1}, \[IndentingNewLine]k\  = \ 
          Max[Most[\ Divisors[c]]]; \[IndentingNewLine]While[
          Mod[k, \ p = Prime[i]] \[Equal] 
            0, \ \(i++\)]; \[IndentingNewLine]{k, 
          p}]\[IndentingNewLine]\), "\[IndentingNewLine]", 
    \(buildClassIIIME[
        c_] := \[IndentingNewLine]Module[{k, p, \ 
          d}, \[IndentingNewLine]{k, p} = 
          findPK[c]; \[IndentingNewLine]d\  = \ \(c\/k\) 
            p; \[IndentingNewLine]Table[Floor[n\ k/p], \ {n, 0, \ d - 1}] // 
          Sort]\)}], "Input"],

Cell[BoxData[
    \(horloge[set_, \ c_] := 
      Show[Graphics[{RGBColor[0.789502, \ 1, \ 
              0.413352], \[IndentingNewLine]Disk[{0, 0}, \ 
              1], \[IndentingNewLine]PointSize[
              0.02], \[IndentingNewLine]RGBColor[0.208896, \ 0.149676, \ 
              1], \[IndentingNewLine]Map[
              Point[{Sin[2\ \(\[Pi]\/c\) #], \ Cos[2\ \(\[Pi]\/c\) #]}] &, \ 
              Range[c]], \[IndentingNewLine]RGBColor[0.841627, \ 0.0914931, \ 
              0.0617227], 
            PointSize[0.035], \[IndentingNewLine]Circle[{0, 0}, 
              1], \[IndentingNewLine]Map[
              Point[{Sin[2\ \(\[Pi]\/c\) #], \ Cos[2\ \(\[Pi]\/c\) #]}] &, \ 
              set]}\ ], \ \[IndentingNewLine]Background \[Rule] \ 
          RGBColor[0.204913, \ 0.0961624, \ 
            0.280537], \[IndentingNewLine]AspectRatio \[Rule] 
          Automatic, \[IndentingNewLine]PlotLabel \[Rule] 
          StyleForm["\<modulo \>" <> 
              ToString[c], \[IndentingNewLine]FontFamily \[Rule] "\<Times\>", 
            FontSize \[Rule] 14, 
            FontWeight \[Rule] "\<Bold\>"]\[IndentingNewLine]]\)], "Input"],

Cell[BoxData[
    \(With[{r = 1.5}, 
      Line[{{r, r}, {r, \(-r\)}, {\(-r\), \(-r\)}, {\(-r\), r}, {r, 
            r}}]], \[IndentingNewLine]PlotRange \[Rule] {{\(-1.3\), \ 
          1.3}, \ {\(-1.3\), \ 1.3}}, \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(horloge[buildClassIIIME[#], \ #] &\)\  @@ \ 
        Select[Range[14, \ 60], Not[\ PrimeQ[#]] &];\)\)], "Input"],

Cell[GraphicsData["PostScript", "\<\
%!
%%Creator: Mathematica
%%AspectRatio: 1 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
%%IncludeResource: font Courier
%%IncludeFont: Courier
/Courier findfont 10  scalefont  setfont
% Background color
.205 .096 .281 r
MFill
% Scaling calculations
0.5 0.47619 0.5 0.47619 [
[.5 1.0125 -33.25 0 ]
[.5 1.0125 33.25 13.625 ]
[ 0 0 0 0 ]
[ 1 1 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
1 1 1 r
gsave
.5 1.0125 -94.25 -4 Mabsadd m
1 1 Mabs scale
currentpoint translate
0 21.625 translate 1 -1 scale
/g { setgray} bind def
/k { setcmykcolor} bind def
/p { gsave} bind def
/r { setrgbcolor} bind def
/w { setlinewidth} bind def
/C { curveto} bind def
/F { fill} bind def
/L { lineto} bind def
/rL { rlineto} bind def
/P { grestore} bind def
/s { stroke} bind def
/S { show} bind def
/N {currentpoint 3 -1 roll show moveto} bind def
/Msf { findfont exch scalefont [1 0 0 -1 0 0 ] makefont setfont} bind def
/m { moveto} bind def
/Mr { rmoveto} bind def
/Mx {currentpoint exch pop moveto} bind def
/My {currentpoint pop exch moveto} bind def
/X {0 rmoveto} bind def
/Y {0 exch rmoveto} bind def
63.000 14.625 moveto
%%IncludeResource: font Times-Bold
%%IncludeFont: Times-Bold
/Times-Bold findfont 14.000 scalefont
[1 0 0 -1 0 0 ] makefont setfont
1.000 1.000 1.000 setrgbcolor
0.000 0.000 rmoveto
63.000 14.625 moveto
%%IncludeResource: font Times-Bold
%%IncludeFont: Times-Bold
/Times-Bold findfont 14.000 scalefont
[1 0 0 -1 0 0 ] makefont setfont
1.000 1.000 1.000 setrgbcolor
(modulo) show
111.500 14.625 moveto
(45) show
125.500 14.625 moveto
%%IncludeResource: font Times-Bold
%%IncludeFont: Times-Bold
/Times-Bold findfont 14.000 scalefont
[1 0 0 -1 0 0 ] makefont setfont
1.000 1.000 1.000 setrgbcolor
0.000 0.000 rmoveto
1.000 setlinewidth
grestore
0 0 m
1 0 L
1 1 L
0 1 L
closepath
clip
newpath
.79 1 .413 r
.5 .5 m
.5 .5 .47619 0 365.73 arc
F
.209 .15 1 r
.02 w
.56627 .97156 Mdot
.63126 .95774 Mdot
.69368 .93502 Mdot
.75234 .90383 Mdot
.80609 .86478 Mdot
.85388 .81863 Mdot
.89478 .76628 Mdot
.928 .70875 Mdot
.95288 .64715 Mdot
.96896 .58269 Mdot
.9759 .51662 Mdot
.97358 .45022 Mdot
.96205 .3848 Mdot
.94152 .32162 Mdot
.91239 .2619 Mdot
.87524 .20683 Mdot
.83079 .15746 Mdot
.7799 .11475 Mdot
.72356 .07955 Mdot
.66287 .05253 Mdot
.59901 .03422 Mdot
.53322 .02497 Mdot
.46678 .02497 Mdot
.40099 .03422 Mdot
.33713 .05253 Mdot
.27644 .07955 Mdot
.2201 .11475 Mdot
.16921 .15746 Mdot
.12476 .20683 Mdot
.08761 .2619 Mdot
.05848 .32162 Mdot
.03795 .3848 Mdot
.02642 .45022 Mdot
.0241 .51662 Mdot
.03104 .58269 Mdot
.04712 .64715 Mdot
.072 .70875 Mdot
.10522 .76628 Mdot
.14612 .81863 Mdot
.19391 .86478 Mdot
.24766 .90383 Mdot
.30632 .93502 Mdot
.36874 .95774 Mdot
.43373 .97156 Mdot
.5 .97619 Mdot
.842 .091 .062 r
.5 Mabswid
[ ] 0 setdash
newpath
.5 .5 .47619 0 365.73 arc
s
.035 w
.5 .97619 Mdot
.63126 .95774 Mdot
.75234 .90383 Mdot
.85388 .81863 Mdot
.928 .70875 Mdot
.96896 .58269 Mdot
.97358 .45022 Mdot
.91239 .2619 Mdot
.83079 .15746 Mdot
.72356 .07955 Mdot
.59901 .03422 Mdot
.46678 .02497 Mdot
.33713 .05253 Mdot
.2201 .11475 Mdot
.08761 .2619 Mdot
.03795 .3848 Mdot
.0241 .51662 Mdot
.04712 .64715 Mdot
.10522 .76628 Mdot
.19391 .86478 Mdot
.30632 .93502 Mdot
% End of Graphics
MathPictureEnd
\
\>"], "Graphics",
  ImageSize->{288, 288},
  ImageMargins->{{0, 0}, {0, 0}},
  ImageRegion->{{0, 1}, {0, 1}},
  ImageCache->GraphicsData["Bitmap", "\<\
CF5dJ6E]HGAYHf4PAg9QL6QYHg<PAVmbKF5d0`40004P000182000`400?l00000o`00003ooooooolQ
ooooo`00oold64LPoc@HA`7ooooo003ooc@HAb3o=1Q70Oooool008Co=1Q71OoG5`nFoc@HA`7ooooo
0023oc@HA`OoeaL?UOld64L1ooooo`00P_ld64L9omLG3`co=1Q70ole9_n5oc@HA`7ooooo0022oc@H
A`WoeaL?2old64L5ocDVo`go=1Q71OoG5`mboc@HA`7ooooo0022oc@HAa[oeaL?2old64L7omLG3g7o
=1Q70Oooool007?o=1Q70ole9_l5oc@HAa3oeaL?2oo:ofT5ocDVo`03ol[oJOoG5`ooeaL?00KoeaL?
