Patterns of Continued Fractions with a Positive Integer as a Gap

Transcription

1 IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: , -ISSN: X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet Arts College, Trchy -, Id) (Mthemtcs, Seethlshm Rmswm College, Trchy, Id) Abstrct : I ths er we detfy the tters of cotued frctos of rtol umbers wth Here we try to fd mmum umber of dstct tters tht wll gve rse to the remg tters Keywords: Cotued frcto lgorthm, Cotued frctos, Euclde lgorthm, Euler s formul, Smle cotued frctos Subject Clssfcto: MSC A5, A7, A55, 3B7, 4A5 Nottos:,,, Cotued frcto exso, 3 Iteger rt of the rtol umber m m 3 () Euler s totet fucto 4 gcd(, m ) Gretest commo devsor of d m I Itroducto The org of cotued frctos s trdtolly lced t the tme of the creto of Eucld's Algorthm Due to ts close reltosh to cotued frcto the creto of Eucld's Algorthm sgfes the tl develomet of cotued frctos Euler showed tht every rtol c be exressed s termtg smle cotued frcto He lso rovded exresso for e cotued frcto form He used ths e re rrtol Ay fte smle cotued frcto reresets rtol exresso to show tht e d umber Coversely y rtol umber c be exressed s fte smle cotued frcto d exctly two wys Frst we gve dfferet reresettos of rtol umber s cotued frcto A exresso of the form where b b b b3 3, b re rel or comlex umbers s clled cotued frcto A exresso of the form where, d,, re ech ostve tegers s clled smle cotued frcto b, The cotued frcto s commoly exressed s 3 DOI: 979/ wwwosrjourlsorg Pge

2 The elemets Ptters of Cotued Frctos Wth A Postve Iteger s A G 3 or smly s,,,, 3,,, re clled the rtl uotets If there re fte umber of rtl uotets,, 3 we cll t fte smle cotued frcto otherwse t s fte We hve to use ether Euclde lgorthm or cotued frcto lgorthm to fd such rtl uotets Oe essetl tool studyg the theory of cotued frcto s the study of coverget of cotued frcto Some smle cocets used ths er re gve below If,,, 3, s fte seuece of ostve tegers, excet ( my or my ot be zero), we defe two seuece of tegers h d ductvely s follows, xh h The s roved [5] for y ostve rel umber x,,,,,, x () x h Also t s oted [5] tht f we defe r,,,, for ll tegers, the r () It s observed [6] tht the rtol umber h,,, r (3) s clled the th coverget to the fte cotued frcto II Method Of Alyss I ths er we try to fd some tters of cotued frctos of rtol umber wth g, where s y ostve teger whch reresets the dfferece betwee d Ay rtol umber my be cosdered the form m, where m, d gcd(, m ),,,3, m We dscuss the tters bsed o the vlue of m Four ossble vlues for m re detfed They re gve below If m, m c be te y oe of the vlues,, 3, d hece the umertors of the rtol umbers re cogruet to,,3, modulo, whch flls y oe of the four cses Cse (): Whe m, the cotued frcto of s,, Cse (): Whe m, the cotued frcto of Cse (): Whe m, the cotued frcto of m m m r s,,,,,, mr m m m mr ( ) s,,, Cse (v): Whe m, d m, the cotued frcto of m h h h, h h DOI: 979/578-XXXXX wwwosrjourlsorg Pge

3 Ptters of Cotued Frctos Wth A Postve Iteger s A G m,,,,,,, m mr s m m m mr, mr Proertes observed of the rtol umbers dscussed the bove four cses re: () The umertor of rtol umber exressed cse () s cogruet to modulo d cse () s cogruet to m modulo wheres cse () d cse (v) the umertors re cogruet to ( ) d ( m) modulo resectvely (b) I cse () d cse () the frst three rtl uotets re sme The fourth rtl uotet of cse () s subdvded to d ( ) cse () Smlrly the fourth rtl uotet m subdvded to d cse (v) (the remg rtl uotets re fxed) m of cse () s (c) The sum of the umertors of cse () d cse () s cogruet to zero modulo Smlr results hold for cse () d cse (v) (d) Sce gcd(, m ) d deeds o choce of m, the umber of tters exst s () Sce () s eve we c r the tters such wy tht the sum of the umertors of such rs re dvsble by d we sy tht the rs re relted THEOREM: If the cotued frcto exso cotued frcto exso Proof: Cosder,,,, =,,, reresets the rtol umber,,,, reresets the rtol umber = THEOREM: For y ostve teger, f the cotued frcto exso umber, where ( ) ( ),,, reresets the rtol umber, Proof: Let ( ) the the,, reresets the rtol s ostve teger the the cotued frcto exso d Sce the cotued frctos exso re uue, ( ),, =,, where s ostve teger DOI: 979/578-XXXXX wwwosrjourlsorg Pge

4 Ptters of Cotued Frctos Wth A Postve Iteger s A G =,, =, =, = = ( ) Hece the cotued frcto exso,, reresets the rtol umber ( ) Suose,,,, =,, x, where x =, = = x Usg (),,,,, =,, x x Sce the coverget of d re sme uto, where th s the coverget of d th coverget of we get, x,, x x d Sce,, Hece the cotued frcto exso where s ostve teger s the ( ) ( ),,, reresets the rtol umber, 3 THEOREM: For y ostve teger, f the cotued frcto exso m m r m,,,,,, mr reresets the rtol umber, m m mr ( m ),,, where s ostve teger the the cotued frcto exso ( ) DOI: 979/578-XXXXX wwwosrjourlsorg 3 Pge

5 Ptters of Cotued Frctos Wth A Postve Iteger s A G,,,, m mr,,,, mr reresets the rtol umber m m mr ( m ), where m d gcd(, m ), s ostve teger m Proof: Let Comrg the cotued frctos of x d y, d usg (theorem ) we get y m Hece,,, y,, m Proceedg s the revous theorem we get m,, y m r Hece the cotued frcto exso,,,,,,,, m reresets ( m ) m DOI: 979/578-XXXXX wwwosrjourlsorg 4 Pge m the rtol umber, where s ostve teger III Illustrto The followg tble gves the tters of the cotued frcto of the rtol umbers wth Here the umber of tters exst s ( ) Let t be,,,,,,,, Rtol umber m mr,, x, where x,,, m m mr,,,,, m mr Let y where y,,, m m mr m m m m r, Cotued frcto exso :,,, :,,,5, :,,,3,, :,,,,, :,,,, :,,,,, :,,,,,, :,,,,,, 9 r

6 Ptters of Cotued Frctos Wth A Postve Iteger s A G 9 9 :,,,,4, :,,,, Here the tters, ;, 9; 3, 8; 4, 7,; 5, 6 re relted Hece t s eough to fd the tters to 5 whle the remg tters 6 to re foud by the rocedure stted theorem ( d 3) IV Cocluso The dstct tters of cotued frctos of rtol umber wth re foud here To fd ll () tters of cotued frctos of the rtol umber wth g, t s eough to fd the frst () tters The remg () tters re foud by the relto metoed bove Refereces [] A Y Khch Cotued Frctos, Dovers boos o mthemtcs,997 [] Brezs, Clude, Hstory of Cotued Frctos d Pde Aroxmts, Srger- Verlg: New Yor, 98 [3] Dvd M Burto Elemetry Number Theory seveth edto Mcgrw Hll Educto [4] George E Adrews, Number Theory, WB Suders Comy [5] Iv Nve, Herbert S Zucerm Hugh L Motgomery, A troducto to theory of umbers, Ffth edto Wley Studet Edto [6] Joth Browe Alfvder Poorte Jeffrey Shllt, Wdm Zudl Neveredg Frctos A Itroducto to cotued frctos, Cmbrdge Uversty Press [7] Joseh H Slverm A fredly troducto to Number Theory, fourth edto [8] Nevlle Robbs, Begg Number Theory, Secod edto Nros Publshg House [9] Olds, CD, Cotued Frctos, Rdom House: New Yor, 963 [] Pettofrezzo, Athoy J, Byrt, Dold R, Elemets of Number Theory, Pretce-Hll Ic:Eglewood Clffs, NJ, 97 [] Rose Keeth H, Elemetry Number Theory d ts Alctos, Addso-Wesley Publshg Comy: New Yor, 98 [] Cotued frctos o web: htt://rchvesmthutedu/ tuyl/cofrc/hstoryhtml [3] Cotued frctos o web: htt://rchvesmthssurveycu/hosted stes/rkott/fbocc/cfintrohtm#secto DOI: 979/578-XXXXX wwwosrjourlsorg 5 Pge