PATTERNS IN CONTINUED FRACTION EXPANSIONS A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY SAMUEL WAYNE JUDNICK IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE ADVISOR: PROFESSOR JOHN GREENE MAY 03
Smuel Judc 03
Cotets Lst of Tbles Chter - Itroducto Defto Cotued Frcto Theorem Fte Cotued Frctos The Cotued Frcto Algorthm 3 Chter Proertes d Imortt Reltos 7 Defto - Covergets 7 Theorem The Fudmetl Proertes 7 Theorem 3 The Exso of d 0 Theorem 4 Ifte Cotued Frctos Theorem 6 Lgrge s Theorem 3 Chter 3 - Aroxmto 5 Theorem 9 Dfferece Betwee d ts Covergets 6 Chter 4 Ptters Cotued Frcto Exsos 8 ( ) Theorem 0 The Exso of ( ) ( ) Theorem The Exso of 5 Theorem 8 Preservg the Prtl Quotets of 40 Theorem 9 Preservg the frst Prtl Quotets of x 4 Chter 5 Future Wor 45 Refereces 49
Lst of Tbles Tble 5 Tble 0 Tble 3 Tble 4 44 Tble 5 45 Tble 6 45 Tble 7 46
Chter - Itroducto It s well ow tht y rel umber hs uue (or lmost uue) decml exso Sce we do ot tyclly wrte fte strg of zeros dow, these exsos c be ether fte or fte For stce bse 0, 3/5 hs decml exso 4, /3 hs decml exso 03333 = 03, d hs decml exso 3459 However, bse 3 the decml exsos of 3/5, /3, d re 000, 0, d 0000 resectvely Notce tht ot oly do the decml exsos chge wth dfferet bses, but lso whether the exso s fte or fte Rel umbers hve other terestg exso clled cotued frcto exso I sese, the cotued frcto exso of rel umber s bse deedet Sce these exsos re gve by lstg oegtve tegers, whe we cosder exsos dfferet bses the oly thg tht chges s how we rereset those tegers Whether or ot the exso s fte or fte does ot chge, eve f we do chge the bse For exmle, bse 0, 3/5 hs cotued frcto exso [,4,6], the exso of /3 s [0,3], d the exso for s[3,7,5,, ] I bse 3, the exsos of 3/5, /3, d re [,,0], [0,0], d [0,,0,, ] These exsos re uue, wth oe exceto Cotued frcto exsos re much dfferet th decml exsos d the exso loe c rovde us wth cosderble mout of formto I ths regrd, reresetg umbers s cotued frctos s more beefcl th usg decml system However, t does hve drwbcs s eve oertos such s ddto re extremely dffcult to erform o two cotued frcto exsos [3, 9-0]To uderstd wht ths exso s, we must frst defe cotued frcto Defto : A exresso of the form () 0 b0 b b 3 where, b re rel or comlex umbers s clled cotued frcto A exresso of the form () 0 3
where b for ll, 0 s teger, d 0,,, re ech ostve tegers s clled smle cotued frcto Due to the cumbersome ture of the otto bove, t s more commo to exress () s 0 3 or smly s[ 0,,, 3,] We wll mostly use the ltter of the exressos, d we sometmes refer t to s the cotued frcto exso of umber The terms 0,,, re clled rtl uotets If there re fte umber of rtl uotets, we cll t fte smle cotued frcto, otherwse t s fte I ths er whe we refer to cotued frctos, we relly re referrg to smle cotued frctos, the oly cotued frcto we cosder As exmle of cotued frcto, let s clculte the cotued frcto exso of rtol umber Exmle To fd the cotued frcto exso of 43 9 we c roceed s follows: 43 5 9 9 9 4 3 3 3 3 5 5 5 4 4 3 We c see from ths tht both the lst two exressos mtch exresso () Hece ths exmle shows us tht 43 hs two cotued frcto exsos, [,3,,4] d[,3,,3,] Ths leds us 9 to our frst theorem Theorem [6, 4] Ay fte cotued frcto reresets rtol umber, d y rtol umber c be rereseted s fte cotued frcto Furthermore, ths cotued frcto s uue, rt from the detty[ 0,,,, ] [ 0,,,,,] Although exmle we showed method of clcultg the cotued frcto exso of umber, t would be ce to hve systemtc roch to fdg the exso of y rel umber, ot just rtol oes The cotued frcto lgorthm gves us just tht
The Cotued Frcto Algorthm Suose we wsh to fd the cotued frcto exso of x We roceed s follows Let x0 xd set 0 x0 We the defe x x mer; x 0 0 d set x We roceed ths x x x x,, x x, We ether cotue x x deftely, or we sto f we fd vlue x [5, 9-30] To llustrte ths lgorthm, cosder the followg exmle Exmle We shll clculte the cotued frcto exso of 44 469 83 44 Let x0, so 0 The 83 83 x 603, 44 3 83 x 3 638 6, 83 3 x3 4000 3 4, 3 6 5 x 5 5 4 5 4 4 Sce x 4 5, we re doe Thus we coclude tht 44 [,,6,4,5] 83 As metoed bove, the cotued frcto lgorthm c be led to rrtol umbers s well As coseuece of Theorem, the lgorthm, whe led to rrtol umber, wll cotue deftely Some rrtol umbers, sure roots for exmle, hve cotued frcto 3