0_ld64L9omLG3g3o=1Q70Oooool007;o=1Q70ole9_l6omLG3`Oob_mY2OoG5`l<ol[oJ@?o=BKo2_o:
ofT;omLG3g3o=1Q70Oooool007;o=1Q70ooG5`l2ocDVo`cob_mY1ooG5`lLol[oJ@WoeaL?L?ld64L1
ooooo`00HOld64L5omLG3`Oo=1Q71OoG5`l5ocDVo`gob_mY1OoG5`lMol[oJ@coeaL?KOld64L1oooo
o`00H?ld64L7omLG3`?o=1Q70ooG5`l6ol[oJ@?o=BKo<?o:ofT9omLG3`?ob_mY0ooG5`mZoc@HA`7o
oooo001Ooc@HA`_oeaL??Oo:ofT7omLG3`Oob_mY0ooG5`l2oc@HA`?o=BKoH_ld64L1ooooo`00Gold
64L9omLG3d3ob_mY1OoG5`l;ol[oJ@?oeaL?0ole9_mQoc@HA`7ooooo001Ooc@HA`WoeaL?DOo:ofT2
ocDVo`?oeaL?HOld64L1ooooo`00Gold64L9omLG3e7ob_mY1Ole9_l3omLG3eko=1Q70Oooool005co
=1Q73?oG5`mBol[oJ@?o=BKo1?o:ofT3omLG3e_o=1Q70Oooool005Wo=1Q70ooG5`l4ol[oJ@OoeaL?
GOo:ofT2omLG3`?o=1Q71OoG5`mAoc@HA`7ooooo001Hoc@HA`03omLG3oo:ofWob_mY00Kob_mY1OoG
5`mPol[oJ@03omLG3old64OoeaL?00KoeaL?D?ld64L1ooooo`00DOld64L3ocDVo`;o=1Q70_oG5`m_
ol[oJ@WoeaL?Cold64L1ooooo`00D?ld64L4ocDVo`;oeaL?LOo:ofT9omLG3doo=1Q70Oooool0053o
=1Q70ole9_l00ooG5`oo=BKool[oJ@1aol[oJ@WoeaL?Cold64L1ooooo`00D?ld64L00ole9_ooeaL?
omLG3`02ocDVog;ob_mY2OoG5`m?oc@HA`7ooooo001>oc@HA`?oeaL?0ole9_mcol[oJ@[oeaL?C_ld
64L1ooooo`00COld64L00ooG5`oob_mYol[oJ@1hol[oJ@OoeaL?0_o:ofT00ooG5`oo=1Q7oc@HA`1;
oc@HA`7ooooo001;oc@HA`;oeaL?O?o:ofT5omLG3`Cob_mY0_oG5`m;oc@HA`7ooooo0019oc@HA`;o
eaL?ROo:ofT00ooG5`oo=1Q7oc@HA`18oc@HA`7ooooo0011oc@HA`GoeaL?0_ld64L00ooG5`oob_mY
ol[oJ@2:ol[oJ@;oeaL?B?ld64L1ooooo`00@?ld64L8omLG3hoob_mY0_oG5`m6oc@HA`7ooooo000o
oc@HA`WoeaL?TOo:ofT00ooG5`oo=BKoocDVo`13oc@HA`7ooooo000ooc@HA`WoeaL?T?o:ofT2ocDV
o`;oeaL?00?o=BKooc@HAold64L0@?ld64L1ooooo`00?old64L9omLG3i3ob_mY1?le9_l00ooG5`oo
=1Q7oc@HA`10oc@HA`7ooooo000ooc@HA`WoeaL?T?o:ofT5ocDVo`03omLG3old64Oo=1Q703oo=1Q7
0Oooool003oo=1Q72OoG5`nAol[oJ@?o=BKo0_o:ofT00ooG5`oo=1Q7oc@HA`0noc@HA`7ooooo000n
oc@HA`WoeaL?V?o:ofT2omLG3cko=1Q70Oooool003go=1Q700CoeaL?ol[oJOo:ofWob_mY1OoG5`nK
ol[oJ@03omLG3old64Oo=1Q703_o=1Q70Oooool003co=1Q700?oeaL?ol[oJOo:ofT0Xoo:ofT00ooG
5`oo=1Q7oc@HA`0joc@HA`7ooooo000koc@HA`03omLG3oo:ofWob_mY0:Gob_mY00?oeaL?oc@HAold
64L0>Old64L1ooooo`00>_ld64L00ooG5`oob_mYol[oJ@2Wol[oJ@03omLG3old64OoeaL?00CoeaL?
=?ld64L1ooooo`00>Old64L00ooG5`oob_mYol[oJ@2Yol[oJ@OoeaL?<old64L1ooooo`00=Old64L2
ocDVo`;oeaL?Zoo:ofT9omLG3c;o=1Q70Oooool003Co=1Q70_le9_l00ooG5`oo=BKoocDVo`2[ol[o
J@WoeaL?<_ld64L1ooooo`00=?ld64L00ole9_ooeaL?ocDVo`02ocDVoj_ob_mY2OoG5`lboc@HA`7o
oooo000doc@HA`03ocDVoooG5`oo=BKo00;o=BKoZoo:ofT9omLG3c;o=1Q70Oooool003Co=1Q700Co
eaL?ocDVoole9_oo=BKo[?o:ofT9omLG3c;o=1Q70Oooool003?o=1Q700?oeaL?ol[oJOo:ofT0[oo:
ofT7omLG3c?o=1Q70Oooool003;o=1Q700?oeaL?ol[oJOo:ofT0/Oo:ofT5omLG3`03ol[oJOoG5`oo
=1Q7037o=1Q70Oooool0037o=1Q700?oeaL?ol[oJOo:ofT0^Oo:ofT00ooG5`oo=1Q7oc@HA`0_oc@H
A`7ooooo000`oc@HA`03omLG3oo:ofWob_mY0;_ob_mY00?oeaL?oc@HAold64L0;_ld64L1ooooo`00
;old64L00ooG5`oob_mYol[oJ@2mol[oJ@03omLG3old64Oo=1Q702go=1Q70Oooool002oo=1Q700?o
eaL?ol[oJOo:ofT0__o:ofT00ooG5`oo=1Q7oc@HA`0/oc@HA`7ooooo000^oc@HA`03omLG3oo:ofWo
b_mY0<3ob_mY00?oeaL?oc@HAold64L0:old64L1ooooo`00;?ld64L2omLG3lCob_mY00?oeaL?oc@H
Aold64L0:_ld64L1ooooo`00:old64L00ooG5`oob_mYol[oJ@35ol[oJ@03omLG3old64Oo=1Q702Wo
=1Q70Oooool002Wo=1Q700?o=BKoomLG3ole9_l0aoo:ofT00ole9_ooeaL?ocDVo`0Yoc@HA`7ooooo
000Xoc@HA`;o=BKo00?oeaL?ocDVoole9_l0aOo:ofT2ocDVo`03omLG3ole9_oo=BKo02So=1Q70Ooo
ool002So=1Q700?o=BKoomLG3ole9_l00_le9_o5ol[oJ@?o=BKo00?oeaL?ocDVoold64L09old64L1
ooooo`00:?ld64L00ole9_ooeaL?ocDVo`02ocDVolGob_mY1?le9_l00ooG5`oo=1Q7oc@HA`0Voc@H
A`7ooooo000Xoc@HA`04omLG3ole9_oo=BKoocDVolOob_mY0ole9_l00ooG5`oo=1Q7oc@HA`0Voc@H
A`7ooooo000Woc@HA`03omLG3oo:ofWob_mY0<gob_mY00?oeaL?oc@HAold64L09Old64L1ooooo`00
9old64L00ooG5`oob_mYol[oJ@3=ol[oJ@03omLG3old64Oo=1Q702Go=1Q70Oooool002Ko=1Q700?o
eaL?ol[oJOo:ofT0coo:ofT00ooG5`oo=1Q7oc@HA`0Toc@HA`7ooooo000Uoc@HA`03omLG3oo:ofWo
b_mY0=7ob_mY00?oeaL?oc@HAold64L08old64L1ooooo`009Old64L00ooG5`oob_mYol[oJ@3Aol[o
J@03omLG3old64Oo=1Q702?o=1Q70Oooool002Co=1Q700?oeaL?ol[oJOo:ofT0doo:ofT00ooG5`oo
=1Q7oc@HA`0Roc@HA`7ooooo000Soc@HA`03omLG3oo:ofWob_mY0=Gob_mY00?oeaL?oc@HAold64L0
8Old64L1ooooo`007Old64L6omLG3mWob_mY1OoG5`lNoc@HA`7ooooo000Loc@HA`OoeaL?f?o:ofT7
omLG3ago=1Q70Oooool001_o=1Q72OoG5`oFol[oJ@WoeaL?7?ld64L1ooooo`006old64L9omLG3mKo
b_mY2OoG5`lLoc@HA`7ooooo000Koc@HA`WoeaL?e_o:ofT9omLG3aco=1Q70Oooool001_o=1Q72OoG
5`oFol[oJ@WoeaL?7?ld64L1ooooo`006old64L9omLG3mKob_mY2OoG5`lLoc@HA`7ooooo000Loc@H
A`OoeaL?f?o:ofT7omLG3ago=1Q70Oooool001go=1Q71OoG5`oJol[oJ@GoeaL?7_ld64L1ooooo`00
7Old64L00ooG5`oob_mYol[oJ@3Qol[oJ@03omLG3old64Oo=1Q701_o=1Q70Oooool001go=1Q700?o
eaL?ol[oJOo:ofT0hOo:ofT00ooG5`oo=1Q7oc@HA`0Koc@HA`7ooooo000Moc@HA`03omLG3oo:ofWo
b_mY0>7ob_mY00?oeaL?oc@HAold64L06old64L1ooooo`007?ld64L00ooG5`oob_mYol[oJ@3Sol[o
J@03omLG3old64Oo=1Q701[o=1Q70Oooool001_o=1Q700?oeaL?ol[oJOo:ofT0iOo:ofT00ooG5`oo
=1Q7oc@HA`0Ioc@HA`7ooooo000Koc@HA`03omLG3oo:ofWob_mY0>Gob_mY00?oeaL?oc@HAold64L0
6Old64L1ooooo`006_ld64L00ooG5`oob_mYol[oJ@3Wol[oJ@03omLG3old64Oo=1Q701So=1Q70Ooo
ool001[o=1Q700?oeaL?ol[oJOo:ofT0ioo:ofT00ooG5`oo=1Q7oc@HA`0Hoc@HA`7ooooo000Joc@H
A`03omLG3oo:ofWob_mY0>Oob_mY00?oeaL?oc@HAold64L06?ld64L1ooooo`006Old64L00ooG5`oo
b_mYol[oJ@3Yol[oJ@03omLG3old64Oo=1Q701Oo=1Q70Oooool001Oo=1Q70_le9_l00ooG5`oob_mY
ol[oJ@3Yol[oJ@03omLG3ole9_oo=BKo01Oo=1Q70Oooool001Ko=1Q70ole9_l00ooG5`oo=BKool[o
J@3Xol[oJ@03ocDVoooG5`oo=BKo00;o=BKo5_ld64L1ooooo`005_ld64L2ocDVo`03omLG3ole9_oo
=BKo0>Wob_mY0_le9_l00ooG5`oo=BKoocDVo`0Foc@HA`7ooooo000Foc@HA`03ocDVoooG5`oo=BKo
00;o=BKojOo:ofT3ocDVo`03omLG3ole9_oo=1Q701Go=1Q70Oooool001Oo=1Q700?oeaL?ocDVoole
9_l0joo:ofT2ocDVo`03omLG3old64Oo=1Q701Go=1Q70Oooool001Ko=1Q700?oeaL?ol[oJOo:ofT0
koo:ofT00ooG5`oo=1Q7oc@HA`0Doc@HA`7ooooo000Foc@HA`03omLG3oo:ofWob_mY0>oob_mY00?o
eaL?oc@HAold64L05?ld64L1ooooo`005_ld64L00ooG5`oob_mYol[oJ@3_ol[oJ@03omLG3old64Oo
=1Q701Co=1Q70Oooool001Go=1Q700?oeaL?ol[oJOo:ofT0l?o:ofT00ooG5`oo=1Q7oc@HA`0Doc@H
A`7ooooo000Eoc@HA`03omLG3oo:ofWob_mY0?7ob_mY00?oeaL?oc@HAold64L04old64L1ooooo`00
5Old64L00ooG5`oob_mYol[oJ@3aol[oJ@03omLG3old64Oo=1Q701?o=1Q70Oooool001Go=1Q700?o
eaL?ol[oJOo:ofT0lOo:ofT00ooG5`oo=1Q7oc@HA`0Coc@HA`7ooooo000Doc@HA`03omLG3oo:ofWo
b_mY0??ob_mY00?oeaL?oc@HAold64L04_ld64L1ooooo`005?ld64L00ooG5`oob_mYol[oJ@3col[o
J@03omLG3old64Oo=1Q701;o=1Q70Oooool001Co=1Q700?oeaL?ol[oJOo:ofT0loo:ofT00ooG5`oo
=1Q7oc@HA`0Boc@HA`7ooooo000@oc@HA`GoeaL?m_o:ofT00ooG5`oo=1Q7oc@HA`0Aoc@HA`7ooooo
000?oc@HA`OoeaL?mOo:ofT00ooG5`oo=1Q7oc@HA`0Aoc@HA`7ooooo000>oc@HA`WoeaL?m?o:ofT0
0ooG5`oo=BKoocDVo`0Aoc@HA`7ooooo000>oc@HA`WoeaL?loo:ofT2ocDVo`03omLG3ole9_oo=BKo
013o=1Q70Oooool000ko=1Q72OoG5`ocol[oJ@;o=BKo00?oeaL?ocDVoole9_l04?ld64L1ooooo`00
3_ld64L9omLG3o?ob_mY0_le9_l00ooG5`oo=BKoocDVo`0@oc@HA`7ooooo000>oc@HA`WoeaL?m?o:
ofT00ole9_ooeaL?ocDVo`0Aoc@HA`7ooooo000?oc@HA`OoeaL?moo:ofT00ooG5`oo=1Q7oc@HA`0?
oc@HA`7ooooo000@oc@HA`GoeaL?n?o:ofT00ooG5`oo=1Q7oc@HA`0?oc@HA`7ooooo000Aoc@HA`03
omLG3oo:ofWob_mY0?Wob_mY00?oeaL?oc@HAold64L03old64L1ooooo`004Old64L00ooG5`oob_mY
ol[oJ@3iol[oJ@03omLG3old64Oo=1Q700oo=1Q70Oooool0017o=1Q700?oeaL?ol[oJOo:ofT0nOo:
ofT00ooG5`oo=1Q7oc@HA`0?oc@HA`7ooooo000@oc@HA`03omLG3oo:ofWob_mY0?_ob_mY00?oeaL?
oc@HAold64L03_ld64L1ooooo`004?ld64L00ooG5`oob_mYol[oJ@3kol[oJ@03omLG3old64Oo=1Q7
00ko=1Q70Oooool0013o=1Q700?oeaL?ol[oJOo:ofT0noo:ofT00ooG5`oo=1Q7oc@HA`0>oc@HA`7o
oooo000@oc@HA`03omLG3oo:ofWob_mY0?_ob_mY00?oeaL?oc@HAold64L03_ld64L1ooooo`004?ld
64L00ooG5`oob_mYol[oJ@3kol[oJ@03omLG3old64Oo=1Q700ko=1Q70Oooool0013o=1Q700?oeaL?
ol[oJOo:ofT0n_o:ofT5omLG3`go=1Q70Oooool000oo=1Q700?oeaL?ol[oJOo:ofT0n_o:ofT7omLG
3`co=1Q70Oooool000ko=1Q700?o=BKoomLG3ole9_l0n_o:ofT9omLG3`_o=1Q70Oooool000go=1Q7
0_le9_l00ooG5`oo=BKoocDVo`3iol[oJ@WoeaL?2old64L1ooooo`003Old64L2ocDVo`03omLG3ole
9_oo=BKo0?Wob_mY2OoG5`l;oc@HA`7ooooo000=oc@HA`;o=BKo00?oeaL?ocDVoole9_l0nOo:ofT9
omLG3`_o=1Q70Oooool000ko=1Q700?o=BKoomLG3ole9_l0n_o:ofT9omLG3`_o=1Q70Oooool000oo
=1Q700?oeaL?ol[oJOo:ofT0n_o:ofT7omLG3`co=1Q70Oooool000oo=1Q700?oeaL?ol[oJOo:ofT0
