Chiang Mai J. Sci. 2014; 41(2) 457 ( 2) ( ) ( ) forms a simply periodic Proof. Let q be a positive integer. Since

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1 56 Chag Ma J Sc 0; () Chag Ma J Sc 0; () : Cotrbutd Papr Th Padova Sucs Ft Groups Sat Taș* ad Erdal Karaduma Dpartmt of Mathmatcs, Faculty of Scc, Atatürk Uvrsty, 50 Erzurum, TURKEY *Author for corrspodc; mal: sattas@ataudutr ad duma@ataudutr Rcvd: 5 March 0 Accptd: Novmbr 03 ABSTRACT I ths papr, w study th Padova sucs modulo m Also, w df th Padova orbt of a -grator group G for a gratg par ( xy, ) Gth w xam th lgths of th prods of th Padova orbt Furthrmor, w obta th Padova lgths of th polyhdral groups (,, ), (,, ), (,, ), (,, ) ad th bary polyhdral groups,,,,,,,,,,, for gratg par ( xy, ) Kywords: Padova suc, Padova lgth, polyhdral group INTRODUCTION Th Padova umbrs ad thr proprts hav b studd by som authors, s for xampl, [,9,] I ths mar, th study of Fboacc sucs groups bga wth th arlr work of Wall [0] Th thory s xpadd to 3-stp Fboacc suc by Özka, Ayd ad Dkc [8] Lü ad Wag [] cotrbutd to study of th Wall umbr for th k-stp Fboacc suc Campbll, Doost ad Robrtso [] xpadd th thory to som smpl groups Thy dfd th Fboacc lgth of th Fboacc orbt ad th basc Fboacc lgth of th basc Fboacc orbt a -grator group Kox [6] xpadd th thory to k-acc (k-stp Fboacc) sucs ft groups Dvc ad Karaduma [] dfd th basc k-acc sucs ad th basc prods of ths sucs ft groups Dvc ad Karaduma [3] xtdd th cocpt to Pll sucs Dvc [] studd o Th Pll-Padova sucs ad th Jacobsthal-Padova sucs ft groups Now w xtd th cocpt to th Padova sucs A suc of group lmts s prodc f, aftr a crta pot, t cossts oly of rpttos of a fxd subsuc For xampl, x, x,, s prodc wth prod A suc s smply prodc wth prod k f th frst k lmts th suc form a rpatg subsuc For xampl, x, x, x 3, x, x,, s smply prodc wth prod Th Padova suc s th suc of tgrs P( ) dfd by tal valus P( 0) = P = P = ad rcurrc rlato P = P( ) + P( 3) () Th Padova suc s,,,,,3,,5,,9,, Th Padova umbrs ar gratd by a matrx Q, 0 0 Q=0 0, 0

2 Chag Ma J Sc 0; () 5 Th powrs of Q gv P( 5) P( 3) P( ) Q P( ) P( ) P( ) P( 3) P( ) P( ) = 3 () For mor formato o ths suc, s [5] I ths papr, th usual otato p s usd for a prm umbr THE PADOVAN SEQUENCE MODULO m Rducg th Padova suc by a modulus m, w ca gt a rpatg suc, dotd by ) { } ) ) ) P = { P 0, P, P (, ), P ),, } whr P ) P od m) It has th sam rcurrc rlato as () - 3) = 0, P (m) (kp(m) - ) =, P (m) (kp(m) - ) = 0, P (m) (kp(m)) -, P (m) (kp(m) + ) -, P (m) (kp(m) + ) - So, w ar do For gv a matrx A= a j wth a ( k+ ) ( k+ ) j s bg tgrs, Aod m ) mas that vry trs of A ar rducd modulo m, that s, α Aod m) ( aj od m) ) Lt Q α = { Q od p ) 0 p } b a cyclc group ad Q p α dot th ordr of Q p α From (), t s asy to s that α kp( p ) = Q p α Thorm Lt t b th largst postv tgr such that kp( p) = kp( p ) Th t α α t kp p = p kp p for vry α t I partcu- α α lar, f kp( p) kp( p ), th kp( p ) = p kp( p) holds for vry α > ( Thorm { P ) ( )} forms a smply prodc suc Proof Lt b a postv tgr Sc + kp( p ) + Q I od p ), that s, kp ( p + ) Q I od p ), Proof Th suc rpats sc thr ar w gt that kp( p ) dvds kp( p + ) O th 3 kp oly a ft umbr m of trpls of trms othr had, wrtg ( p ) Q = I + ( aj p ), w hav possbl, ad th rcurrc of trpl rsults ( kp p p p p p + Q = I + ( a p j )) = ( aj p ) I od p ), = 0 rcurrc of all followg trms From whch ylds that kp( p + ) dvds kp( p ) p Thrfor, kp( p ) = kp( p ) or kp( p ) = kp( p ) p, dfto of th Padova suc w hav + + P( ) = P P( 3) so f ) P ( + ) ( j+ ), ad th lattr holds f ad oly f thr s ) P ( + ) ( j+ ) ad ) P ( j), th a a j whch s ot dvsbl by p Sc ) P ( j+ ), ) P ( j+ ) ad t t kp ) P ( j) ( p ) kp( p + ( t ) ), thr s a a + j whch s ( 0), whch mpls that th suc { P ) t+ t+ ot dvsbl by p, thus, kp ( p ) kp( p ) ( )} s smply prodc Th proof s fshd by ducto o t Lt th otato kp ) dot th smallst prod of { P ) t ( )}, calld th prod of th Thorm 3 If m= p ( t ) whr p 's ar = Padova suc modulo m dstct prms, th kp) = Icm kp( p ), th last commo multpl of th kp ( Exampl { P 3 ) ( p ) ( )} = {,,,,, 0,,,, 0, 0,, 0,,,,,,, } Thus, kp ( 3) = 3 As a cosuc of Proof Th statmt, kp( p ) s th lgth th Thorm W gv th followg, ( p ) of th prod of { } P, mpls that th ( p ) suc { } P rpats oly aftr blocks Corollary of lgth u kp p, u N; ad th statmt, { P m ( kp ), ) P m ( kp ) 6, ), P m ( kp )), kp ) s th lgth of th prod { P ) ( )}, ) ) P ( kp) +, ) P ( kp) + ) } ={, m,, 0, 0,, 0,, ( p ) mpls that { }, } I fact, bcaus {P ()} P rpats aftr kp ) for all (m) s smply prodc, valus Thus, kp ) s of th form u kp( p w hav P (m) (0) =, P (m) () =, P (m) ) () =,, for all valus of, ad sc ay such umbr P (m) (kp(m) - ) =, P (m) (kp(m) - 6) = -, P (m) gvs a prod of { P ) ( )} Th w gt that (kp(m) - 5) =, P (m) (kp(m) - ) = 0, P (m) (kp(m) = Icm kp m kp p

3 58 Chag Ma J Sc 0; () Cojctur Thr xsts a j wth 0 j s som applcatos of th Cojctur Tabl Tabl : Th lgth of kp(p) such that kp( p ) dvds ( p 3 p j )It ca b p kp(p) trsult THE PADOVAN LENGTH OF GENERATING PAIRS IN GROUPS Lt G b a group ad xy, G If vry lmt of G ca b wrtt as a word u u u3 u um um x y x y x y whr u Z, m, th w say that x ad y grat G ad that G s a -grator group Lt G b a ft -grator group ad X b th subst of G G such that ( xy, ) Xf ad oly f G s gratd by x ad y W call ( xy, ) a gratg par for G Dfto 3 For a gratg par ( xy, ) G, w df th Padova orbt P ( G) = { x} by xyy,, x = x, x = y, x = y, x = x x, 0 + Thorm 3 A Padova orbt of a ft group s smply prodc Proof Lt b th ordr of G Sc thr

4 Chag Ma J Sc 0; () 59 3 dstct 3-tupls of lmts of G, at last o of th 3-tupls appars twc a Padova orbt of G Thus, th subsuc followg ths 3-tupls Bcaus of th rpatg, th Padova orbt s prodc Sc th Padova orbt prodc, thr xst atural umbrs ad j, wth > j, such that x+ = xj+, x+ = xj+, x+ 3 = xj+ 3 By th dfg rlato of a Padova orbt, w kow that x = ( x ) ( x+ ) ad xj = ( xj ) ( xj+ ) Hc, x = xj, ad t th follows that x = x = x, x = x = x, x = x = x j j j 0 j+ j j+ j+ j j+ Thus, th Padova orbt s smply prodc W dot th lgth of th prod of th Padova orbt Pxyy,, ( G ) by,, ( G ) ad w call th Padova lgth of G wth rspct to gratg par ( xy, ) Thorm 3 Th Padova lgth of Z Z m (whr Z = x ad Z m = y ) uals th last commo multpl of Proof By th dfto of drct product, w m gt th followg prstato x, y : x = y =, xy = yx Hr th start th Padova orbt s x = x, x = y, x = y, x = xy, x = y, x = xy, x = xy, x = xy, x = x y, Now th proof s fshd wh w ot that th Padova orbt wll rpat wh x = α x, x = α + y, xα + = y Examg ths statmt mor dtal gvs P( α 5) P( α ) x y = x = x0, P( α ) P( α) x y = y = x, P( α 3) P( α+ ) x y = y = x Th last o-trval tgr satsfyg th abov codtos occurs wh α = Icm kp, kp) APPLICATIONS Dfto Th polyhdral group ( lm,, ) for l, m, >, s dfd by th prstato l m x, y, z : x = y = z = xyz = or l m x, y : x = y = ( xy) = Th polyhdral group ( lm,, ) s ft f ad oly f th umbr k = lm + + = m + l + lm lm l m s postv Its ordr s lm k For mor formato o ths groups, s [3, pp6-68] Dfto Th bary polyhdral group lm,,, for l, m, >, s dfd by th prstato l x, y, z : x = y m = z = xyz or x, y : x l = y m = ( xy) Th bary polyhdral group lm,, s ft f ad oly f th umbr k = lm + + = m + l + lm lm s postv Its l m ordr s lm k For mor formato o ths groups, s [3, pp68-] Now w obta th Padova lgths of th polyhdral groups (,, ), (,, ), (,,,), (,,) ad th bary polyhdral groups,,,,,,,,,,, for gratg xy par (, ) Thorm Th Padova lgth of th polyhdral group (,, ) s Proof From Thorm 3 t s asy to s that LP xyy,, ((,,) ) = sc (,, ) Z Z ad kp = Thorm Th Padova lgth th polyhdral group (,,) for > s as follows:, 0 mod 8, LP,, (),,, mod 8, xyy =, mod 8,, othrws

5 60 Chag Ma J Sc 0; () Proof W frst ot that th group dfd by x, y : x = y = ( xy) =, x =, y = ad xy = Th Padova orbt s x, y, y, xy, y, x, xy, y x, y, y, xy, y, xy, xy, xy, y, y, Usg th abov, th Padova orbt bcoms: x0 = x, x = y, x = y,, 3 5 x = y x, x8 = y, x9 = y,, 8 9 x = xy, x5 = y, x6 = y,, x = xy, x + = y, x + = y,, x = y x, x = y, x = y, So w d a N such that 8 = u for u N If 0 mod 8, thr ar two subcass: Frst cas: If 0 mod 8, th = So w 8 gt,, ((,,) ) = Scod cas: If mod 8, th = So 8 w gt,, ((,,) ) = If mod 8, th = So w gt,, ((,,) ) = If mod 8 or 6 mod 8 th = So w gt,, ((,,) ) = If s odd th = So w gt,, ((,,) ) = Thorm 3 Lt G b ayo of th polyhdral groups (,,) ad (,, ) for > Th, 0 mod,,, ( G) =, mod,, othrws Proof Frstly, lt us cosdr th polyhdral group (,,) W frst ot that th group dfd by x, y : x = y = ( xy) =, x =, y = ad xy = If 0 mod,,, ((,,) ) = If mod,,, ((,,) ) = If s odd,,, ((,,) ) = Scodly, lt us cosdr th polyhdral group (,, ) W frst ot that th group dfd by x, y : x = y = ( xy) =, x =, y = ad xy = If 0 mod,,, ((,, ) ) = If mod,,, ((,, ) ) = If s odd,,, (,, ) = Th proofs ar smlar to th proof of Thorm ad ar omttd Exampl For =, Th Padova lgth of th polyhdral groups (,, ) s W frst ot that th group dfd by x, y : x = y = ( xy) =, x =, y = ad xy = So, th Padova suc th (,, ) s x, y, y, xy,, xy, y, x, xy, x, y, y,, ad th Padova lgth of th polyhdral groups (,, ) s Thorm Th Padova lgth of th bary polyhdral group,, s Proof W frst ot that th group tdfd by x, y : x = y = ( xy), x =, Th Padova orbt s x y y xy y x xy x y y x y y x xy x y y 3 3 3,,,,,,,,,,,,,,,,, So w gt xyy LP,,,, = Thorm 5 Th Padova lgth of th bary polyhdral group,, for > s as follows:, 0 mod,,, (,, ) =, mod,, othrws Proof W frst ot that th group dfd by x, y : x = y = ( xy), x =, y = ad xy = Th Padova orbt s x, y, y, xy, y, x, xy, y x, x y, x y, x y, y, xy, x y, xy, y, y,

6 Chag Ma J Sc 0; () 6 Usg th abov, th Padova orbt bcoms: x = x, x = y, x = y,, 0 x= yxx, = xy, x= xy,, x = xy, x = y, x = y,, x = xy, x = y, x = y,, x = y xx, = xy, x = xy, So w d a N such that 8 = v f for v N If 0 mod, thr ar two subcass: Frst cas: If 0 mod 8, th = So w gt,, (,, ) = Scod cas: If 0 mod 8, th = So w gt,, (,, ) = If mod, th = So w gt,, (,, ) = If s odd, th = So w gt,, (,, ) = Thorm 6 Lt G b ayo of th bary polyhdral groups,, ad,, for > Th LP xyy,, ( G ), s v, =, s odd Proof Frstly, lt us cosdr th bary polyhdral group,, W frst ot that th group dfd by x, y : x = y = ( xy), x =, y = ad xy = If 0 mod,,, (,, ) = If mod,,, (,, ) = If s odd,,, (,, ) = Scodly, lt us cosdr th bary polyhdral group,, W frst ot that th group dfd by x, y : x = y = ( xy), x =, y = ad xy = If 0 mod,,, (,, ) = If mod,,, (,, ) = If s odd,,, (,, ) = Th proofs ar smlar to th proof of Thorm 5 ad ar omttd Exampl For = 3, th Padova lgth of th bary polyhdral group,,3 s W frst ot that th group dfd by x, y : x = y = ( xy) 3, x =, y = ad xy = 6 So, th Padova suc th,,3 s x, y, y, xy, y, yxy, xy 3, y(xy), y(xy) 5, x(xy), (xy), xy 3, x(xy), (xy), x(xy), x(xy), (xy), y, y, xy, (xy) 5 y, (xy), y, (xy), y, yx 3,, (xy), x, yx 3, (xy), x, (xy), x, x, yx, (xy) 5, xyx, (xy), x, (xy), x,, yx 3, (xy), x, yx 3, x, y, y,, ad th Padova lgth of th bary polyhdral groups,,3 s REFERENCES [] Campbll C M, Doost H ad Robrtso E F, Fboacc Lgth of Gratg Pars Groups, Brgum GE t al, ds, Applcatos of Fboacc Numbrs, Vol 3, Kluwr Acadmc Publshrs, 990: - 35 [] Dvc Ö ad Karaduma E, O th basc k-acc sucs ft groups, Dscrt Dy Nat Soc, 0; [3] Dvc Ö ad Karaduma E, Th Pll sucs ft groups, Utl Math, to appar [] Gl TB, Wr MD ad Zara C, Complt padova sucs ft flds, Th Fboacc Quartrly, 00; 5: 6-5 [5] Suchtml [6] Kox S W, Fboacc sucs ft groups, Th Fboacc Quartrly, 99; 30: 6-0 [] Lü K ad Wag J, k-stp Fboacc suc modulo m, Utl Math, 00; : 69-8 [8] Özka E, Ayd H ad Dkc R, 3-stp Fboacc srs modulo m, Appl Math, Comp, 003; 3: 65-

7 6 Chag Ma J Sc 0; () [9] Sato S, Taaka T ad Wakabayash N, Combatoral rmarks o th cyclc sum formula for multpl zta valus, J Itgr S, 0; : [0] Wall DD, Fboacc srs modulo m, Am Math Mothly, 960; 6: [] d Wgr BMM, Padua ad psa ar xpotally far apart, Publcacos Matmàtus, 99; : [] Dvc Ö, Th Pll-Padova sucs ad Th Jacobsthal-Padova sucs ft groups, Utl Math, to appar [3] Coxtr HSM ad Mosr WOJ, Grators ad Rlato for Dscrt Groups, 3 rd Ed, Sprgr, Brl, 9