exsos tht exhbt ce erodc behvor Other umbers such s e hve evdet tters tht occur ther exsos, d yet others such s hve exsos tht do o ot er to follow y tters Below re some exmles log wth ther cotued frcto exsos 3 [,,,,,,,] [,,] 7 [,,,, 4,,,, 4,,,, 4,] [,,,, 4] e [,,,,, 4,,, 6,,,8,] [3, 7,5,, 9,,,,,,] It s dffcult to rove the bove exsos of or e, however the ext exmle llustrtes tht oe c fd the exso of 3 wth ese Exmle 3 We follow the cotued frcto lgorthm Let x0 3 Sce 3, 0 Now, 3 3 x, 3 x 3 3, 3 3 x3 x 3 3 Sce x3 xths clerly forces x4 x, x5 x,, x x, x x, d so the corresodg rtl uotets lterte betwee d deftely Therefore, 3 [,,] Uses of Cotued Frctos Cotued frctos costtute mjor brch of umber theory becuse they hve my lctos wth the feld Frst of ll, they rovde us wth method to fd the best rtol 4
roxmtos of rel umber the sese tht o other rtol wth smller deomtor s better roxmto [3, 6-8] Cotued frctos llow oe to fd solutos of ler Dohte eutos wth ese See [6, 3-46] Also the cotued frcto exso of c be used to fd solutos to Pell s euto, x y For more formto o Pell s Euto d cotued frctos, refer to [] Furthermore, cotued frctos c be ut to use the fctorzto of lrge tegers [5, 46] We c lso me use of cotued frctos to hel rove tht y rme of the form 4 c be exressed uuely s the sum of two sures [6, 3-33] Motvto of our roblem Ths er ws sred by the followg uesto Suose we strt wth some umber x whch hs ow exso [ 0,,,] d we dd to t decresg seuece of ostve vlues r The s the vlue of x r roches x wht hes to the corresodg cotued frcto exso? It turs out tht some terestg tters become evdet To llustrte ths cosder the followg: Exmle 4 Let x 9 whch hs exso[,3,,3,4] Now the followg tble gves 9 the exsos of the umbers for0 Cotued Frcto Exso 0 [, 3,, 0,,,, ] [, 3,, 4,, 5,, ] [, 3,, 7,,, 3,, ] 3 [, 3,, 9,,,,,, 3] 4 [, 3,, 30, 3,,, 3, 5] 5 [, 3,, 30,, 3, 7,, 7] 6 [, 3,, 3, 56] 7 [, 3,, 3, 7,, 3, 4] 5
8 [, 3,, 3, 5, 3, 3,, 3] 9 [, 3,, 3, 4,,, 3, 3,, 3] 0 [, 3,, 3, 4, 3,, 3, 3,, 3] [, 3,, 3, 4, 7,, 3, 3,, 3] [, 3,, 3, 4, 5,, 3, 3,, 3] Tble Loog t the tble we see tht the [,3,,] tter ers ech exso d whe 6 ech tter strts wth[,3,,3,], the frst 4 rtl uotets of x For 8, the etre exso of x ers the begg However ths s t the oly terestg thg to me ote of For 9 we lso see tht the oly rtl uotet tht chges s the oe mmedtely followg the 4 whle the rest of the rtl uotets re[,3,3,,3] If we loo t ths orto reverse order, we see tht [3,,3,3,] [3,,3,4] whch exctly mtches ll but the frst term of the exso of x I ths er, we wll loo to other terestg tters tht rse cotued frcto exsos d exl whe recsely ths tter occurs d why t does A rtculrly ce result tht cme bout from ths vestgto c be foud chter 4 It gves ( ) exlctly the cotued frcto exso for rtol umber of the form for ( ) oegtve teger, gve the exso of We use here to me uc ote regrdg the Theorems dscussed ths er The frst e theorems re commoly foud y textboo o cotued frctos Theorems 0 d those tht follow re troduced ths er Tht Theorem 7 exsts, however, s hted t [5, 38] 6
Chter - Proertes d Imortt Reltos Oe essetl tool studyg the theory of cotued frctos s the study of the covergets of cotued frcto Defto : Let x [ 0,,,,,] The reduced frctos gve below re clled the covergets of x d re defed by: 0 0, 0 0, 0,, 0 3 Theorem [5, 33] Let 0,,, deote the umertors of the covergets of some umber x whle 0,,, deotes the deomtors Now defe cotued frcto lgorthm The the followg reltos hold 0,, 0 d defe x s the for 0 ( ) for x x x x v x x for 0 Proof We rove () d () Proerty (v) follows drectly from roerty () Proof of (): Ths result follows by mg use of relto () d ducto To estblsh bss for ducto, we use the gve tl vlues to show the relto holds for, 0, d 00 ( ) 7
0 ( ) 0 0 0 0 ( ) ( ) 0 0 0 0 Now suose t holds for some teger m 3 The, m m m m ( ) ( ) m m m m m m m m m m m m ( ) m m m m ( ) m by our ducto ssumto ( ) m So t holds for m+ s well Proof of (): Ag we roceed by ducto Recll from the cotued frcto lgorthm tht x0 x d x x x x For 0, x0 x 0 x x 0 x 0 For, 0 0 0x x 0 x 0 0 x 0 x 0x x 0 0 x 0 So the result holds for 0 d Assume t holds for some umber, the we hve 8
x x x x x x ( x ) ( x ) ( x ) ( x ) x x x by the ducto ssumto So the result holds for + s well Alyg roerty () of Theorem c gve us effcet wy of clcultg the covergets of cotued frcto f we ow the rtl uotets Exmle 5 demostrtes ths Exmle 5 Cosder 380 [,3,5,7,9] We c clculte the covergets by usg the followg 05 tble: - - 0 3 4 3 5 7 9 0 4 5 380 0 3 6 5 05 Notce tht f we follow the rrows the dgrm bove, to fd 3 5we multly 7 by d dd 4 Smlrly, to fd 4 05 we multly 9 by 5 d dd 6 to t We c lso use the tble bove to llustrte roertes () d () from Theorem For roerty (), we see tht 3 3 (5) 5(6) To demostrte roerty () for sy, x d x 3, frst observe tht 380 05 39 x x0 so x d x Now, 05 380 39 05 3 64 05 39 9