noo:ofT5omLG3`go=1Q70Oooool000oo=1Q700?oeaL?ol[oJOo:ofT0oOo:ofT00ooG5`oo=1Q7oc@H
A`0=oc@HA`7ooooo000?oc@HA`03omLG3oo:ofWob_mY0?gob_mY00?oeaL?oc@HAold64L03Old64L1
ooooo`003old64L00ooG5`oob_mYol[oJ@3mol[oJ@03omLG3old64Oo=1Q700go=1Q70Oooool000oo
=1Q700?oeaL?ol[oJOo:ofT0oOo:ofT00ooG5`oo=1Q7oc@HA`0=oc@HA`7ooooo000?oc@HA`03omLG
3oo:ofWob_mY0?gob_mY00?oeaL?oc@HAold64L03Old64L1ooooo`003_ld64L00ooG5`oob_mYol[o
J@3ool[oJ@03omLG3old64Oo=1Q700co=1Q70Oooool000ko=1Q700?oeaL?ol[oJOo:ofT0ooo:ofT0
0ooG5`oo=1Q7oc@HA`0<oc@HA`7ooooo000>oc@HA`03omLG3oo:ofWob_mY0?oob_mY00?oeaL?oc@H
Aold64L03?ld64L1ooooo`003_ld64L00ooG5`oob_mYol[oJ@3ool[oJ@03omLG3old64Oo=1Q700co
=1Q70Oooool000co=1Q71OoG5`oool[oJ@03omLG3old64Oo=1Q700co=1Q70Oooool000_o=1Q71ooG
5`onol[oJ@03omLG3old64Oo=1Q700co=1Q70Oooool000[o=1Q72OoG5`okol[oJ@;o=BKo00?oeaL?
oc@HAold64L03?ld64L1ooooo`002_ld64L9omLG3o[ob_mY0ole9_l00ooG5`oo=BKooc@HA`0<oc@H
A`7ooooo000:oc@HA`WoeaL?n_o:ofT2ocDVo`03omLG3ole9_oo=BKo00go=1Q70Oooool000[o=1Q7
2OoG5`ojol[oJ@;o=BKo00?oeaL?ocDVoole9_l03Old64L1ooooo`002_ld64L9omLG3o_ob_mY00?o
=BKoomLG3ole9_l03_ld64L1ooooo`002old64L7omLG3ogob_mY00?oeaL?oc@HAold64L03Old64L1
ooooo`003?ld64L5omLG3okob_mY00?oeaL?oc@HAold64L03Old64L1ooooo`003old64L00ooG5`oo
b_mYol[oJ@3mol[oJ@03omLG3old64Oo=1Q700go=1Q70Oooool000oo=1Q700?oeaL?ol[oJOo:ofT0
oOo:ofT00ooG5`oo=1Q7oc@HA`0=oc@HA`7ooooo000?oc@HA`03omLG3oo:ofWob_mY0?gob_mY00?o
eaL?oc@HAold64L03Old64L1ooooo`003old64L00ooG5`oob_mYol[oJ@3mol[oJ@03omLG3old64Oo
=1Q700go=1Q70Oooool000oo=1Q700?oeaL?ol[oJOo:ofT0oOo:ofT00ooG5`oo=1Q7oc@HA`0=oc@H
A`7ooooo000?oc@HA`03omLG3oo:ofWob_mY0?gob_mY00?oeaL?oc@HAold64L03Old64L1ooooo`00
3old64L00ooG5`oob_mYol[oJ@3mol[oJ@03omLG3old64Oo=1Q700go=1Q70Oooool000oo=1Q700?o
eaL?ol[oJOo:ofT0oOo:ofT00ooG5`oo=1Q7oc@HA`0=oc@HA`7ooooo000@oc@HA`03omLG3oo:ofWo
b_mY0?_ob_mY00?oeaL?oc@HAold64L03_ld64L1ooooo`004?ld64L00ooG5`oob_mYol[oJ@3hol[o
J@GoeaL?3old64L1ooooo`004?ld64L00ooG5`oob_mYol[oJ@3gol[oJ@OoeaL?3_ld64L1ooooo`00
3old64L00ole9_ooeaL?ocDVo`3gol[oJ@WoeaL?3Old64L1ooooo`003_ld64L2ocDVo`03omLG3ole
9_oo=BKo0?Kob_mY2OoG5`l=oc@HA`7ooooo000>oc@HA`;o=BKo00?oeaL?ocDVoole9_l0m_o:ofT9
omLG3`go=1Q70Oooool000ko=1Q70ole9_l00ooG5`oo=BKool[oJ@3eol[oJ@WoeaL?3Old64L1oooo
o`003old64L2ocDVo`03omLG3oo:ofWob_mY0?Gob_mY2OoG5`l=oc@HA`7ooooo000Aoc@HA`03omLG
3oo:ofWob_mY0?Kob_mY1ooG5`l>oc@HA`7ooooo000Aoc@HA`03omLG3oo:ofWob_mY0?Oob_mY1OoG
5`l?oc@HA`7ooooo000Aoc@HA`03omLG3oo:ofWob_mY0?Wob_mY00?oeaL?oc@HAold64L03old64L1
ooooo`004_ld64L00ooG5`oob_mYol[oJ@3gol[oJ@03omLG3old64Oo=1Q7013o=1Q70Oooool001;o
=1Q700?oeaL?ol[oJOo:ofT0moo:ofT00ooG5`oo=1Q7oc@HA`0@oc@HA`7ooooo000Boc@HA`03omLG
3oo:ofWob_mY0?Oob_mY00?oeaL?oc@HAold64L04?ld64L1ooooo`004_ld64L00ooG5`oob_mYol[o
J@3gol[oJ@03omLG3old64Oo=1Q7013o=1Q70Oooool001;o=1Q700?oeaL?ol[oJOo:ofT0moo:ofT0
0ooG5`oo=1Q7oc@HA`0@oc@HA`7ooooo000Coc@HA`03omLG3oo:ofWob_mY0?Gob_mY00?oeaL?oc@H
Aold64L04Old64L1ooooo`004old64L00ooG5`oob_mYol[oJ@3eol[oJ@03omLG3old64Oo=1Q7017o
=1Q70Oooool001?o=1Q71OoG5`ocol[oJ@03omLG3old64Oo=1Q7017o=1Q70Oooool001;o=1Q71ooG
5`oaol[oJ@03omLG3old64Oo=1Q701;o=1Q70Oooool0017o=1Q72OoG5`o^ol[oJ@;o=BKo00?oeaL?
oc@HAold64L04_ld64L1ooooo`004Old64L9omLG3ngob_mY0ole9_l00ooG5`oo=BKooc@HA`0Boc@H
A`7ooooo000Aoc@HA`WoeaL?kOo:ofT2ocDVo`03omLG3ole9_oo=BKo01?o=1Q70Oooool0017o=1Q7
2OoG5`o]ol[oJ@;o=BKo00?oeaL?ocDVoole9_l04old64L1ooooo`004Old64L9omLG3nkob_mY00?o
=BKoomLG3ole9_l05?ld64L1ooooo`004_ld64L7omLG3noob_mY00?oeaL?oc@HAold64L05?ld64L1
ooooo`004old64L5omLG3o3ob_mY00?oeaL?oc@HAold64L05?ld64L1ooooo`005_ld64L00ooG5`oo
b_mYol[oJ@3_ol[oJ@03omLG3old64Oo=1Q701Co=1Q70Oooool001Oo=1Q700?oeaL?ol[oJOo:ofT0
kOo:ofT00ooG5`oo=1Q7oc@HA`0Eoc@HA`7ooooo000Goc@HA`03omLG3oo:ofWob_mY0>gob_mY00?o
eaL?oc@HAold64L05Old64L1ooooo`005old64L00ooG5`oob_mYol[oJ@3]ol[oJ@03omLG3old64Oo
=1Q701Go=1Q70Oooool001So=1Q700?oeaL?ol[oJOo:ofT0joo:ofT00ooG5`oo=1Q7oc@HA`0Foc@H
A`7ooooo000Ioc@HA`03omLG3oo:ofWob_mY0>Wob_mY00?oeaL?oc@HAold64L05old64L1ooooo`00
