On the periodic continued radicals of 2 and generalization for Vieta s product

Transcription

1 O the erodc cotued radcal of ad geeralzato for Veta roduct Jayatha Seadheera Abtract I th aer we tudy erodc cotued radcal of We how that ay erodc cotued radcal of coverge to q, for ome ratoal umber q deed o the cotued radcal Furthermore we how that f r a erodc eted radcal of, whch ha eted root, the the lmt ot of the equece ( ) q r have the form α, where α a algebrac umber Th reult gve a et of ubequece coverge toα, for eachα Alo we how that lmt of thee ub equece ca be rereeted a Veta le eted radcal roduct Hece th reult geeralze the Veta roduct for Several teretg examle are llutrated Itroducto Oe obectve of th aer to etablh bac theorem to udertad the behavor ad tructure of cotued radcal (fte eted radcal) of The other obectve to geeralze well ow Veta roduct for I mathematc lterature we ca fd everal attemt related to the frt obectve Amog thee, roblem 76, 77, 78 of Polya ad Zego excellet boo [] the romet oe I the at htory, tme to tme everal author cocered about th toc [],[3] Edouard Luca ued eted radcal of to develo h method for rmalty tetg [] Alo the recet aer[5],[6],[7] ad [8] are o thee le I th aer, ecto cocetrate o the frt obectve Mot of the reult ecto ca be foud above referece But for the ae of comletee ad for ome geeralzato, we rerove thoe reult At the ed of ecto, we gve a mle crtero to clafy all erodc cotued radcal of Bacally ecto cocetrate o the ecod obectve I ome ee t the atural exteo of ecto Surrgly th atural exteo beautfully coect wth geeralzato of Veta roduct Th geeralzato retrcted to the erodc cotued radcal of For the dfferet d of geeralzato, ecally related wth elltc fucto, refer the aer [9], [] I 593 Frech mathematca Fraco Veta gave the beautful fte roduct formula for ug oly ad quare root () It ad that th wa the frt exact aalytc exreo gve for, a well a the frt recorded ue of a fte roduct Mathematc

2 I h aer [9], Aaro Lev wrte Gve the mlcty, elegace, ad age of Veta roduct t urrg that there eem to have bee few attemt at fdg mlar formula I that aer he gave the followg Veta le roduct for () