On certain algorithms in the practice of geometry and the theory of numbers

Pub . Ma . UAB Vol . 29 Ns 1 Ab il 1985 ON CERTAIN ALGORITHMS IN THE PRACTICEOF GEOMETRY ANDTHE THEORY OF NUMBERS 0 . In oduc ion Pe e Hil on and Jean Pede sen [4e demons a ed in 111 and 131 a sys ema ic me hod o olding a s aigh s ip o nape , by wha we called a pn-címaxc bold¿nq p coceduke,  o app oxima e, o any desi ed deg ee o accu acy, a egula con exs-gon and ce ain egula s a s-gons, p o ided ha  s E F, he se o 6o1d¡nq numbm .  He e is de inéd o be he se o all in ege s s o he o m 2x x 1 , whe e x > 1, y > 2 . 2_1 O cou se, suchnumbe s s a eodd . By in oducincT SecokLdaxq olds on he s ipo pape we showed how i is possible o app oxima e egula 2 k s-gons, whe e  s E F and  k > 1  (and we included, o he sake o comple eness, he  exac  cons uc ions o he egula  2 k -gons, k >2) . The only emaining numbe s > 3 a e hose o he o m 2 k a,  whe e a is odd,  : P 1  and  no  a olding numbe and  k > 0 . Howe e , he me hod o app oxima ing hose egula polygons can be desc ibed by a seguence o s eps as ollows (consul [ll o de ails) . Fi s , sincewe know ha , o any odd numbe a, 2(D(a) = 1 mod a, whe e i(a) is he Eule o ien unc ion, i ollows ha a is a ac o o some elemen o F, say s, wi h s = a£ . We canuse he p ima y olding p ocedu e o ob- ain a s ip o pape sui able o app oxima ing a egula s-gon . I we hen in oduce k' seconda y old lines a each poin ha would ha ebeen a e ex o he egula s-gon, we, canuse a longe s ip o his olded ape o cons uc a egu- la 2 k s-áon . We hen glue his 2 k s-gon o a pieceo pape and old on he lines connec ina e e y 9 h e ex o p oduce he desi ed 2 k a-gon . In [2] and [3], we in oduced an al- go i hm o inding he op imal sEF such ha als . In summa y, he abo e p ocedu es (using p ima y andse- conda y olds) p o ided us, in conjunc ion wi h he algo i hm e e ed o abo e, wi h a sys ema ic me hod ha could be used o app oxima e egula con ex s-gons o all s % 3 . The same p ocedu es p oduced many egula  S ax  s-gons, whe e  s E F .  In ac , as discussed and p o edin [2), o a gi en s = (x,y) E F,  he exac numbe o s a s-gons p oduced by he p ima y olding p ocedu e is 2 4'(y)xy . Fu he , hesecould be explici ly desc ibed . In [2] we aised he ques ion as o whe he by gene a- lizing in a na u al way he p ima y olding, we migh be able o a oid he gluings ep desc ibed abo e, and also be able o old a,Pl egula s a polygons . In his pape we answe ha ques ion, in he a i ma i e . Gi en  a,b odd wi h  a <2 and a p ime o b,  we des- c ibe in Sec ion 1 a genena í .zed p ima y olding p ocedu e which app oxima es a egula s a {a}-gon . The e a e, hen, e y ob iousseconda y p ocedu es whichallow us o emo e he es- ic ion ha bo h a and b be odd . Thegene aliza ion con- sis s in allowinga p ocedu e o a bi a y pe iodici y . The p ó- cedu es in p e ious pape s ha e all been o pe iod 1 o 2 . An in e es ing aspec o he con en o his pape , and he o he pape swe e e o, is he way he geome y mo i a es he numbe heo y, and he subsequen in e ac ion be ween he wo opics . Indeed, al houqh he  Quabí-Onde c Theonem  o Sec ion 2 would s and on i s own me i s as an in e es ingpieceo num- be heo y, i is