6Old64L00ooG5`oob_mYol[oJ@3Yol[oJ@03omLG3old64Oo=1Q701Oo=1Q70Oooool001[o=1Q700?o
eaL?ol[oJOo:ofT0ioo:ofT00ooG5`oo=1Q7oc@HA`0Hoc@HA`7ooooo000Joc@HA`03omLG3oo:ofWo
b_mY0>?ob_mY1OoG5`lJoc@HA`7ooooo000Joc@HA`03omLG3oo:ofWob_mY0>;ob_mY1ooG5`lIoc@H
A`7ooooo000Joc@HA`03ocDVoooG5`oo=BKo0>7ob_mY2OoG5`lHoc@HA`7ooooo000Ioc@HA`;o=BKo
00?oeaL?ocDVoole9_l0h?o:ofT9omLG3aSo=1Q70Oooool001Wo=1Q70_le9_l00ooG5`oo=BKoocDV
o`3Pol[oJ@WoeaL?6?ld64L1ooooo`006Old64L3ocDVo`03omLG3ole9_oob_mY0=oob_mY2OoG5`lH
oc@HA`7ooooo000Joc@HA`?o=BKo00?oeaL?ol[oJOo:ofT0g_o:ofT9omLG3aSo=1Q70Oooool001go
=1Q700?oeaL?ol[oJOo:ofT0goo:ofT7omLG3aWo=1Q70Oooool001ko=1Q700?oeaL?ol[oJOo:ofT0
goo:ofT5omLG3a[o=1Q70Oooool001ko=1Q700?oeaL?ol[oJOo:ofT0goo:ofT00ooG5`oo=1Q7oc@H
A`0Loc@HA`7ooooo000Noc@HA`03omLG3oo:ofWob_mY0=oob_mY00?oeaL?oc@HAold64L07?ld64L1
ooooo`007old64L00ooG5`oob_mYol[oJ@3Mol[oJ@03omLG3old64Oo=1Q701go=1Q70Oooool0023o
=1Q700?oeaL?ol[oJOo:ofT0foo:ofT00ooG5`oo=1Q7oc@HA`0Noc@HA`7ooooo000Poc@HA`03omLG
3oo:ofWob_mY0=_ob_mY00?oeaL?oc@HAold64L07_ld64L1ooooo`008Old64L00ooG5`oob_mYol[o
J@3Iol[oJ@03omLG3old64Oo=1Q701oo=1Q70Oooool0027o=1Q700?oeaL?ol[oJOo:ofT0fOo:ofT0
0ooG5`oo=1Q7oc@HA`0Ooc@HA`7ooooo000Roc@HA`GoeaL?eOo:ofT00ooG5`oo=1Q7oc@HA`0Poc@H
A`7ooooo000Qoc@HA`OoeaL?doo:ofT00ooG5`oo=1Q7oc@HA`0Qoc@HA`7ooooo000Poc@HA`WoeaL?
coo:ofT3ocDVo`03omLG3old64Oo=1Q7027o=1Q70Oooool0023o=1Q72OoG5`o>ol[oJ@?o=BKo00?o
eaL?ocDVoold64L08_ld64L1ooooo`008?ld64L9omLG3lkob_mY0_le9_l00ooG5`oo=BKoocDVo`0S
oc@HA`7ooooo000Poc@HA`WoeaL?c_o:ofT2ocDVo`03omLG3ole9_oo=BKo02?o=1Q70Oooool0023o
=1Q72OoG5`o?ol[oJ@03omLG3ole9_oo=BKo02Co=1Q70Oooool0027o=1Q71ooG5`o?ol[oJ@03omLG
3old64Oo=1Q702Go=1Q70Oooool002;o=1Q71OoG5`l00old64OoeaL?ol[oJ@3<ol[oJ@03omLG3old
64Oo=1Q702Ko=1Q70Oooool002Wo=1Q700?oeaL?ol[oJOo:ofT0bOo:ofT00ooG5`oo=1Q7oc@HA`0W
oc@HA`7ooooo000Yoc@HA`03omLG3oo:ofWob_mY0<Wob_mY00?oeaL?oc@HAold64L09old64L1oooo
o`00:_ld64L00ooG5`oob_mYol[oJ@37ol[oJ@03omLG3old64Oo=1Q702So=1Q70Oooool002[o=1Q7
0_oG5`o7ol[oJ@;oeaL?:_ld64L1ooooo`00;?ld64L00ooG5`oob_mYol[oJ@33ol[oJ@03omLG3old
64Oo=1Q702[o=1Q70Oooool002go=1Q700?oeaL?ol[oJOo:ofT0_?o:ofT6omLG3bgo=1Q70Oooool0
02go=1Q700?oeaL?ol[oJOo:ofT0^oo:ofT7omLG3bgo=1Q70Oooool002ko=1Q700?oeaL?ocDVoole
9_l0^Oo:ofT9omLG3bco=1Q70Oooool002go=1Q70_le9_l00ooG5`oo=BKoocDVo`2hol[oJ@WoeaL?
;?ld64L1ooooo`00;Old64L3ocDVo`03omLG3ole9_oob_mY0;Oob_mY2OoG5`l/oc@HA`7ooooo000]
oc@HA`Co=BKo00?oeaL?ol[oJOo:ofT0]_o:ofT9omLG3bco=1Q70Oooool002ko=1Q70ole9_l00old
64OoeaL?ol[oJ@2fol[oJ@WoeaL?;?ld64L1ooooo`00<old64L00ooG5`oob_mYol[oJ@2eol[oJ@Oo
eaL?;Old64L1ooooo`00<old64L00ooG5`oob_mYol[oJ@2eol[oJ@KoeaL?;_ld64L1ooooo`00=?ld
64L00ooG5`oob_mYol[oJ@2col[oJ@03omLG3old64Oo=1Q703;o=1Q70Oooool003Go=1Q70_oG5`na
ol[oJ@;oeaL?=Old64L1ooooo`00=old64L00ooG5`oob_mYol[oJ@2]ol[oJ@03omLG3old64Oo=1Q7
03Go=1Q70Oooool003So=1Q700?oeaL?ol[oJOoG5`l01?oG5`nWol[oJ@03omLG3old64Oo=1Q703Ko
=1Q70Oooool003Wo=1Q71ooG5`nUol[oJ@03omLG3old64Oo=1Q703Oo=1Q70Oooool003So=1Q72OoG
5`nOol[oJ@?o=BKo00?ob_mYomLG3old64L0>Old64L1ooooo`00>?ld64L9omLG3ikob_mY1?le9_l0
0ooG5`oo=1Q7oc@HA`0ioc@HA`7ooooo000hoc@HA`WoeaL?W_o:ofT2ocDVo`;oeaL?00?o=BKooc@H
Aold64L0>Old64L1ooooo`00>?ld64L9omLG3ikob_mY00?o=BKoomLG3ole9_l00_le9_lkoc@HA`7o
oooo000hoc@HA`WoeaL?W_o:ofT01?oG5`oo=BKoocDVoole9_lloc@HA`7ooooo000ioc@HA`SoeaL?
WOo:ofT00ooG5`oo=1Q7oc@HA`0noc@HA`7ooooo000joc@HA`GoeaL?0_ld64L2omLG3iWob_mY0_oG
5`m1oc@HA`7ooooo0013oc@HA`03omLG3oo:ofWob_mY09Gob_mY00?oeaL?oc@HAold64L0@Old64L1
ooooo`00A?ld64L2omLG3i?ob_mY0_oG5`m4oc@HA`7ooooo0016oc@HA`03omLG3oo:ofWob_mY08So
b_mY1OoG5`l2ol[oJ@03omLG3old64Oo=1Q704Co=1Q70Oooool004Oo=1Q70_oG5`n7ol[oJ@SoeaL?
Aold64L1ooooo`00BOld64L2omLG3`;o=BKoP_o:ofT9omLG3dOo=1Q70Oooool004Wo=1Q70_le9_l0
0ooG5`oo=BKoocDVo`21ol[oJ@WoeaL?Aold64L1ooooo`00BOld64L3ocDVo`;oeaL?POo:ofT9omLG
3dOo=1Q70Oooool004Wo=1Q71Ole9_l00ooG5`oob_mYol[oJ@1nol[oJ@WoeaL?Aold64L1ooooo`00
