Solving the Pell Equation

500 Pages·2009·0.54 MB·English

by  Michael J. Jacobson Jr.

3.2 heT InndainasictitahmaeM 33
areellwesdcribedybSelenius 05 adn ehtretedetsnireared ohsdlu octlusn siht
orkw ofr urthfer details and rereenfces .eWllwi oyln ,hcteks htwi aoitiddanl
ofniraomit,n atarainvreht(e are era)lvesof eht aglormhti rehe.
eWlliwaemussahtt Q,,AB (cid:3) Z adnahtt( ,AB )=1ni;2(2.3)sihtemasn
ahtt ( B,Q ) = 1. sA eht qienuhetc orf sogvlni 22(.2) asw ,wnkon eht setp of
gnidn(cid:147) an geterni P hsuc ahtt Q | BP + A could be easily edhievac yb the
aatktuk rposecs. tI swofoll ahtt escni ( B,Q ) = 1, ew sutm evah Q | P 2 (cid:138) D
adn Q | PA + DB . By gnittup R =1 ni 2(2.4,)ew see rfom 2(2.5) ahtt
(cid:8) (cid:9) (cid:8) (cid:9)
PA + DB 2 A + BP 2 P 2 (cid:138) D
(cid:138) D = . 2(2.7)
Q Q Q
romFsihtelpmisaoitnoresbvewcaneolpvedehtcilcycohtemdofrsognivleht
lelP eqauoit.n
enGiv egtersni n,A ,B ,Q , adn P where ( A ,B ) = 1 hscu
n (cid:138) 1 n (cid:138) 1 n n n (cid:138) 1 n (cid:138) 1
ahtt
(cid:25) (cid:25)
(cid:25) A 2 (cid:138) DB 2 (cid:25) = Q ,
n (cid:138) 1 n (cid:138) 1 n
dn(cid:147)ybehtaatktukrposesc a peovsiti 15 geterni P hsucahtt |P 2 (cid:138) D | si
n +1 n +1
aminlim adn Q | ( P B + A .) Ptu Q = |P 2 (cid:138) D |/Q ,
n n +1 n (cid:138) 1 n (cid:138) 1 n +1 n +1 n
A P + DB B P + A
A = n (cid:138) 1 n +1 n (cid:138) 1 , B = n (cid:138) 1 n +1 n (cid:138) 1 . 2(2.8)
n Q n Q
n n
By 2(2.7)ew get
(cid:25) (cid:25)
(cid:25) A 2 (cid:138) DB 2 (cid:25) = Q , 2(2.9)
n n n +1
adn( A ,B )=1.ehTalrettrtluseswofollaeylisyboresbvgniahtt |A B (cid:138)
n n n n (cid:138) 1
B A | = 1. eTh ohtemd retansiemt ,nehw ofr soem n , Q = 1 , 2 , 4
n n (cid:138) 1 n +1
bace,esu as ew evah alpxedeni e,abvo Braamhgatpu ahd arleayd nwsoh woh
ot esovleht ellP eqauoitn ocne yan soiotuln of 2(2.6) si .wnkon
oCredisnehteaxelpmof D =67.eWbegnihtwi n =0, A =1, B =0,
(cid:138) 1 (cid:138) 1
Q =1, adn P =0.eWwon ammusrezi niaelbT21. eht sooituln of eht ellP
0 0
equationybthis process,calledhet alavarkac eht( crielcor cilccyohtem)dyb
eht aiIdn.sn
lbaeT..21 alavkaarCrof D =76
n P Q A B P (mod Q )
n n n(cid:138) 1 n(cid:138) 1 n+1 n
0 0 1 1 0 1
1 8 3 8 1 1
2 7 6 14 5 5
3 5 7 09 11 2
4 9 2 122 72
43 2 ylraEyrtoHsifo thellePnqoEtuia
ecniS221 2 (cid:138) 67 • 27 2 = (cid:138) 2,ewget T =221 2 +1=48842, U =27 • 221=5967
as a sooituln of eht ellP auqeoitn T 2 (cid:138) 67 U 2 =1. Corecngnin siht ,euqinhcet
leaHkn 25 ats,det (cid:141)tI si bodney all rpa;esi ti si recatylni eht tsen(cid:147) gniht ahtt
aswedeivhacni eht ehtory of bemrusn beofre Lagragne(cid:142).,ofanrneuylUtteht
aidnIsn did ont edivrpo a rpoof ahtt eht cilccy ohtemd odluw ysaawl or.kw
Theyerewt,entconitseems,intheemp riacilwgdeleonkahtttiysaawlsdemee
ot od os, adn yeht desu ti ot eosvl eht ellP auqeoitn ofr D = 61 , 67 , 97 , 103.