ha d o imagine howone would ha e disco e ed i wi hou he geome ic mo i a ion . Mo eo e , al hough ou ge- ne alized p ima y olding p ocedu e ob ia es he need o glue a cons uc ed N-qon o a oieceo pape in o de o cons uc an M-gon, wi h MIN, he numbe heo y gene a ed by he gluing echnique, desc ibed in [2l and [3], s ands in i s own igh , and is in no sense supe seded by he mo e sophis ica ed pape - olding p ocedu eso his a icles, no subsumed in he numbe heo y ha a ises om hose mo e sophis ica ed p ocedu es . In Sec ion 1 we desc ibe he pace - oldingp ocedu e which enables us o cons uc a bi a y s a polygons . We ha e sough , by including his sec ion, o make he en i e pape ea sonablysel -con ained, hough we a eno ac ually ad oca ing he neglec o ou ea lie pape son his subjec . Sec ion 2 openswi h he de ini ion o a symbol bal a 2 k1k 2 a k which may be ega ded as encoding he ins uc ions o olding a s ino ape o o m a s a {á} -qon, wi h ai ,b odd, and i a l < 2 .  The "code" is desc ibed in a ypical case in Sec ion 1 and, in gene al, in Appendix 1 (Sec ion 4) . Howe e , his sym- bol also cons i u es an in e es ing algo i hm o de e mining he quad i-ohdelc o 2 mod b, ha is, he smalles posi i ein e ge  4  such ha  2 2 = l mod b .  Indeed,  i  a l is p ime o b, hen he quasi-o de is k = 1 k i and he pa i yo k  i=1 de e mines whe he 2 = 1 o 2k - -1 . O cou se, he quasi-o de , ein o ced wi h he in o ma ion p o ided by he pa i y o , p o idesmuchmo e in o ma ion han he o de o 2 mod b . Examples a e gi en in Appendix 2 (Sec ion 5) o show how o apply he algo i hm o ob ain he symbol (0 .1) and hen how, in a gi en case, o ob ain, om he symbol, he ac o complemen a y o b in 2 k ± 1 . In Sec ion 2 we desc ibe he symbols, p o e some basic D ope ies, and enuncia e he Quasi-O de Theo em . The heo em is p o ed in Sec ion 3, whe e we also ob ain some e inemen s o he heo em o u he numbe - heo e ical in e es . We e- ma k ha an independen p oo o he Quasi-O de Theo em was shown o us by Ge ald P es on . This p oo was based on he no- ion o Hasse unc ions (see, o example, [41) ; howe e , he di ec ion o p oo does no ake us h ough Theo em 2 .5, which has an immedia e applica ion o pape - olding . The pape closeswi h he wo appendices al eady e e- ed o ; in he i s we go back o he geome ical signi ican- ce o he symbols,and, in he second, we discuss, as examples, Fe ma and Me senne non-p imes . A ea u e o he ea lie pape s [2] and [31 missing om he p esen pape was he aene aliza ion om 'base 2' -- he onlybase o geome ical in e es , since we modes ly con ine ou sel es o  b .ihec :ng  angles -- o 'base ' ,  whe e  is an a bi a y posi i ein ege * 1 . I appea s ha his gene aliza ion leads o in e es ing di icul ieswhenwe y o in oduce he analogs o ou symbols in base , since, in his gene al con ex , hey may ail o exis o a gi en b . We p opose o de o e a sequel [61 o he s udy o gene alized symbols and he (gene alized) quasi-o de p oblem . 1 . How o old egula s a polycTons Fi s we suppose ha app op ia e bold, o cAe"e, lines ha e beenmade on ou s aigh s ipo pape and we des- c ibe he ac ual cons uc ion n ocess o oldinga {a}-gon l , whe e a and b a e mu uallyp imein ege swi h a<b . SuoDose, as illus a ed in Figu e 1, ha we ha e a s aigh s ip o pape ha has c easesalongs aigh lines emana ing omma ked e ices Ai,i=0,1, . . ., a he opand bo om ed- ges, and ha , o a ixed k, hose a he pa icula e ices A nk, n=0,1,2, . . . . b,  which a e on he op edge,  o miden ical angles b . Suppose u he ha hese e ices a e equally spaced (we desc ibebelow howyou migh ob ainsuch a s ip) . Figu e 1 (a) shows he beginning o he s ip . I we old his s ipon AnkAnk+2 (as shown in Figu e 1(b)) and hen on AnkAnk+l  (as shown in Figu e 1(c)),  he di ec ion o he op edge o he apewillbe o a ed h ough an angleo 2(b i ) and he ane will be o ien ed he same way, wi h espec o he cen e o he polygon being delinea ed by i s op edge . We call hese wo olds h ough A nk, in ha o de , a 2(b )- iuíA a A nkl and obse e ha , i a 2(b )- wis is pe o med a A nk o n = 0, 1, 2, . . ., b-1, he op edgeo he ape will ha e u ned h ough an angle o 2aw and he poin A bk will hen be coinciden wi h A o . Thus he op edge o he ape will ha e isi ed e e y a h e ex o a bounding egula con- ex b-gon, and hence de e mines a egula s a la}-gon . 1 A closedsequenceo b edges ha isi , in o de , e e y a h e ex (mod b) o a bounding egula con ex b-gon . We include he egula con ex  b-gon as he special case  a =1 . 36 Fígu e i A k l~ a, A ik+) . We now explain how we ob ained he desi ed c ease lines in he s ip o ape in he i s place . Recall ha we a e seeking o cons uc a s a {e}-gonwhe e a, b a e mu ually p imeposi i e in ege s wi h  a < 2 .  We assume i s ha a, b a e odd . Thuswe wish o ha e a s ip o pape on which he angle b u appea s a egula in e als along he op edge . We designa e he di ec ion om le o igh as he 4oAmAd di ec ion on he ape . We beginby ma kinga poin A o on he op o he ape and making an .íní íal c ease line going in he downwa d o wa d di ec ion om A o o' A 1 a he bo om o ape, and abz(une ha he angle i makes wi h he op edge is a  we call his he  pu a í e angle . The we con inue o b 3 7 o m new c easelines acco ding o he ollowing ou ules : (1) The i s new c ease lineemana es om he e ex A 1 . (2) Each new c ease line goes in he o wa d di ec ion along he s ip o pape . (3) Each new c ease line always bí6ec 6 he angle be - ween he las c ease line and he edge o he ape om which i emana es . (4) The bisec ion o angles a any e ex con inues un il a c ease line p oduces a pu a i eangle o he o m b  whe e a' is an odd numbe ; hen he olding s ops a ha e ex and commences a he in e sec ion poin o ha las c ease linewi h he o he side o he ape . Le us conside he exampleb = 11, a = 3 . Then we cansee ha i we begin wi h an angle o 11 a Ao (as shown in Figu e 2(a)) and adhe e o he abo e uleswe will ob ain a s ipo apewi h he angles and c eases (do ed li- nes) indica ed in Figu e 2(b) . Adhe ing o he no a ion o he p ima y oldingp ocedu es in [11, [21 and [31, we could w i e his mo e gene alized olding p ocedu e as As be o e, his no a ion means ha i we begin olding on he s ip o pape a he placewhe e he e is one c ease line slo- ping upwaAdb hen he i s d l e e s o he one bisec ion (p oducing a line in a downwa d di ec ion) a A l0n ( o an = 0,1,2, . . .) on he op o he ape ; he u 3 e e s o 3 8 {d 1 u 3 d 1u1 d 3 u 1 }  .  (1 .1) he 3 bisec ions (p oducing c eases in an upwa d di ec ion) ma- de a he bo om o he ape h ough AlOn+l ; e c . Howe e , he oldingp ocess is duplíca ed hal way h ough, so i su ices o w i e jus he i s h ee exponen s in (1 .1) . In ac , we can deno e (1 .1) e enmo e simply as {1,3,1}  (1 .2) wi h he unde s andina ha we old dk luk2 d k3 u k4 . . . wi h he k i , k 2 , k 3 , . . . cycling, in o de , epea edly h ough he alues 1, 3, 1, . . . We call (1 .1) o (1 .2) a oA pe ú .od 3 . No e ha , in his p ocedu eswe ha e hi he o conside ed o pe iod 1 ({d n u n }) _a b p ima y oldingp ocedu e e minology, he p ima y olding in 11, 2, 3] we e all o pe iod 2  (Id m u n }, m * n) . I is easy o see ha , s a ing wi h any pu a i e angle a < z),  we will alwaysob ain (a, b odd, mu ually p ime, by ou ules a p ima y oldina 'p oduces' his pu a i e angle a i e angle 11 angle 11 n a indeed, ou c ease lines could ha ebeen used o old a s a 11 { 3} -gon, heycould also ha e been used o old a 11-gon and a s a  { 5}-L}This ea u e o ou wi h i s c easelines ob iously applies in gene al : o he b-gons will be a ailable o us om he ape yielding he p ocedu e k 1 ,k 2 . . .,k which angle . We alsono e ha , s a ingwi h he 11 n a he op o he ape, we p oduced a pu- i a he bo on o he ape, hen a pu a i e he op o he ape, and so on . Thus i , con ex ape u nished s a s a a < 2,  he e is always a comple ely de e mined unique symbol like he one abo e (we do no need a,b ela i ely p ime) . App op ia ely in e p e ed, we can use his symbol o ead o he oldingp ocedu e ha p oduces he angle o  a along b he op edaeo he ape, so ha a symbol such (1 .3) encodes a olding p ocedu e o p oducinga s a {á}-gon, and also ells us wha o he s a polygons we can ob ain om he same ape (o cou se, o each symbol a diag am simila o Figu e 6 can be d awn o illus a e he ela i e posi ions o he angles an ) b Be o ewe close hissec ionwe would like o poin ou ha he oldingp ocess desc ibed abo e is he mob e6bící .en one possible . Tha is, he e could no be any olding p ocedu e ia^ s a oY h .i .s ype ha wouiü p ocu e .11G cÑ, +~- , ' ed a u u  o iy 7 on s wi_ h - . . . . . . ewe olds . I is also op imal om he poin o iew o "di - icul yo execu ion", o i keeAs he numbe o bisec ionsa each e ex o a minimum . These las commen s a e explained as ollows . I he olding p ocedu e {kl,k2# . . .0k } p oduces he angle  , hen (see (2 .3) and (2 .4) bl 2 k ±1, whe e b k = E k . I we adop he p ocedu es desc ibed in his sec- i=1 i ion we willha e a p ocedu e  {£l, 92 ,.... Qs}  such ha s R =  E  Q .  is he  smaUu  numbe  m  such ha  bl 2 m ±l, j=1 ha is, he quo-6í-ondeA o 2 mod b . Mo eo e , willbe a mul ipleo . s and, sui ably cycling he Qj , each k i is a mul iple o 2 1 . 46 All hese ac s a e con ained in henumbe - heo e ical esul s o he nex wo sec ions . 2 . S - ~ n bols a nd he quasi-o de o 2 mod b By he symbol b a l a 2 . . . a k l k 2 . . . k b =  a l +  2 k ia i+l ,  i =  1, 2,  . we unde s and ha b is an odd posi i e in ege , ha a l is an odd posi i e in ege < 2, i = 1,2, . . ., ,  and ha k l ,k21 . . .k a e posi i e in ege s such ha ' ,  a +1  =  a l .  (2 .2) Le us aa ee whe e con enien , o de ine al o all in ege s i by making a l pe iodic in i, wi h pe iod , and simila - ly o k i . We no e ha , gi en odd posi i ein ege s a, b wi h  a < 2,  he e is always a symbol (2 .1) wi h  a l = a,  and ha he symbol is uniqueup o .í ~on ; he ewe say ha (2 .1) a ises by i e a ion i he e exis s si such ha a l+s = a i' ki+s =k i ' o all i . A p ope i e a ion, ha is, one in which s ~ ,  is called a  hepe c don . Gi en b,kl, . . .,k , he equa ions (2 .2) ha euniquesolu- ions, in he "unknowns" a ., namely i Ba i = bA i , i = 1, 2, . . ., ,  (2 .3) whe e  B  =  2 k -  (-1) ,  k  =  E  ki ,  (2 .4) i=1 and  A .=2 1 ' -k i-1 -2k-ki-l-kl-2+ . . .+(-1) 2ki-( - 1) i  i=1,2, . . ., . (2 .5) We no e, o u u e use, ha A i ¿s índependen oj ki _ l . We also ema k ha he solu ions (2 .3) o he equa ions (2 .2) alwaysexis , bu ha ( o a gi en odd posi i e in ege b) he numbe s al gi en by (2 .3) may ail o be in ege s . Howe e , we ha e immedia ely P oposi ion 2 .1  (i)  The 6olu c :ou ob (2 .2) a ce na íonal numbM  al sa ís1yíng  0 < a l<2 ; (ii) íñ any a l .í .6 an ¡ n egeh, hen aie al ah .e odd .ín e .geA6 . P oo (i) I is clea o m (2 .4) and (2 .5) ha B, A i a e odd posi i ein ege s . Thus om (2 .3), each a l is a posi i e a ionalnumbe . Now 2k¡ai+l = b - a l < b,  since a l > 0 . Since  al+1  is posi i e and  k i > 1,  we in e ha  a l+1 < 2' ki-1 To p o e (ii), obse e ha a l-l = b - 2  a . . Thus i a l is an in ege ,  ai_1  is an odd in ege , and he esul ollows by ini e induc ion . 48 As an applica ion, conside B, A i , gi enby (2 .4), (2 .5) . As al eadyobse ed, B and A i a e odd posi i e in ege s o all i . Mo eo e , i ollows immedia ely om ki (2 .3) ha he solu ion o he equa ionsB = x i+2 x i+l , i = 1,2, . . . . ,xi+l = x l ,  is  x i = A i ,  so ha k . B = A i + 2 1Ai+ .l .  (2 .6) is a s mbol . Thus, by P oposi ion 2 .1, A l A 2 . . . A k 1 k 2 . . . k B (2 .7) we will also need he ollowing elemen a y p oposi ions ; he i s is p o ed in [21 . P oposi ion 2 .2  In . he bymbal  (2 .1) , gcd (b,a i )  -í .6 Lndependen Ul 1 . P oposi ion 2 .3  iñ, ín . he bymbal  (2 .1) , ki > n, . hen al+1 < ñ . 2 P oo This is ob ious om (2 .2) . P oposi ion 2.4  (Pe iodici y lemma)  11, .i .n (2 .1), heh .e exis . s an  s  euch ha  s i  and  k i+s - ki joh  aP,C  i,  hen  al+s = al dan aCQ .  i . P oo I is clea om (2 .5) ha i ki+s =ki o all i, hen Ai+s -_ Ai o all i . The esul now ollows om (2 .3) . The pe iodici y lemmaasse s ha i he sequence k1,k2, . . .,k is a epea ing sequence, hen he symbol (2 .1) is ob ainéd by he same epe i ion . I he e is no p ope epe i ion, we say ha he symbol (2 .1) is neduced añd w i e b a l a 2 k 1 k 2 a k (2 .8) Then a gene al symbol (2 .1) is ob ainedby nepea íng  a unique educed symbol ; and a educed symbol (2 .8) is ob ained by compkUsb .íng a gene al symbol . Gi en posi i e odd in ege s a wi h a < 2,  he e is a unique educed symbol (2 .8) = a . and b wi h We come now o ou main p elimina y esul . Theo em 2 .5  Le  k l,k 2 ,. . . . k  be poeí í e .íníegeu a .~í h E  k i = k > 2 .  Then, bon a g .í en odd .ín egeA  al < .2  ,  we ha e i=1 k  ala2 . . .a  al a 2 . . .a -1 a 2 -1  .í6 and on y -í~  2 k+l -1 k 1k2 . . . .k  k1 k 2 . . .k -1 k +l in eí heA ccue,  la e en . P oo Assume he le -hand symbol . Then, by (2 .3), I we e odd, we wouldha e 2 k -lla i , an e iden con adic ion . Thus is e en and al = A i , o all i . So (2 k -  ( - 1) )a i =  (2 k -  1)A i . Loe now sol e he equa ions 2k+1 - 1 = X i + 2ki Xi+l' whe e  k' i=k i , 1 <-i -< -1, k = k + 1,  so ha  E ki =k+1 =k', i=1 sa , o ob ain (compa e (2 .6)) x i = A!, wi h (compa e (2 .5)) A1-2k'-k!_  ~ .1 - 2k'-k1!_1-k1!_2 + . . .+ . (-1) 2kl  -  (-1)  (2 .9) Thus we ob ain he symbol Howe e , we see om (2 .9), ecalling ha Al is independen o k , ha Al = A 1 = al , es ablishing he exis ence o he igh -hand symbol o he heo em . The con e se is p o edsimi- la ly . The e is acompanion heo em as ollows ; we need no gi- e an exnlici n oo . Theo em 2 .5 *  Le  kl,k2, . . . ,k  be pos .í i e ín egeu wí h Ek i = k ? 1 .  Then, Son a gí en odd ín egeA  al < 2k-1,  we ha e i=1 A ' A' . . . A' - A' 1 2  1 k l k 2  ...  k -1  k +l 2 k +1 a l a2 k l k 2 . . . . . . a k is and o ney íS al a2 . . a -1 a 2 k+1 +1 kl k 2 . . k -1 k +1 In "eA case,  .í s odd . Quasi-O de Theo em  Le-  b  be an odd pos .í c : e íw egen, and .le  a . i  be an odd pos .í í e .ín egen wí h  a l< 2  and a  p~u :me xo b . Then í6  b T4e p o e his heo em in he nex sec ion bu we may imme dia elyanounce he ollowing co olla y, ela ing o he ohden o 2 mod b . Co olla y 2 .6  eU .i h che dame hupo . hehes as ín . he 9 .ua~sí-Onde Theo em, «úe ha e (i)  .í~   íz e en,  hen he anden o 6  2 mod b  .í s  k  and, e en í6  k .í 5 e en,  2 k/2 P--1 mod b ; (ii)  í6   .í6 odd, hen he oAden o6  2 mod b  í s  2k,  and 2 k -1 mod b . 3 . P oo o he Main Theo em p o e We a e now eady o s a e ou main heo em . a l a 2 . . . a k l k 2 . . . k wí h  E k i = k, we ha e i=1 (i)  k  .í,b - he m .Lní .mal Q sueh ha  bi~~±1, (ii)  b 12 k -1 ,¿~   .í s e en,  bl2 k+1  í5   .i s odd . We i s s udy a special case o he main heo em and Theo em 3.1  Le  Q > 2 .  Then í~  ~ -1 we ha e  E  Qi i=1 P oo We a gue by induc i n on Q , he case Q = 2 being _i l ialsince  3 Cl~ .  Thus we assume he heo em o  Q > 2  and p o e i o Q +1 .  Le hypo hesis, we ha e 2' -" -1 I  =1  and  2 1 =l,  he conclusion is i ially ue . I no , i ollows om he pe iodici y lemma ha , o some  i, Qi > 2 . Wi hou eal loss o gene ali ywe may assume ha Q > 2 so ha , by P oposi ion 2 .3, al < 2 Q-1 . Thus, by ou induc i e A al a2 ... as 2 2 - 1 (3 .2) k1 k 2 ... ks s wi h E k i IQ . By epe i ion, i necessa y, we ind he i=1 symbol a l a2 ... a 29 - 1(3 .3) k 1 k2 . . . k wi h Ek i = Q . By Theo em 2 .5 we deduce he symbol i=1 Pl i e ki = k i , 1 -< i -< -1, k =k +1 .  Then 1 E 1 ki = Q+1 . Comp essing, i necessa y, we ob ain u wi h  E k' I (Q+1) .  By he uniqueness o he educed symbol, as i=1 1 a unc ion o b and a o , we in e ha (3 .5) is iden ical wi h (3 .1), so ha he induc i e s ep is achie ed and he heo em is p o ed . The e is, o cou se, a companion heo em, wi h almos ¡den ical p oo , namely,  . Theo em 3 .1 * Le  Q > 1 .  Then we ha e  E  Q¡  I  Q . i=1 11  11  11 a l a 2 ... a -1  a k 1 k 2 ...  k -1  k +l a la2 . . . a 2 1 a a" . . . all 12  u k ' k' ... k' 1 2  u 2 2 Q J P oo o he Quasi-O de Theo em Fi s le (3 .4) (3 .5) Thus, by Theo em 3 .1 o 3 .1*, klk 0 . wi h no es ic ionon gcd(a l,b) . Le  E k . = k and le k  be he minimal Q such ha 1=1 1  k  0 bl2 Q ± 1 .  I 2 0 ± 1 = bq,  hen, ob iously, k  a l q a 2 q . . . a q 2 0 ±1 Now suppose ha a l is p ime o b . Then, by (2 .3) and (2 .4), (2k -  (_,) )a ¡ = bAl . Since b is p ime o ai , we ha e bl2 k - (-1) . Since klk 0 , he minimali y o k 0 implies ha k = k 0 . Mo eo e i is plain ha bl2 k -1 i is e en and bl2 k +1 i is odd . Rema ks . (i) No e ha we ha ep o ed ha , i we emo e om he hypo heseso he Quasi-O de Theo em he condi ion ha al be p ime o b, and i k is dejíned as he minimal Q Q  such ha bl2 ± 1, hen  E k i ¡k . I we w i e quo(b) o i=1 he cguasi-o de o ,? . mod b, hen his says ha i a la2 . . . a  b  I , hen E kil áuo(b) . Mo eo e , he k l k 2 . . . k i=1 .immedc :a eey ansla able in o old- heo e ic language! Fo i ells us ha , i we know how o old ou s ipo pape o p o k  k+1 duce a s a  { 2 a l }-gon, hen, o p oduce a s a  {2  a -1}-gon, we in oduce one mo e old line p ecisely a hose e ices on he op edgeo he ape which a e des ined o become e ices o ou polygon . 5 . Appendix 2 : á ew we ll-chosen examples whe e, by (2 .5) We no e ha , i I al a 2 ... a b wi h a l = 1, hen, by (2 .3), 2 k - ( -1) = bAl, A  =  20 -1  -  Z a -2  +  . 1 J Ek . =k, i=1 1 (5 .2) wi h  a . =  E k . .  (5 .3) i=1 1 Mo eo e , by ou main heo em, k = quo (b) . Le us apply his o case b = 641 . We ob ain, by ou algo i hm, 641 [15 159 241 25 77 141 125 129 72 1 4 3 2 2 2 9 Thus we in e , since k = 32, = 9, ha and, om (5 .2) auo(641) = 32 and, indeed, ha  232 + 1 =- - 0 mod641 . Mo eo e , we know om (5 .1) 2 32 + 1 = 641Al, (5 .4) A 1= 2 23 - 2 21 + 219 - 217 + 214 - 2 10 +29 - 27 +1 = 6700417 . This is, o cou se, Eule 's amous ac o iza ion showing 5 ha 22 + 1 is no a (Fe ma ) p ime . 4 Only he pape - olding ana ic would ake he iew ha he p incipalin e es o (5 .4) is ha i shows how o old he egula con ex 641-gon andce ain s a 641-gons . As a second example, conside he symbol 23 He e k = 11, = 6, so ha 1 11 3 5 9 7 1 2 21 1 4 4 See, o example, he on co e o [5] . quo(23) = 11, 2 11 - 1  0 mod 23, and, againby (5 .2), he complemen a y ac o is Re e ences Rebu el 16 d'oc ubne de¡ 1984 Depa men o Ma hema ics Uni e si y o San aCla a San aCla a Cali o nia 95053 U .S .A . A1 = 27 - 2 6 +25-23 + 2 - 1 = 89 Thus 2 11 - 1 = 23 " 89 and is no a (Me senne) P ime . [1] Pe e Hil on andand JeanPede sen,"App oxima ing any egu la polygonby olding pape : An in e play o geome y, ana lysis_andnumbe heo y", Ma hema ics Magazine, Vol . 56 . Nó 3, 1983 (141 - 155) . [2] -------------------------, "Regula polygons, s a polygons and numbe heo y", Coxe e Fes sch i , Ma h . Sem . Giessen 164, 1984, (217 - 244) . [3] -------------------------, "Folding egula s a polygons an l ni  ymbe heo %T" The Ma hema ical In ell i a en c e . Vol . 7 (1), 1985 (15 - 26) . [4] K .R . Ma hews and A .M . Wa s, "A gene aliza ion o Hasse's gene aliza ion o he Sy acuse algo i hm", Ac a A i hme ica XLIII, 1983 (75 - 83) . [5] Ma hema ical In elligence , Vol . 6 . Ns 3, 1984, on co e . [6] Pe e Hil on and Jean Pede sen, "On gene alized symbols, o_ de s and quasi-o de s" ( o appea ) .