B_ld64L3ocDVo`;o=1Q70ooG5`mlol[oJ@[oeaL?Aold64L1ooooo`00D_ld64L00ooG5`oob_mYol[o
J@1gol[oJ@;oeaL?0_ld64L7omLG3dSo=1Q70Oooool005?o=1Q700?oeaL?ol[oJOo:ofT00oo:ofT5
omLG3fgob_mY00?oeaL?oc@HAold64L00old64L5omLG3dWo=1Q70Oooool005Co=1Q70ooG5`l00oo:
ofWoeaL?omLG3`05omLG3f[ob_mY0_oG5`mDoc@HA`7ooooo001Goc@HA`WoeaL?H_o:ofT3ocDVo`04
ol[oJOoG5`ooeaL?omLG3eKo=1Q70Oooool005Oo=1Q72OoG5`mQol[oJ@Co=BKo00?oeaL?oc@HAold
64L0Eold64L1ooooo`00Eold64L9omLG3f7ob_mY0_le9_l2omLG3`03ocDVoold64Oo=1Q705Oo=1Q7
0Oooool005Oo=1Q72OoG5`mPol[oJ@?oeaL?0ole9_mIoc@HA`7ooooo001Goc@HA`coeaL?COo:ofT5
omLG3`Sob_mY0ooG5`l2oc@HA`?o=BKoF_ld64L1ooooo`00F?ld64L7omLG3`Co=1Q70ooG5`m9ol[o
J@OoeaL?1?o:ofT3omLG3f;o=1Q70Oooool005Wo=1Q71OoG5`l8oc@HA`;oeaL?0_o:ofT3ocDVod7o
b_mY3?oG5`mUoc@HA`7ooooo001Xoc@HA`?oeaL?0ole9_m0ol[oJ@WoeaL?J?ld64L1ooooo`00JOld
64L2ocDVo`?oeaL?@?o:ofT9omLG3fSo=1Q70Oooool006Wo=1Q71Ole9_l4omLG3a_ob_mY1OoG5`lK
ol[oJ@[oeaL?J?ld64L1ooooo`00J_ld64L3ocDVo`Go=1Q71?oG5`l6ol[oJ@?o=BKo3Oo:ofT7omLG
3`gob_mY0ole9_l6ol[oJ@CoeaL?00?o=1Q7omLG3ooG5`l01ooG5`mXoc@HA`7ooooo001foc@HA`Ko
eaL?1?le9_l;ol[oJ@WoeaL?2oo:ofT4ocDVo`KoeaL?1_ld64L7omLG3fWo=1Q70Oooool007_o=1Q7
00?o=BKoomLG3ooG5`l01_oG5`l7ol[oJ@WoeaL?1oo:ofT8omLG3`03ocDVoold64Oo=1Q700[o=1Q7
1OoG5`mZoc@HA`7ooooo001koc@HA`Go=BKo1?ld64LGomLG3`Co=1Q71Ole9_mkoc@HA`7ooooo001l
oc@HA`?o=BKo3?ld64L9omLG3`co=1Q70ole9_mloc@HA`7ooooo002;oc@HA`WoeaL?Rold64L1oooo
o`00S?ld64L7omLG3hco=1Q70Oooool008go=1Q71OoG5`n=oc@HA`7ooooo003ooc@HAb3o=1Q70Ooo
ool00?oo=1Q78?ld64L1ooooo`00oold64LPoc@HA`7ooooo003ooc@HAb3o=1Q70Oooool00?oo=1Q7
8?ld64L1ooooo`00oold64LPoc@HA`7ooooo003ooc@HAb3o=1Q70Oooool00?oo=1Q78?ld64L1oooo
o`00oold64LPoc@HA`7ooooo001`oc@HA`Cooooo00Go=1Q7oooooooooooooooooc@HA`03ooooo`;o
=1Q70oooool4oc@HA`;ooooo00Co=1Q7oooooooooooooooo0_ld64L2ooooo`03oc@HAooooooooooo
00Gooooo0_ld64L3ooooo`Oo=1Q70_ooool2oc@HA`CoooooM?ld64L1ooooo`00LOld64L2ooooo`;o
=1Q70_ooool2oc@HA`;ooooo0_ld64L2ooooo`03oc@HAooooooooooo00;o=1Q70_ooool01?ld64Oo
ooooooooooooool2oc@HA`;ooooo00Co=1Q7oooooooooooooooo0_ld64L2ooooo`;o=1Q70_ooool0
0old64Ooooooooooo`06oc@HA`;ooooo0_ld64L2ooooo`;o=1Q700?ooooooc@HAold64L0LOld64L1
ooooo`00LOld64L2ooooo`;o=1Q70_ooool2oc@HA`;ooooo00?o=1Q7ooooooooool00old64L4oooo
o`?o=1Q70_ooool2oc@HA`;ooooo0_ld64L2ooooo`;o=1Q70_ooool00old64Ooooooooooo`03oc@H
A`;ooooo0_ld64L6ooooo`Ko=1Q700?ooooooc@HAold64L0L?ld64L1ooooo`00LOld64L2ooooo`;o
=1Q70_ooool2oc@HA`;ooooo00?o=1Q7ooooooooool00old64L4ooooo`?o=1Q70_ooool2oc@HA`;o
oooo0_ld64L2ooooo`;o=1Q70_ooool00old64Ooooooooooo`03oc@HA`;ooooo0_ld64L6ooooo`Ko
=1Q700?ooooooc@HAold64L0L?ld64L1ooooo`00LOld64L2ooooo`;o=1Q70_ooool2oc@HA`;ooooo
00?o=1Q7ooooooooool00old64L4ooooo`?o=1Q70_ooool2oc@HA`;ooooo0_ld64L2ooooo`;o=1Q7
0_ooool00old64Ooooooooooo`03oc@HA`;ooooo0_ld64L00ooooooo=1Q7oc@HA`02ooooo`Go=1Q7
0ooooomboc@HA`7ooooo001aoc@HA`?ooooo00Go=1Q7oooooooooooooooooc@HA`02ooooo`;o=1Q7
0_ooool00old64Ooooooooooo`02oc@HA`;ooooo00Co=1Q7oooooooooooooooo0_ld64L2ooooo`;o
=1Q70_ooool2oc@HA`;ooooo0_ld64L2ooooo`03oc@HAooooooooooo00Co=1Q700Cooooooc@HAooo
oooooooo0old64L4ooooog?o=1Q70Oooool0073o=1Q70oooool00old64Ooooooooooo`02oc@HA`;o
oooo1?ld64L3ooooo`Co=1Q70_ooool01?ld64Oooooooooooold64L3ooooo`04oc@HAooooooooooo
ooooo`;o=1Q70_ooool3oc@HA`?ooooo1_ld64L3ooooo`?o=1Q70ooooomdoc@HA`7ooooo0028oc@H
A`;ooooo2_ld64L2ooooo`co=1Q70oooool3oc@HA`03oooooold64Oo=1Q707Co=1Q70Oooool008So
=1Q70_ooool:oc@HA`;ooooo3Old64L2ooooo`Co=1Q71?oooomboc@HA`7ooooo0027oc@HA`?ooooo
2Old64L3ooooo`ko=1Q700?ooooooc@HAold64L00_ld64L4ooooog;o=1Q70Oooool00001\
\>"],
  ImageRangeCache->{{{0, 287}, {287, 0}} -> {-1.11612, -1.05001, 0.00777786, \
0.00777786}}]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Derivation of all ME", "Subsubsection"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Union[
      With[{c = 12}, \[IndentingNewLine]Table[
            Floor[n\ c/d], \ {d, c - 1}, \ {n, 0, \ d - 1}] // 
          Sort]]\)], "Input"],

Cell[BoxData[
    \({{0}, {0, 6}, {0, 4, 8}, {0, 3, 6, 9}, {0, 2, 4, 7, 9}, {0, 2, 4, 6, 8, 
        10}, {0, 1, 3, 5, 6, 8, 10}, {0, 1, 3, 4, 6, 7, 9, 10}, {0, 1, 2, 4, 
        5, 6, 8, 9, 10}, {0, 1, 2, 3, 4, 6, 7, 8, 9, 10}, {0, 1, 2, 3, 4, 5, 
        6, 7, 8, 9, 10}}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(horloge[#, \ 12] &\)\  /@ \ Rest@%;\)\)], "Input"],

Cell[BoxData[
    \(\(Show[GraphicsArray[Partition[%, \ 5]]];\)\)], "Input"],

Cell[TextData[{
  "Notice for instance that collection (0 1 6 7) verifies\n\t\t",
  Cell[BoxData[
      FormBox[
        RowBox[{\((0\ 6)\), "\[Subset]", 
          StyleBox[\((0\ 1\ 6\ 7)\),
            Background->RGBColor[0, 1, 1]], " ", "\[Subset]", 
          " ", \((0\ 1\ 3\ 4\ 6\ 7\ 9\ A)\)}], TraditionalForm]]],
  " \nwhich pretty well describes its position among the ME species."
}], "Text"]
}, Closed]],

Cell[CellGroupData[{

Cell["Derivation of all prototypes", "Subsubsection"],

Cell[BoxData[
    FormBox[
      RowBox[{\(prototype(c_, k_, d_)\), ":=", 
        RowBox[{"Sort", "[", 
          RowBox[{"Table", "[", 
            RowBox[{
              
              InterpretationBox[\(\((\(c\ n\)\/k + \[LeftFloor]n\/k\
\[RightFloor])\)\ mod\ c\),
                Mod[ 
                  Plus[ 
                    Times[ n, c, 
                      Power[ k, -1]], 
                    Floor[ 
                      Times[ n, 
                        Power[ k, -1]]]], c],
                Editable->False], ",", \({n, 0, d - 1}\)}], "]"}], "]"}]}], 
      TraditionalForm]], "Input"],

Cell[BoxData[
    \(\(horloge[prototype[12, 4, 6], \ 12];\)\)], "Input"],

Cell[BoxData[{
    \(\(SetOptions[Graphics, \ 
        DisplayFunction \[Rule] Identity];\)\ \), "\n", 
    \(\(Table[
        horloge[prototype[12, a, \ b], \ 12], \ {b, 2, 
          12}, \[IndentingNewLine]\t{a, 6}];\)\)}], "Input"],

Cell[BoxData[{
    \(\(Show[GraphicsArray[%], \ 
        DisplayFunction \[Rule] $DisplayFunction];\)\), "\n", 
    \(\(SetOptions[Graphics, \ 
        DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["Little movie", "Subsubsection"],

Cell[BoxData[
    \(\(\(Do[
      If\ [Not[PrimeQ[n]], \[IndentingNewLine]\t
        horloge[buildClassIIIME[n], \ n]], \ {n, 6, 45}]\)\(\ \)\)\)], "Input"]
}, Closed]]
}, Open  ]]
},
FrontEndVersion->"5.1 for Macintosh",
ScreenRectangle->{{0, 1440}, {0, 829}},
WindowSize->{801, 743},
WindowMargins->{{3, Automatic}, {Automatic, 4}},
Magnification->1,
StyleDefinitions -> "Classroom.nb"
]

(*******************************************************************
Cached data follows.  If you edit this Notebook file directly, not
using Mathematica, you must remove the line containing CacheID at
the top of  the file.  The cache data will then be recreated when
you save this file from within Mathematica.
*******************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1754, 51, 57, 0, 62, "Title"],
Cell[1814, 53, 521, 11, 100, "Author"],
Cell[2338, 66, 23639, 389, 300, 1942, 115, "GraphicsData", "PostScript", \
"Graphics"],

Cell[CellGroupData[{
Cell[26002, 459, 444, 17, 107, "Section"],
Cell[26449, 478, 328, 14, 28, "Text"],
Cell[26780, 494, 1424, 49, 169, "Exercise"]
}, Open  ]],

Cell[CellGroupData[{
Cell[28241, 548, 212, 8, 59, "Section"],
Cell[28456, 558, 497, 14, 64, "Exercise"]
}, Open  ]],

Cell[CellGroupData[{
Cell[28990, 577, 483, 15, 115, "Section"],

Cell[CellGroupData[{
Cell[29498, 596, 262, 6, 64, "Exercise"],
Cell[29763, 604, 175, 3, 54, "DisplayFormula"]
}, Open  ]],
Cell[29953, 610, 1761, 56, 232, "Exercise"]
}, Open  ]],

Cell[CellGroupData[{
Cell[31751, 671, 34, 0, 59, "Section"],

Cell[CellGroupData[{
Cell[31810, 675, 32, 0, 43, "Subsubsection"],
Cell[31845, 677, 622, 13, 229, "Input"],
Cell[32470, 692, 1145, 20, 273, "Input"],
Cell[33618, 714, 228, 4, 63, "Input"],

Cell[CellGroupData[{
Cell[33871, 722, 136, 2, 47, "Input"],
Cell[34010, 726, 22906, 416, 296, 3402, 171, "GraphicsData", "PostScript", \
"Graphics"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{
Cell[56965, 1148, 45, 0, 35, "Subsubsection"],

Cell[CellGroupData[{
Cell[57035, 1152, 167, 4, 63, "Input"],
Cell[57205, 1158, 287, 4, 79, "Output"]
}, Open  ]],
Cell[57507, 1165, 74, 1, 47, "Input"],
Cell[57584, 1168, 76, 1, 47, "Input"],
Cell[57663, 1171, 403, 9, 82, "Text"]
}, Closed]],

Cell[CellGroupData[{
Cell[58103, 1185, 53, 0, 35, "Subsubsection"],
Cell[58159, 1187, 608, 17, 61, "Input"],
Cell[58770, 1206, 72, 1, 47, "Input"],
Cell[58845, 1209, 235, 5, 79, "Input"],
Cell[59083, 1216, 210, 4, 63, "Input"]
}, Closed]],

Cell[CellGroupData[{
Cell[59330, 1225, 37, 0, 35, "Subsubsection"],
Cell[59370, 1227, 156, 3, 63, "Input"]
}, Closed]]
}, Open  ]]
}
]
*)



(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)