39 4 x 0 64 380 39 x 0 05 3 64 d 3 64 4 x 9 380 64 x3 05 6 3 9 The followg s terestg d useful result Theorem 3 [6, 6] If [ 0,,,, ] the [,,, ] for Also f 0 0 the 0 [,,,, ] If 0 0 the for [,, 4, 3, ] Proof Mg use of Theorem () for we hve: 0 0 0 0 0 For we hve: 0 0 0 Now ssume the result holds for some teger m So, m m [ m, m,, ] m m Now, m m m m m m m m m m m 0 m
m m m by the ducto ssumto Thus the result holds for m so t holds for by ducto The roof for the [,,,, 0 ] cse s smlr Theorem 4 [6, 70] Every fte cotued frcto[ 0,,,,,] uuely reresets rrtol umber y Coversely, f y s rrtol umber the ts cotued frcto exso s fte It s mortt to ote few thgs regrdg Theorem 4 Frst, s metoed erler, from Theorem we mmedtely ow tht rrtol umber wll hve fte cotued frcto exso Next, we eed to clrfy wht s met whe we sy the exresso [,,,,,] 0 reresets rrtol umber Syg tht y = 0 lm [,,,,,] mes tht y At ths stge, we should rovde some justfcto for ths Frst, observe tht for y oegtve teger, ( ) ( ) By Theorem () ( ) ( ) by Theorem ()
I smlr mer, t c be show tht These results tell us tht the eve covergets form cresg seuece whle the odd covergets form decresg seuece Tht s, (3) 0 4 5 3 d 0 4 5 3 Alyg Theorem () for y oegtve teger we lso hve, ( ), (4) so the eve covergets re less th the odd covergets Combg (3) d (4) ow tells us tht: (5) 0 4 5 3 0 4 5 3 Now the seueces d re both mootoc d bouded, d therefore re coverget Furthermore, they re subseueces of Flly, observe tht lm lm 0 sce s Hece, the seuece Cuchy seuece d therefore coverges to some rrtol umber, sy y s
Now sce coverges to y so do d Ths leds us to the followg: Theorem 5 [6, 63] The eve covergets of the cotued frcto exso of y re ll less th y d they form cresg seuece The odd covergets of y re ll greter th y d they form decresg seuece Tht s, 0 4 5 3 y 0 4 5 3 Imortt Remr: Suose s some rel umber d [ 0,,,,,] If we let [,,,,, ] where 0 d f we defe logously to x from the cotued frcto lgorthm, the Theorem () stll holds The roof gve ws deedet of whether x ws rtol or rrtol Tht s, The ushot of ths s tht f we wrte y rel umber s [ 0,,,,, ] the we c stll ly ll the roertes of Theorem just s f were eve though c be y rel umber greter th or eul to We refer to s comlete uotet [5, 3] Theorem 6 (Lgrge s Theorem) [, 44] Ay udrtc rrtol umber hs cotued frcto exso whch s erodc from some ot owrd Coversely, f we strt wth cotued frcto exso tht s evetully erodc, the t reresets udrtc rrtol umber We wll just rovde setch of the roof For more detled roof see [, 44-45] Proof Setch Suose x s rel umber wth cotued frcto exso tht s evetully erodc wth erod legth of l Tht s, x [ 0,,,,,, l] If x [,,, l] the x [ l, l,, l] s well Now by Theorem () we hve, x x x x x l l l l 3
It s cler from bove tht x s rrtol d tht x stsfes udrtc euto wth tegrl coeffcets By substtutg udrtc rrtol s well x x from Theorem (v), t becomes evdet tht x s x To rove the coverse, suose tht x s some udrtc rrtol Hece, x stsfes the udrtc euto Ax Bx C 0, x for some tegers A, B, d C Oce g by Theorem rts () d (v), x x for 0 d x euto x x Thus x s udrtc rrtol s well d so t stsfes the A x B x C 0, where the tegers A, B,d resectvely From here, t c be show tht A, B, d C re ech defed terms of the tegers A, B, d C C re ll bouded by some costt, sy m, deedet of Therefore, there c oly be fte umber of dfferet trles A, B, C, d hece we c fd 3 dstct dces, sy,,d, 3 such tht A, B, C A, B, C A, B, C So 3 3 3 x, x, d x 3 re three roots of the udrtc euto corresodg to ths trle, whch mes tht two of them must be the sme Sce s determed drectly from x, f ot o x x sy, the ts exso must be erodc from tht 4
Chter 3 Aroxmto It s ofte very rctcl to roxmte rrtol umbers wth rtol umbers It s cler tht gve y rrtol umber, we c roxmte t wth rtol umber to y desred ccurcy The more ccurte roxmto we desre, the lrger the deomtor of the rtol must be The covergets of cotued frcto rovde us wth method to fd rtol umbers tht roxmte rrtol umbers whle hvg s smll deomtor s ossble I fct, o other rtol umbers wth smller deomtors c roxmte rrtol umbers better th ts covergets For exmle, cosder 3459653 whch hs exso[3,7,5,,9,,,,,,] We c esly fd tht 3459 39699 3459 000000 5000 mtches the frst 7 dgts of the decml exso of However, f we loo t the covergets 355 of we see tht oe of them s [3, 7,5,] whch hs decml exso 34599 3 whch s ctully ot oly better roxmto for but lso hs deomtor tht s gret del smller No rtol umber wth deomtor smller th 3 c rovde better roxmto to I ths er, we vestgte uttes of the form r Whe r hes to be rrtol, we c use the roxmto roertes of the covergets of r to hel ssst us studyg the cotued frcto exso of r Some clssc theorems o cotued frctos d roxmto re gve below If you wsh to ler more bout roxmto usg cotued frctos, see [4] or [, 54-76] Theorem 7 [6, 7] Let y be rrtol umber d, The be successve covergets of y y y Furthermore, t lest oe, sy, stsfes the eulty: 5
y Corollry If x s rrtol, the there exsts fte umber of rtols such tht x The ext theorem s extremely terestg d somewht surrsg, s t tells us tht f rtol umber roxmtes rrtol umber well eough, the t must be oe of ts covergets Theorem 8 [5, 37-38] For y rel umber, f The s ecessrly oe of the covergets of the cotued frcto exsos of The followg theorem gves uer d lower bouds o the dstce betwee rrtol umber d y of ts covergets Theorem 9 [5, 37] If s rrtol the for y 0, ( ) Proof Let [ 0,,,, ] where [,,] The by Theorem rts () d () we hve, d so ( ) ( ) ( ) Now observe tht 6
( ), d lso tht ( ) ( ) Ths comletes the roof Corollry [5, 37] If s rrtol the, 7
Chter 4 - Ptters Cotued Frcto Exsos As revously metoed, the gol of ths er s to vestgte how the cotued frcto exso of r chges s the vlue of r chges We beg ths chter by vestgtg the cotued frcto exsos of vrous fte seres Let s frst cosder the smle geometrc seres 3 0 where s teger greter th The rto of terms ths seres s so we ow ths seres coverges to I the ext exmle we show how the cotued frcto exsos of the rtl sums c be used to derve ths Exmle 6 The frst two rtl sums of let s fd the exso of clerly hve exsos [] d [, ] 0 Now, ( )( ) [,, ] I the sme mer, we shll ow fd the exso of the thus rtl sum, where 3 Observe tht, th 0 8
( )( ) (6) Therefore, Sce [,, ] 0 coverges to 0 s, from (6) we see tht the smle geometrc seres, s exected It s useful to ote tht exmle 6 we used ute obvous but useful fct tht geerl, lm [,,,, m] [,,, ] m 0 0 I the ext exmle, we cosder the exso of the seres 3, for secfc vlues of d Although the exso of ths seres hs smlrtes to the oe bove, the term comlctes thgs Exmle 7 Suose of the seres 9 [0,3,5, 6, 4] The followg tble gves the frst 5 rtl sums 4 9
Cotued Frcto Exso [0, 3, 5, 6, 5, 3, 6, 5, 3] 3 [0, 3, 5, 6, 5, 3, 8, 3, 5,,,,, 4, 3] 4 [0, 3, 5, 6, 5, 3, 8, 3, 6607,,,,, 4, 3] 5 [0, 3, 5, 6, 5, 3, 8, 3, 75,,,,, 4, 3] 6 [0, 3, 5, 6, 5, 3, 8, 3, 674959,,,,, 4, 3] Tble We see tht the exso becomes fxed excet for oe rtl uotet whe 3 The ofxed rtl uotets re 5, 6607, 75, d 674959 Observe tht: 5 6(), 6607 6(4 ), 75 6(4 4 ), 674959 3 6(4 4 4 ) From ths we see tht the o-fxed rtl uotet corresodg to the th rtl sum( 3 ) 3 tes the form 6 We c use the formto from Tble to tell us wht the seres 0 goes to fty Thus, coverges to Smlr to exmle 6, the o-fxed rtl uotets go to fty s 355 coverges to[ 0,3,5, 6,5,3,8,3] As metoed 4333 ror to exmle 7, the exsos tht er Tble re much more dffcult to redct th the exsos tht er the rtl sums of geometrc seres However bsed o severl exmles, we were ble to me some cojectures tht redct cert tters tht wll er We me o ttemt to rove these ths er, but they er Chter 5 I exmle 4, we geerted tble of cotued frcto exsos for umbers of the form 9 for0 We oted rtculr tter tht occurred for vlues of 9 where oly sgle rtl uotet chged Prt of the tble s gve g below 0
Cotued Frcto Exso 8 [, 3,, 3, 5, 3, 3,, 3] 9 [, 3,, 3, 4,,, 3, 3,, 3] 0 [, 3,, 3, 4, 3,, 3, 3,, 3] [, 3,, 3, 4, 7,, 3, 3,, 3] [, 3,, 3, 4, 5,, 3, 3,, 3] Tble 3 A secto of the tble revels tht the tter of terest occurs t the ext teger fter 8, 9 the sure of the deomtor of our orgl umber, A cler tter s lso evdet the oly o-fxed uotets of the exsos, e, 3, 7, d 5 They re ll of the form where s the dfferece betwee d 8 Ths s ot true geerl Tht s, f we costructed smlr tble wth dfferet tl umber, the o-fxed uotets my ot hve the form However, the lest commo deomtor of our orgl umber, x, s 9 d so ts sure s 8 3 8 Whe for stce, the deomtor of the resultg umber s ( ) whch s 8 multles of 8, whle the o-fxed rtl uotet s8 7 Ths s ctully the ey to the vlue of the o-fxed term geerl s we see the followg theorem Theorem 0 Suose [ o,,,, ] where d The for, ( ) [ o,,,,,,,,,,,, ] ( ) d ( ) [ o,,,,,,,,,,,, ] ( ) Proof: Sce cotued frcto exsos re uue, cosder the followg cses
Cse : Suose x [ o,,,,,,,,,, ], we wll show tht x ( ) ecessrly c be wrtte the form Observe tht the frst covergets of ( ) d x re exctly the sme, so x [,,,,, ] where o [,,,,,, ] for re ll covergets of x Now suose Sce [ o,,,, ], by Theorem 3, [,, ] Hece, Now sce x, by substtuto we get the followg:
x ( ) ( ) ( ) ( ) ( ) ( ) By Theorem () ( ) ( ) ( ) ( ) ( ) ( ) Cse : Now let y [ o,,,,,,,,,, ] We ow show tht y c be ( ) wrtte the form Ths tme let the covergets of ( ) be deoted by for whle we deote the covergets of y by Notce tht from the exso of y, ' ' for, ' ' ' d ', d ' d ' We c esly clculte ' d ' : ( ) d smlrly, ' ' Now g let s suose o y [,,,,,, ] where [,,,, ] The, g usg Theorem 3 we hve:, whch mles 3
' ' Oce g we use substtuto to the euto y ' ' d we get: y ( ) ( ) ( ) By Theorem () ( ) ( ) ( ) ( ) From the roof of Theorem 0, we see tht the exso chges bsed o whether s eve or odd The followg corollres tell us exctly how ech cse lys out Corollry Whe s eve, [,,,,,,,,,,,, ] ( ) o [,,,,,,,,,,,, ] ( ) o Corollry Whe s odd, [,,,,,,,,,,,, ] ( ) o 4
[,,,,,,,,,,,, ] ( ) o Theorem 0 tells how the exsos of rtol umbers chge s we dd d subtrct tegrl multles of However, t does ot tell us wht hes whe tht multle s The ext theorem tes cre of tht Theorem Suose [ 0,,, ] where, 0,d The, ( ) [ o,,,,,,,,,, ] d ( ) [ o,,,,,,,,,, ] Proof: We roceed the sme mer s the roof of Theorem 0 x [,,,,,,,,,, ] We the ote tht the Cse : Suose o covergets of x d re the sme u through Let s deote the th coverget of x by ' The we see from Theorem () tht ' ' ' Now f x [ 0,,,,, ] ( ) Smlrly, where [,,, ] the by Theorem 3 5
' ( ) By Theorem (), x ' ( ) d substtutg we get: ( ) ( ) x ( ) ( ) ( )( ) ( )( ) ( ) ( ) by Theorem () ( ) ( ) Cse : Suose y [ o,,,,,, ] where [,,,, ] Ths tme, ( ) d so ' ' Thus, by Theorems d 3,, d so ( ) y ' ' ( ) Substtuto the yelds, y ( ) ( ) 6
( )( ) ( ) ( )( ) ( ) By Theorem () ( ) Corollry Whe s eve, [ o,,,,,,,,,, ], [ o,,,,,,,,,, ] Corollry Whe s odd, [ o,,,,,,,,,, ], [ o,,,,,,,,,, ] To hel clrfy Theorems 0 d let s cosder some exmles Exmle 8 Suose 79 [,3,4,5,7,] [,3,4,5,8] 557 Sce s eve, from Theorem we c coclude wthout y comutto tht 79 [,3, 4,5,9,7,5, 4,3] 557 557 7
Ag, we do t eed to do y comutto to see tht from Theorem 0 whe 7, 79 557 (7 )557 [,3, 4,5,8, 7,,7,5, 4,3] So fr ths er we hve strted wth the cotued frcto exso of d the observed how the cotued frcto exso of r chged s we ced vrous vlues for r However, sometmes we strt wth cert form of cotued frcto exso, d fd out wht vlue of r would corresod to ths exso For stce, loog t Theorems 0 d, we see tht the exsos gve re very close to beg ldrome A ldrome s word, hrse, or umber tht s the sme red bcwrds d forwrds, such s 3 or wow These exsos led to the followg uesto If we gore the frst rtl uotet, for wht vlue of r does the exresso r Tht s, exso of the form 0 [,,,,,,,, ] or 0 hve cotued frcto exso whch s ldrome? [,,,,,,, ] [,,,,,, ], 0, for some ostve teger Ths uesto s ot dffcult to swer usg the sme techue used for the roofs of Theorem 0 d It turs out tht to fd ths vlue of r we eed to ow wht s Theorem Suose 0 [,,,, ] where The ( ) [ 0,,,,,,,,, ] ( ) Proof Let x [ 0,,,,, x ] where x [,,,, ] The, x by Theorem 3 8
Now by Theorem () d substtuto, x x x ( ) ( ) ( ) Now observe tht x ( ) ( ) ( ) ( ) ( ) ( ) ( ) by Theorem () ( ) Therefore, [ 0,,,,,,,,, ] ( ) s desred Theorem 3 Suose 0 [,,,, ] where The ( ) [ 0,,,,,,,, ] ( ) Proof Oce g let x [ 0,,,,, x ] where x [,,, ] The by Theorem 3, x By Theorem (), 9
x x x so tht ( ) ( ) x ( ) ( ) ( ) ( ) ( ) ( ) by Theorem () ( ) We ow see tht [ 0,,,,,,,, ] ( ) Theorem 4 Suose 0 [,,,, ] where The ( ) [ 0,,,,,,, ] ( ) Proof Let x [ 0,,,,, x ] so tht The by Theorem () we hve, x [,,, ] by Theorem 3 30
x x x Thus, x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) by Theorem () Therefore, ( ) [ 0,,,,,,, ] ( ) s desred Although Theorems, 3, d 4 hve cotued frcto exsos tht re more lesg to the eye th the exsos gve Theorems 0 d, the ext exmle shows tht they re ot s turl I order to ly Theorems, 3, d 4, we eed to ow the vlues of d sometmes Exmle 9 Notce tht [3,,5,7,8] [3,,5,7,7,] To roceed, we must frst c oe 643 of these exsos tht we desre to wor wth Let s wor wth [3,,5,7,7,] ths tme We strt by observg tht sce 5 s the lst term, 5 Next we fd 5 The 4 3
covergets of 643 whe 3 we see tht 3 7 38 73 949 re,,,,, d 79 564 643 so 4 564 Thus by Theorem 5 ( ) [3,,5,7,7,,3,,7,7,5, ] 643 643(3 643 564) By Theorem 3, 5 ( ) 564 643 643(643 564 ) [3,,5,7,7,,,7,7,5,] Flly by Theorem 4, 5 ( ) 79 [3,,5,7,7,,7,7,5,] 643 643564(643 79) There s ctully secfc cse where Theorems d 3 c be thought of s beg just s turl s Theorems 0 d If [0,,, ] d [,, ] s ldrome, the t follows tht To see ths, let x [,,, ] so tht 3 0, d thus x x Usg ths d the fct tht x s ldrome, Theorem 3 tells us x [,,, ] d hece,, gvg Therefore we c relce ech wth thus elmtg the eed to fd the covergets to ly these theorems I fct, oe could ccomlsh ths by frst lyg oe of the Theorems, 3, or 4 to rtol umber less th From there, the umertor of the resultg rtol umber would serve s llustrted the ext exmle Exmle 0 Suose 4 x [0,3, 4] We frst ly Theorem 3 so tht the rtl uotets 3 fter the frst become ldrome Dog so gves 4 3 [0,3, 4, 4,3] 55 3 3(3 3 ) 78 Next, we wll ly Theorem wth 4 to 55 78 Sce the cotued frcto
exso of 55 78 meets the codtos metoed bove, we ow tht Thus 55 [0,3,4,4,3,4,3,4,4,3], whch g hs exso wth the 78 78(4 78 55) form zero followed by ldrome Theorems 0 d hve some terestg lctos regrdg fte seres I rtculr, we wll show tht cert tyes of fte seres coverge to rrtol umber by observg tht ther cotued frcto exsos re fte However, we wll frst wt to defe some tools to d us the roof We strt by defg fucto, L, o the rel umbers If x, the Lx ( ) s eul to the umber rtl uotets the cotued frcto exso of x whe the lst rtl uotet s ot eul to oe We sy Lx ( ) f x s rrtol Tht s, f, L([,,,, ]) 0 d L([,,,]) 0 For exmle, f x [3,, 5, 7, ] [3,, 5, 7,,] the Lx ( ) 5 If y, the Ly ( ) Next, we shll defe the vector V Suose r [ o,,,,, ], The V wll rereset ll the rtl uotets of r excet the frst two d the lst Tht s, V [,, ] R R Now let s deote the reverse of V by V Tht s, V [,,, ] It s mortt to R lso ote tht L( V ) L( V ) We wll ow use these tools to rove the followg theorem 33
Theorem 5 Let be ozero rtol umber The the fte seres to rrtol umber Furthermore, ths rrtol umber s ot udrtc rrtol Proof: Suose o [,,,, ] where coverges d s eve (The cse where s odd s [,, V, ] R very smlr) Now let V0 (, 3,, ) so V0 (,, ) d thus o 0 We ow tht 0 L V0 d L 0 By Theorem, [,,,,,, ] R o V0 V0 R Now let V V0,,, V0 so tht [ 0,, V, ] Note t ths stge, f the the fl rtl uotet of would be coverted to Ths would oly force us to use the secod Corollry of Theorem 0 sted of the frst The roof would be erly detcl Cotug o we see tht, 0 L V L V ( ), d so L 3 Oce g by Theorem, [,,,,,, ] R 0 V V so tht R Defe V V,,, V [,, V, ] Hece, 0 34
L V L V ( ) d L 3 Suose we cotue to defe V3, V4,, V s we dd bove Tht s, V0 (, 3,, ) R V j Vj,,, Vj for j 0 We shll ow rove by ducto tht for ech 0 the followg holds: L V, d d L We hve lredy show tht t holds for 0,, Suose t holds for ech of the tegers from 0 u to some teger 3 m 0 Vm R m The Vm Vm,,, Vm d [,,, ] By our ducto ssumto, m m L V d L m m m R Oce g from Theorem 0 we ow [ 0,, Vm,,, Vm, ] Sce m R V m Vm,,, Vm t follows tht [ 0,, Vm, ] So, m m m m L V L V ( ) d L m m 3 m 35
Hece, the result holds for ech teger 0 Ths shows us tht Therefore, we c coclude tht L s coverges to rrtol umber by Theorem 4 Also, sce ths fte cotued frcto exso s clerly ot erodc, Theorem 6 tells us tht the rrtol umber whch the seres coverges to cot be udrtc rrtol ( ) Usg Theorems 0 d, we ow the recse form of the exso of for y ostve teger I smlr mer to Theorem, we could use ths owledge to rove tht y fte seres of the form coverges to rrtol umber b where To see Theorem 5 cto, cosder the followg exmle b s y seuece wth b b for ll Exmle Cosder the seres 4 for some teger Usg 0 Theorem 0, we c clculte the cotued frcto exsos of the frst severl rtl sums of ths seres wthout dog y comutto They re gve below (7) 0 [0, ], [0,, ], [0,,,, ], 0 0 0 (8) 3 0 [0,,,,,,,, ], (9) 4 0 [0,,,,,,,,,,,,,,,, ] From (7) we see tht the frst rtl sum hs two rtl uotets d hece we ly the odd cse of the corollry of Theorem We the see tht 36
L 3 d L 3 5 so we swtch to the eve cse of the corollry 0 0 d from the o we cotue to use ths cse (8) shows us tht 3 L 5 9 d 0 from (9) we see tht 4 L 0 9 7 From these clcultos, t s ret tht L d hece 0 0 coverges to rrtol umber ( ) I the ext exmle we cosder the seres 3 3 7 5 d show tht t 3 3 3 3 0 coverges to rrtol umber smlr to the revous exmle However, ths tme we ly Theorem 0 to fd the exso of the rtl sums Ths yelds more terestg tter the exso Exmle By lyg Theorem 9 to the rtl sums of the seres followg: 0 3 ( ) we get the (0) 0 ( ) ( ) ( ) 3 [0,3], 3 [0,,,,3], 3 [0,,,,3,,,,,, ] 0 0 0 () 3 0 ( ) 3 [0,,,,3,,,,,,,,,,,,,,,3,,,], () 4 ( ) 3 [0,,,,3,,,,,,,,,,,,,,,3,,,,,,,,,3,,,,,,,,,,,,,,,3,,,] 0 We frst ly the odd cse of Theorem 0 to get the secod exso (0) Sce 3 3 3 3 ( )(3 ) our vlue of Theorem 0 s Note tht the clculto of ech successve rtl sum the vlue of we re usg s For exmle let s loo t how we go from the secod exso to the thrd exso (0) The rtol umber rereseted by the thrd 37