It asw ont litnu eht alet 1930s ahtt a rpoof ahtt eht cyccil ohetmd odluw
syaawl rpoecud a aeulv of Q = 1 asw rpoduced yb angar.Ayy 35 eH ondet
i (cid:2)
ahtt siht rpossec codlu be detrerpneseas eht apxeoisnn of D otnia ypet of
serimegalureudnitconrfacoitnhciwhodluwysaawlbe peroici.d
eWonet ahtt fi a(s si recaytlni eht case ofr n =0)
P B (cid:13) A om(d Q ) ,
n n (cid:138) 1 n (cid:138) 1 n
eh,nt yb,2(2.8)
P B (cid:138) A = B ( P 2 (cid:138) D ) /Q
n +1 n n n (cid:138) 1 n +1 n
(cid:13) 0 (omd Q ) .
n +1
,suhT yb oitcudnin ew yam aemuss ahtt Q | ( P B (cid:138) A .) ecniS Q |
n n n (cid:138) 1 n (cid:138) 1 n
( P B + A ) yb corsnoitctun adn ( Q ,B ) | ( A ,B ,) ew get
n +1 n (cid:138) 1 n (cid:138) 1 n n (cid:138) 1 n (cid:138) 1 n (cid:138) 1
( Q ,B )=1 adn
n n (cid:138) 1
P (cid:13)(cid:138) P (omd Q ) .
n +1 n n
Hence,
P = q Q (cid:138) P 2(3.0)
n +1 n n n
ofr soem q (cid:3) Z . If ewwon begni htwi n =0 adn en(cid:147)ed
n
(cid:2)
P + D
(cid:8) = i ( > 0) ( i =0 , 1 , 2 ,... ) ,
i Q
i
(cid:22) =(sgin P 2 (cid:138) D ) ,
i+1 i+1
ew get
(cid:2)
P + D (cid:22) Q
(cid:8) = i+1 = (cid:2) i+1 i .
i+1 Q D (cid:138) P
i+1 i+1
By ,2(3.0)
(cid:2)
D (cid:138) P
i+1 = (cid:8) (cid:138) q ;
Q i i
i
hence,
(cid:22)
(cid:8) = i+1 . 2(3.1)
i+1 (cid:8) (cid:138) q
i i
eWwongitseaetvnieht rpomelb of ehtaeulvof q .
n
(cid:2)
eeohmrT 2.2. fI ew utp q = (cid:5) ( P + D ) /Q (cid:6) , neth 0 <q (cid:7) q (cid:7) q +1 .
n n n
3.2 heT InndainasictitahmaeM 53
.foorP Ptu P = qQ (cid:138) P , P (cid:2) = ( q +1) Q (cid:138) P adn onet ahtt P (cid:13) P (cid:2) (cid:13)
n n n n (cid:2) (cid:2)
P om(d Q .) yB oitin(cid:147)end of q ,ewevah P < D adn P (cid:2) > D .
n +1 n
If q <q , ehnt
n
(cid:2)
0 <P = q Q (cid:138) P <P < D .
n +1 n n n
Hence, |D (cid:138) P 2 | = D (cid:138) P 2 , |D (cid:138) P 2 | = D (cid:138) P 2 . cneiS P <P , ew get
n +1 n +1 n +1
D (cid:138) P 2 >D (cid:138) P 2 , hwcihsi pmioelbissyb soitcelen of P .
n +1 n +1
If q >q +1, enht
n
(cid:2)
P = q Q (cid:138) P >P (cid:2) > D .
n +1 n n n
In this case, |D (cid:138) P 2 | = P 2 (cid:138) D , |D (cid:138) P (cid:2)2 | = P (cid:2)2 (cid:138) D , adn P 2 (cid:138) D >
n +1 n +1 n +1
P (cid:2)2 (cid:138) D , hwhic is also pimossible. (cid:18) (cid:17)
By eThorem 22. adn ,2(3.1)ew see ahtt
(cid:8) > 1 ( i =0 , 1 , 2 ,... ) .
i+1
siTh eamsn ahtt eht exrpessoin 2(3.1) can be sued ot egvisu
(cid:2) (cid:22)
1
D = q + , 2(3.2)
0 (cid:22)
2
q +
1 (cid:22)
3
q +
2
q 3 + . .
.
(cid:2)
a semiregular 45 edunitconrafocitn exapsnoin of D .
A berumn of iscmonceptions utincon e ot crialucet ocn recgnin eht ccilcy
ohetm.d Oen of ehste si ahtt ti asweredvresidcoyb Lagragne. siTh, as eel-S
suin ahs deptoni o,tu si ont eht ac.es Lagragne amed esu of selpmideunitcon
rfaocisnt,hciwhodluwont encessarylibe eht saemas eht serimegalur-nitcon
deu rfaoitcsn ylticilpmi deyolpmeyb eht cilccy ohtem.d Onetf eht aglormhti
siadretttubiotaraahBskI,ItubasoidtennemybarahSknalku,hS 55 araBhask
amed on alcmi ot bgnie eht orgianiotr of eht ohtem,d adn asa,avedyJaowh
edorkw ni eht 10ht ruytcen or eareirl, ahd eredvsidco a tarainv of eht -hect
,euqin ti smees ahtt ti tsumevahbnee eolpvededhcumearreil ahtn eht emit
of ara.ahBsk,aniyFllrehte si eht b,feile perahsp eud ot,arennyT 65 ahtt eht
cyclic method esdervi rfom rGeek in(cid:148)uences. There seems, in spite of a-nT
renys(cid:144)aan,sisylotbeelttilsodilecnedivenippusortof.sihtehTselpmiafctsi
that,astionedenmearlier,ewdonotr eallywknowhatthereeGksknewabout
ehtllePaeuqoit.nahWtewodw,onkre,vewohsiahttehtaidnInohtemsd-sid
yalpa otsihry of aetsydteolnempvedadntrnemeen(cid:147)pu ot adn gnidulcni eht
reyvocsidofehtcilcycohtem,dadnsihteryvrtsognylgusgestsahttes(cid:144)laHkn 75
position that the Indians edolvev the hniqueect yb eshemseltv si het correct
oen.
63 2 ylraEyrtoHsifo thellePnqoEtuia
4.2 tamreF dna Hsi srosseccuS
ehT sotry of ehtllePeqauoitnrsseemu htiw ehtahgnellec 85 siseud ni 1657 ot
·rF ecieln ni apracliutr adn aeacihmmtatisn ni geenralybera.mtFeramtFahd
omts,ylekilrhtoguh sih ,hraeesrc coem ot rocegezin eht atladnufnem anrute
of eht lleP auoeqit.n eH assk ofr a rpoof of eht gniwofoll:tnaetmset
enviGyan p[]eosvitibremun [ D ] erwahvet ahtt si ont a sauqr,e rehte
areaoslnegvianetin(cid:147)nibremunofauqsreshscuaht,t fi ehtsauqresi
deilpitlum otni eht engvibermun adn ytinu si adedd ot eht rpo,tcud
eht rtluse si a auqsre.
It next requests a general rule yb whic h sooitulsn of eht rpomelb ocdlu be
eerenitddm a,dn as exealpsm, asks orf soisontul enwh D =109 , 149 , 433.
ehT sotry of woh eht ocesdn aprt of siht ahgnellec asw redeawsn yb
erkrBocnu adn asillW ahs bnee eryv ellw otdl yb eliW 95 adn aMoheny 06
and eends on elaboration here. Instead, ew will tentcon esourselv with giv-
ing a somewhat tereni(cid:3)d accoun t rofm ahtt veddirpo yb eliW 16 concerning
ers(cid:144)krBocnueuqinhcet ofr osgnivl eht ellP auqeoit.neW aehpmezis aht,t a-l
ohtguhers(cid:144)krBocnuohtemdsitaneleviuqot wahtewllwircsedbie,ehdidont
kniht abotu ti ni etiuq siht.yaw
Let P,Q,R (cid:3) Z , ewhre Q (cid:12)=0,
P 2 (cid:138) QR = D > 0 ,
adn D is not anegtralni square. Put
F ( X,Y )= QX 2 (cid:138) 2 PXY + RY 2 2(3.3)
adnelt (cid:15) adn (cid:15) (cid:2) onedeteht rezosof F ( ,x 1.)cniSe D siontaauqsre,ewwkon
ahtt (cid:15),(cid:15) (cid:2) (cid:12)(cid:3) Q . erkBrocnu ssmee ot evah desu eht gniwofoll r,tleus aohtlguh
eh sevdirpoon rpoof of .ti
orpPosiiton 2.3. seoSupp (cid:15)> 1 nda (cid:15) (cid:2) < 0 .fI F ( X,Y )=1 ,erehw X,Y (cid:3) Z
nda X >Y > 1 , neth (cid:5) (cid:15) (cid:6) <X/Y < (cid:5) (cid:15) (cid:6) +1 .
.foorP ecniS F ( X,Y )=1,ewyamasesmu ahtt X = qY + Z , ewrhe 0 <Z <
Y . sloA,
|Q || X (cid:138) (cid:15) (cid:2)Y || X (cid:138) (cid:15)Y | =1 . 2(3.4)
ecniS (cid:15) (cid:2) < 0,ewget |X (cid:138) (cid:15) (cid:2)Y | = X (cid:138) (cid:15) (cid:2)Y >X > 1.slAo, X (cid:138) (cid:15)Y =( q (cid:138) (cid:15) ) Y + Z ;
su,ht fi q (cid:138) (cid:15) < (cid:138) 1, ehnt X (cid:138) (cid:15)Y < (cid:138) Y + Z (cid:7) (cid:138) 1, adn fi q (cid:138) (cid:15) > 0, ehnt
X (cid:138) (cid:15)Y >Z (cid:4) 1. In either case, |X (cid:138) (cid:15)Y | > 1,hciwhsi pmiosselbiyb.2(3.4)
tI swofollahtt (cid:15) (cid:138) 1 <q <(cid:15) or q = (cid:5) (cid:15) (cid:6) . (cid:18) (cid:17)
fI ew etutssibtu X = qY + Z ni 2(,3.3)ew gte
F (cid:2)( Y,Z )= Q (cid:2)Y 2 (cid:138) 2 P (cid:2)YZ + R (cid:2)Z 2 ,
4.2 tmreaFnda sHirsossSeucc 73
where Q (cid:2) = q 2 Q (cid:138) 2 qP + R , P (cid:2) = P (cid:138) qQ , R (cid:2) = Q , adn
P (cid:2)2 (cid:138) Q (cid:2)R (cid:2) = D . 2(3.5)
It is easy ot wsho ahtt
(cid:2) (cid:2)
P (cid:2) (cid:138) D 1 P (cid:2) + D 1
= (cid:6) , = (cid:6) .
Q (cid:2) P + D (cid:138) q Q (cid:2) P (cid:138) D (cid:138) q
Q Q
us,Thfi (cid:11) adn (cid:11) (cid:2) areehtzerosof F (cid:2)( ,x 1,) enht (cid:11) =1 / ( (cid:15) (cid:138) q ,) (cid:11) (cid:2) =1 / ( (cid:15) (cid:2) (cid:138) q .)
If (cid:15)> 1, (cid:15) (cid:2) < 0, adn q = (cid:5) (cid:15) (cid:6) , ehnt (cid:11) > 1, (cid:11) (cid:2) < 0.
htiW ehste rpaneirmily aoisnt,osberv ew can won go on ot esdcrbie
ers(cid:144)krBocnu reyv gnioinesu .euqinhcet eW ppusose T,U si a sooituln of
T 2 (cid:138) DU 2 =1 adn tup Q =1, P =0, R = (cid:138) D , X = T , adn X = U .eW
0 0 0 0 (cid:2) 1 (cid:2)
evah F ( X ,X )= Q X 2 (cid:138) 2 P X X + R X 2 =1 adn (cid:15) = D , (cid:15) (cid:2) = (cid:138) D
0 0 1 0 0 0 0 1 0 1 0 0
are eht zeros of F ( ,x 1.) gnPittu q = (cid:5) (cid:15) (cid:6) adn gnitutssibtu q X + X ofr
0 0 0 0 1 2
X ni F ( X ,X ) ew get F ( X ,X )=1 0( <X <X .) reHe,
0 0 0 1 1 1 2 2 1
Q = q 2 Q (cid:138) 2 q P + R , P = P (cid:138) q Q , R = Q .
1 0 0 0 0 0 1 0 0 0 1 0
eWtup (cid:15) =1 / ( (cid:15) (cid:138) q ,) q = (cid:5) (cid:15) (cid:6) ,adn X = q X + X 0( <X <X ) adn
1 0 0 1 1 1 1 2 3 3 2
ocetupm F ( X ,X ) =(1,) .cte nI a,ftc fi F ( X ,X )=1 0( <X <X ,)
2 2 3 i i i+1 i+1 i
ewtup
1
(cid:15) = , 2(3.6)
i (cid:15) (cid:138) q
i(cid:138) 1 i(cid:138) 1
q = (cid:5) (cid:15) (cid:6) , adn
i i
X = q X + X 2(3.7)
i i i+1 i+2
ni F ot oatnbi F ( X ,X )=1 hwti
i i+1 i+1 i+2
Q =( P 2 (cid:138) D ) /Q , P = P (cid:138) q Q , R = Q ,
i+1 i+1 i i+1 i i i i+1 i
yb2.(3.5)
sA eht ecnseuqe { X } is a strictly ecdreasi ng or(f increasing i ) sequence
i
of peovsitiegters,ni siht rpocesssutm coem ot a ahtl htwi X =1, X =0
j j +1
ofr soem j (cid:4) 0.oTdn(cid:147) T adn U , all that is necessaryis to proceedardkwbac
sguni 2(3.7) ocne all ehtaeulsvof q ,q ,q ,...,q evahbnee .denimreted
0 1 2 j (cid:138) 1
eWlliw won yfilpmeex 26 eht rposecs ofr eht case of
T 2 (cid:138) 13 U 2 =1 . 2(3.8)
reHe,
(cid:26) (cid:2) (cid:27)
F ( X ,X )= X 2 (cid:138) 13 X 2 , q = 13 =3;
0 0 1 0 1 0
(cid:28) (cid:2) (cid:29)
3+ 13
F ( X ,X )= (cid:138) 4 X 2 +6 X X + X 2 , q = =1;
1 1 2 1 1 2 2 1 4
83 2 ylraEyrtoHsifo thellePnqoEtuia
(cid:28) (cid:2) (cid:29)
1+ 13
F ( X ,X )=3 X 2 (cid:138) 2 X X (cid:138) 4 X 2 , q = =1;
2 2 3 2 2 3 3 2 3
(cid:28) (cid:2) (cid:29)
2+ 13
F ( X ,X )= (cid:138) 3 X 2 +4 X X +3 X 2 , q = =1;
3 3 4 3 3 4 4 3 3
(cid:28) (cid:2) (cid:29)
1+ 13
F ( X ,X )=4 X 2 (cid:138) 2 X X (cid:138) 3 X 2 , q = =1;
4 4 5 4 4 5 5 4 4
(cid:28) (cid:2) (cid:29)
3+ 13
F ( X ,X )= (cid:138) X 2 +6 X X +4 X 2 , q = =6;
5 5 6 5 5 6 6 5 1
(cid:28) (cid:2) (cid:29)
3+ 13
F ( X ,X )=4 X 2 (cid:138) 6 X X (cid:138) X 2 , q = =1;
6 6 7 6 6 7 7 6 4
(cid:28) (cid:2) (cid:29)
1+ 13
F ( X ,X )= (cid:138) 3 X 2 +2 X X +4 X 2 , q = =1;
7 7 8 7 7 8 8 7 3
(cid:28) (cid:2) (cid:29)
2+ 13
F ( X ,X )=3 X 2 (cid:138) 4 X X (cid:138) 3 X 2 , q = =1;
8 8 9 8 8 9 9 8 3
(cid:28) (cid:2) (cid:29)
1+ 13
F ( X ,X )= (cid:138) 4 X 2 +2 X X +3 X 2 , q = =1;
9 9 01 9 9 01 01 9 4
F ( X ,X )= X 2 (cid:138) 6 X X (cid:138) 4 X 2 .
01 01 11 01 01 11 11
eWeosberv ahtt F ( X ,X ) = 1 can be easily edhievac with X = 1
01 01 11 01
adn X =0.eWcan wondn(cid:147)
11
X = q X + X =1 , X = q X + X =2 , X =3 ,
9 9 01 11 8 8 9 01 7
X =5 , X =33 , X =38 ,
6 5 4
X =71 , X =109 , X =180 ,
3 2 1
X =649 ,
0
adn tup T =649, U =180 as a sooitunl of 2.(3.8)
erkrBocnudesu sih ohtemd ot dn(cid:147) osoitulsn of realsvetluc(cid:2)id llePeauq-
oit,sn gnidulcni x 2 (cid:138) 433 y 2 = 1. siTh asw a amojr eaft of caaclluoit,n as
eht aeulv of y si a bermun of 19 gid.sti er,evwoH rehtien eh onr asillW onr
·rF eceiln asw aelb ot vedirpo a rpoof ahtt eht ellP eqoaiutn codlu ysaawl be
edsovl on-rn(tviaiyll) ofr yan peosviti on-nsqaureaeulv of D .eramtF 36 otok
oncite of siht adn asetdt ahtt eh ahd hscu a rpoof y(cid:141)b emasn of netesced
ylud adn arpporpaiylet (cid:142)a.deilpp,ofnUranutyletreamtFdedivrpo on rurefht
ofrniaoimtncocnergninsihrpoofahntsih.taomnHnf 46 a,dnhtiwgraerets-cu
cess,eilW 56 evahdaetpmettotrocertsntcuawhtreams(cid:144)tFohtemdtgimhevah
b.nee elihW ew yamreven reayll wonk ahwt siht a,sw ti si ersselehtven eryv
eylkil ahtteramtFdid evah a rpoo.f eTh afct ahtt eh seelcetd 109, 149, adn
4.2 tmreaFnda sHirsossSeucc 93
433 ofr aesulv of D as ahgnellec eaxselpm si apralucitryl esgugevits baceesu
eht correspgonidnellP eqaouistnevahalrgeaeulsvof t adn u .
ehT ohtemd of erkBrocnuasw omde(cid:147)id adn dednetxe yb Erelu, owh re-
aziled a,htt as si taappren rfom 2(3.6,) eudnitcon rfaocisnt codlu be sued ot
evdirpo an tecnie(cid:2) aglomhrti ofr sovlgni eht ellP eqoaiut.n er,evwoH enev
rhotguh eh ahd evdsied all of eht pmitoratn otosl, eh sutj ellf sohrt of v-rpo
gni ahtt sih ohetmdodluworkwofryanon-nsqaure D . sAtionedenm earlier,
eht teoelnmpedv of hscu a qienuhetc asw r(cid:147)st oden yb Lagragne ni a reraht
ysmulcork,whciwhehalrete.dvrpmioorFrurefhtofniramoitnonsihtapr-cit
alurylretgnietsniaprtofaamehmacittl,otsihryehtrearedsirreefredots(cid:144)lieW
book.In ehtenxtearhtpcewlwliedscrbieLagragnes(cid:144)ohetmd ofsgunielpsmi
deunitocnrfaoitcsn ot eosvleht ellPauqeoit.n 66
04 tsNeonda rsneeecRfe
Ntose dna secneRrefe
1 1nK.0o[]
2 2.W1h][i
3 9,1]Dc[i.loVII, h.C.21
4 37seL,[]pp..644(cid:133)124Aerombeilsseccaeucorsrofemosfothsisi79seL,[]p..001
5 2,7]arF[.loVII, p.,785 teno.342
6 2,7]arF[.loVI, p..904
7 08hoT[,].loVII,p..602(cid:133)302
8 thaT ,si a th(cid:147)fndaxithsfo bthofo theselma ndafo the.selmaef
9 99Ac[]r ndan2e0.L][
01 0uK,8r[] p.;4212,7]arF[.loVII,p.,885 teno;54278jDi[,] p.,893 teno.2
11 uK0r[8.]
21 2,7]arF[.loVI, pp..804(cid:133)704
31 See9,1]cDi[.loVII,pp..343(cid:133)243
41 See,]0t8A[mp.(cid:2)551 ro2,1]Ha[ep..023
51 roF the lniagiro keerG noiresv fo the uimlhocs nda a hrnecFn,otialntrsa ees
1,7A]c[rpp..371(cid:133)171
61 ,Seerof,xpmeela]h39S[cnad5t.9]aW[
71 .]u03rW[roFaeromylisaebeilsseccano,isrevees2,1H]a[e pp..323(cid:133)913
81 78Dj[i,] p.,993 tneo.3
91 b6u9a,]P[II. ,1 p.;535(cid:133)43521Ha[e,] p. xxxv.
02 2,7]arF[.loVI, p..904
12 78jDi[,] p.;9932,7]arF[.loVII, p.,095 tneo.652
22 6K,8n[]o p..592
32 0uK,8r[] p..421 Seeosla2,7]arF[.loVII,p.,095 tneo.752
42 6,Hm4][ooBko XII,sneil,031(cid:133)721 p..291
52 S1,t[6r];1.2.62u,9]hT[oBko VI, .2
62 .]0t8A[m
72 46ie,B][ p..942
82 8.9]raV[
92 2,7]arF[p..904
03 roFaytnhegltntemaetrfo,thsieesb,K5]n[7ohC.II.Anotianlastrfothetnaveler
krowfonhoTe si ni78w,]oF[p..85
13 78w,]oF[pp..201(cid:133)101
23 a,5K]7n[o p..731
33 2,1]Ha[e pp..89(cid:133)19
43 See2,1H]a[ex1xxxxi(cid:133)c;K[n a,5]7o pp..931(cid:133)631
53 b5K7n[.o]
63 78w.]oF[
73 b5K7n[,o] pp.;162(cid:133)5527,8]woF[h.C .2
83 b5K7n[,o] p..852
93 1,8]Ha[e.loVI, pp.;212(cid:133)202 ,v][d4W5 pp..971(cid:133)561
04 78w,]oF[p..54
14 78w,]oF[notiSce(b3)..2
24 a,5K]7n[o p.;83178w,]oF[pp..44(cid:133)24
34 78w,]oF[h.C .3
44 78w,]oF[p.,05 pp..55(cid:133)45
54 4n.8a]T[
tsNeondarsneeecRfe 14
64 28,Ss][etraPI.
74 owTulfeussuecros rof tshilairtemaera D2S[6] nda7.6S]ir[
84 4,6]Ha[e p..182
94 71lo,C][ p..363
05 36]Sl[enda 5.7]Sl[e
15 hsTisirenveytlixecpildte,tasbuttismeestobeticpimlinithe kuttaka prossec
tthadulowbe duesto (cid:147)nd P .
n+1
25 5Hn,6[a] p..202
35 See.]A[yy04
45 See §83fo.loVI fo75re.]P[
55 ,]4u5Sh[p. 1 nda p..02
65 7n,3a]T[p.(cid:2).042
75 H5n,[6a] pp..402(cid:133)302
85 2,1]reF[pp..533(cid:133)333 AnhsinlEgnoisrevnac beuondfni 4,6H]a[epp..682(cid:133)582
95 4.8]ieW[
06 4h9aM[.]
16 4,8]ieW[pp..79(cid:133)29
26 hsTi xpmeeal fos(cid:144)rekuonrcBnac be uonfd ni2,1]reF[.loVIII, p..084 It si osla
dtepneirrni2Wh1[i,] pp..55(cid:133)35
36 2,1]reF[p..334
46 49Hf[o.]
56 9;7]ieW[48ie,]W[notSice XIII.
66 roFtnhroea ppesrevticeno tshieesdrsadEw5d0Ew[,] pp..211(cid:133)56
3
duenitnoC nositcarF
1.3 lareneG deutninoCtscnaoriF
elihW eht seuqinhcet of eht aidInn amaamiechitstn adn ohtes of er,krBocnu
Eelru,adnLagragneofrsovlgniehtellPeqaoiutnareter(cid:3)iednotsoemedgree,
yehtacnallbede(cid:147)inuybcoredisngniehtoehtryofwahtareacdellsrimegealur
uedintconracftions.
fI ew evahowt ssecneuqe of getersni { a } ofr n (cid:4) 1 adn { q } ofr n (cid:4) 0
n n
adn soem ceolpxmbemrun (cid:8) , ewtup (cid:8) = (cid:8) adn en(cid:147)ed
0
a
(cid:8) = j +1 .
j +1 (cid:8) (cid:138) q
j j
Then a generaluedintconracftion 1 expansion of (cid:8) can be engiv as
a
1
(cid:8) = q + 3(1.)
0 0 a
2
q +
1 a
3
q +
2 . . . a
i
q +
i(cid:138) 1 a
i+1
q + .
i (cid:8)
i+1
nIehtcaserewhe (cid:8) siriraoitan,l3(1.)sisadiotbe ralugerimes 2 fiehtgniwoofll
oh:dl
1. |a | =1 ( i (cid:4) 1.)
i
2. q (cid:4) 1 ,q + a (cid:4) 1 ( i (cid:4) 1.)
i i i+1
3. q + a (cid:4) 2 yletin(cid:147)ni .noetf
i i+1
orFexael,pm fi (cid:8) si real adn q = (cid:5) (cid:8) +1 / 2 (cid:6) , het nearestegertni to (cid:8) , adn
n n n
sign a =sgi(n (cid:8) (cid:138) q ) ( n (cid:4) 0,) ewevahawht si caelld eht tseraneregntie
n +1 n n
dntnouiecntociarf expansion of (cid:8) .