exso s 3 7 If we th of 3 3 3 Theorem 9 s 3 3 3 the 3 3 d hece we re ddg ( ) to t It s ot dffcult to see tht ths s the cse for ech rtl sum d hece why we get the exsos (0), (), d () Oce g we see tht the legth of these exsos s clerly gog to fty so ( ) 3 reresets rrtol umber 0 Let us oce g refer bc to the exmle 4 We strted wth the rtol umber x 9 whch hs exso[,3,,3,4] Observe tht Tble, oly oce ws greter th 8 dd ech exso strt wth[,3,,3,4,], the etre exso of x The ext few theorems swer the uestos of exctly wht vlue eeds to be dded to x order for ths to occur It turs out tht the swer to ths uesto deeds o the - st coverget d oce g the rty of Theorem 6 If x [ 0,,,, x ] d we th of x s fucto f( x ), deedg o x the o the tervl, we hve f( x ) s A cotuous mootoclly decresg fucto whe s eve A cotuous mootoclly cresg fucto whe s odd Proof: We hve, ( x ) ( x ) F ( x ) ' ( x ) ( x ) (3) ( ) ( x ) by Theorem () 38
From (3) t s ow cler tht whe s eve F s lwys egtve d whe s odd F s lwys ostve, thus the result follows Theorem 7 Suose [ 0,,, ] The the tervl o the rel le wth cotued frcto exso of the form [ 0,,,, b, b,] where b s ostve teger for ech s:, f s odd, f s eve Proof: Suose f ( x) [ 0,,,, x] where x [ b, b,] The f hs dom, d by Theorem 3 f s cresg fucto whe s odd d decresg fucto whe s eve By Theorem (), x f( x) x Sce f () d lm f( x), x the result follows Note tht Theorem 7, ths s oe of the rre cses where we do t ut the restrcto tht So oe hs to choose crefully f the desred form of the exso hs or ot Sce [ 0,,,,] s eul to but hs oe more coverget d rtl uotet th 39
[,,, ] ; t must be uderstood wht the rty of s d wht the vlues of the 0 covergets d re, s they re dfferet deedg o wht your desred form s Mg use of Theorem 7 leds us the to the followg theorem, whch ow swers the uesto of wht uttes we eed to dd to rtol umber order to reserve ll or most of ts rtl uotets Theorem 8 Suose [ 0,,,, ] the ( ) r where 0 r hs ( ) cotued frcto exso of the form[ 0,,,,, b, b,] Proof: From Theorem 7, we ow tht order for the exso of desred form, we eed ( ) r to hve the (4) ( ) r, f s odd, f s eve Observe tht ( ), By Theorem () ( ) ( ) Hece, (5) ( ) ( ) Sce s the uer or lower boud o the tervls bove, we see from (4) d (5) tht y r such tht 0 r wll hve the desred exso form ( ) 40
I vestgtg uttes of the form r for smll vlues of r gve tht [ 0,,, ] we c see from (4) tht whe s odd, ddg y ostve umber r, o mtter how smll, wll result exso tht does ot reserve every rtl uotet of However, recll tht y rtol umber hs recsely two exsos Tht s, f the [,,, ] [,,,,] Wth roer re-dexg ths ow chges the rty of d 0 0 ddg smll eough uttes to ths wll ow reserve every rtl uotet From ths comes bout the followg corollry to Theorem 7 Corollry Suose [ 0,,,, ] where The tervl o the rel le wth cotued frcto exso of the form [ 0,,,,,, b, b 3,] s, whe s odd, whe s eve Also, ( ) s where 0 s ( ) wll hve cotued frcto exso of the form [,,,,,, b, b,] 0 3 We c get other ce result by observg tht sce, t follows tht ( ) We use ths to gve corollry to Theorem 8 4
Corollry Suose [ 0,,, ] Now c, f ecessry, so tht s eve The, r b [ 0,,,,,] f 0 r, the r [ 0,,,, b,] f r 0 If sted we choose the exso wth odd Notce tht Theorem 8 we strted wth rtol umber d gve the rel umber r so tht ( ) r hd cotued frcto exso whch strted wth the etre cotued frcto exso of Ths begs the uesto of wht hes f we relce wth some rrtol umber x I ths cse x hs fte cotued frcto exso so the exso of x rclerly cot beg wth the etre exso of x We sted s for wht vlues of r wll the cotued frcto exsos of x d x rhve the sme frst rtl uotets? Tht s, for wht vlues of r does x [ 0,,,,,,] d x r [ 0,,,, b, b,]? To fd recse swer we c use Theorem 7, however, the Theorem below gves cer result Theorem 9 Suose the rrtol umber x hs cotued frcto exso of the form [ 0,,,,,,] where The the cotued frcto exso of x ( ) r where r 3 wll hve exso of the form[ 0,,,,, b,] Proof: We rove the cse where s eve Accordg to Theorem 7 we ow tht, sce s eve, we eed to rove tht xr, x () x It s cler tht x x Let x [ 0,,,,, x ] so by Theorem r so we just eed to show tht x x r, or r x 4
Observe tht x ( ) x ( ) x ( x )( ) x ( x )( ) by Theorem () x Now the fucto defed by f( x ) s mootoclly cresg o ( x )( ) the tervl, d therefore tes o mmum vlue t f () 0 However, sce we reure the x The f () 3 wor d so clerly y r smller wll lso Corollry If x s rrtol umber d hs exso of the form [ 0,,,,,,] where, the x ( ) r wll hve exso of the form [ 0,,,,, b,] r 6 f Exmle 3 Suose 5 x [,,3,,3,,3,] [,,3] d we wsh to dd some utty r so s to reserve the frst 5 rtl uotets of x We fd tht 4 55, d 3 6, so r 4 ( ) 55 3556 6 8946 Now the frst 7 rtl uotets of x r [,,3,,3,,, ] If we sted c slghtly lrger vlue of r such s 55 556 6930 the the frst 7 rtl uotets of x r re [,,3,,4,,, ] Notce tht ths tme the frst 5 rtl uotets of r re ot reserved re The result gve Theorem 9 s ot otml, sce the otml vlue for such r to reserve the frst rtl uotets would smly be foud by usg Theorem 7 d some lgebr However, ths vlue deeds o the vlue of the tl rrtol umber x whle the vlue for r 43
gve Theorem 9 oly deeds o d It s cler tht we c fd eve smller vlues for r to substtute to Theorem 9 We leve tht for future wor Gve tht the exso of s [ 0,,, ], the followg tble summrzes the cotued frcto exsos of vrous rtol umbers tht were dscussed ths er Rel Number Prty of N Cotued Frcto Exso ( ),,,,,,,,,,, ( ) ( ) ( ) eve 44 0 odd,,,,,,,,,,, 0 eve,,,,,,,,,,, 0 odd,,,,,,,,,,, 0 eve 0,,,,,,,,, odd 0,,,,,,,,, eve 0,,,,,,,,, odd 0,,,,,,,,, ( ) ( ) ( ) ( ) ( ) ( ) eve,,,,,,,,,, 0 odd,,,,,,,,,, eve odd eve odd Tble 4 0 [,,,,,,,, ] 0 [,,,,,,,, ] 0 [,,,,,,, ] 0 [,,,,,,, ] 0
Chter 5 - Future Wor I Exmle 7 we looed t the cotued frcto exsos tht er the rtl sums of the seres for secfc vlues of d From studyg ths exmle mog others, t becme evdet tht much more reserch c be doe o the tters foud the exsos of As metoed t the ed of Exmle 7 we mde severl cojectures regrdg these exsos To hel us llustrte ths, cosder the followg tbles whch gve the exso of for vrous vlues of,, d Exresso Exso / = 37/89 [,,, 5,, 6] = [,,, 5,, 5, 7,, 5, ] = 3 [,,, 5,, 5, 8,,, 3,,, 8, ] = 4 [,,, 5,, 5, 8,,,, 89,,,, 8, ] = 5 [,,, 5,, 5, 8,,,, 800,,,, 8, ] = 6 [,,, 5,, 5, 8,,,, 7979,,,, 8, ] Tble 5 Exresso Exso / = 388/93 [4, 5,, 4, 3] = [4, 5,, 4, 4,, 4,, 5] = 3 [4, 5,, 4, 4,, 9,, 8, 5, 6] = 4 [4, 5,, 4, 4,, 9,, 845, 5, 6] = 5 [4, 5,, 4, 4,, 9,, 78686, 5, 6] = 6 [4, 5,, 4, 4,, 9,,737899, 5, 6] Tble 6 Exresso / = 73/458 [0, 6, 3,,,, 6] Exso 45
= [0, 6, 3,,,, 7, 5,,,, 3, 6] = 3 [0, 6, 3,,,, 7, 5,,, 5, 3, 3,,, 3,,, 6] = 4 [0, 6, 3,,,, 7, 5,,, 5, 3, 835,,, 3,,, 6] = 5 [0, 6, 3,,,, 7, 5,,, 5, 3, 84089,,, 3,,, 6] = 6 [0, 6, 3,,,, 7, 5,,, 5, 3, 3858539,,, 3,,, 6] Tble 7 From Tbles 5, 6, d 7 we see tht ech exso evetully becomes fxed rt from oe rtl uotet If we cut off the cotued frcto of ech rght before the o-fxed rtl uotet, the the vlue of ths rtol umber s wht the seres coverges to Now let s observe the o-fxed rtl uotets gve ech tble I Tble 5 they re 89, 800, d 7979 Smlr to Exmle 7 they c be wrtte s: (6) 89 (89 ) 800 (89 89 ) 3 7979 (89 89 89 ) I Tbles 6 d 7 the o-fxed rtl uotets gve re 8, 845, 78686, 737899 d 3, 835, 84089, 3858539 resectvely Oce g observe tht 8 9() 845 9(93 ) (7) 78686 9(93 93) 73789 3 9 9(93 93 93) d 3 4() 835 4(458 ) (8) 84089 4(458 458 ) 3858539 4(45 3 8 ) 458 458 Frst, otce from (6), (7), d (8) tht the o-fxed rtl uotets hve the form j j ( ) for some j I ech cse bove, c be foud by fdg [gcd(, )] For stce, gcd(37,89), gcd(388,93) 9 d gcd(73, 458) 4 46
Secodly, observe tht the lst set of fxed rtl uotets s eul to ' ' where s the deomtor corresodg to the - th coverget of I other words, t s the sme s the exso of reverse order f we gore the frst rtl uotet I ddto, we sometmes hve to strt wth the form of the exso tht eds For exmle, usg the rtol umbers from Tbles 5, 6, d 7 we see tht 36/89 = [,,, 8,, ] = [,,, 8,,, ], 387/93 = [4, 6, 5], d 7/458 = [0, 6,,, 3, 3] = [0,6,,, 3,, ] resectvely Wrtg these reverse order whle gorg the frst rtl uotet gves [,,, 8,, ] = [,,, 8, ], [5, 6], d [,, 3,,, 6] ll of whch er the tbles bove We ow gve the followg cojecture summrzg the formto reseted bove Cojecture Suose d re rtol umbers greter th d the seres u coverges to some rtol umber [ 0,,, ] v where Further ssume tht the exso of s [ b0, b,, b ] The for some dex, j, where j s 3 or 4: j j3 [ 0,,,,gcd(, ),, b, b, b ] 0 or j j3 [ 0,,,,,gcd(, ), b, b, b ] 0 Also for ech teger m j, we hve: m m3 [ 0,,,,gcd(, ),, b, b, b ] 0 or 47
m m3 [ 0,,,,,gcd(, ), b, b, b ] 0 Future wor would volve determg f ths cojecture s true, d f so, rovdg roof It would lso volve exlorg other tters tht rse the exsos of rtl sums of seres At the ed of Chter 4 we gve result tht gve the rrtol umber x [,,,,,,], the the sum of x d the rtol umber r hs exso tht 0 reserves the frst rtl uotets of x f r s smll eough Future wor ths re would volve ushg the lmts o how lrge vlue of r we c fd tht stll hs ths roerty 48
Refereces [] Hrdy, G H, d Wrght, EM A Itroducto to the Theory of Numbers, 4th ed, Oxford: Clredo Press, 960 [] Jcobse, Mchel, d Hugh Wllms Solvg the Pell Euto New Yor; Lodo: Srger, 008 [3] Khch, A Y Cotued Frctos, 3 rd ed, Chcgo: Uversty of Chcgo Press, 964 [4] Khovs, Alexey N The Alcto of Cotued Frctos d ther Geerlztos to Problems Aroxmto Theory The Netherlds: P Noordhoff, 963 [5] LeVeue, Wllm J Dohte Aroxmto Fudmetls of Number Theory Redg, MA: Dover Publctos, Ic, 977 9-48 [6] Olds, CD Cotued Frctos New Yor: Rdom House, Ic, 963 49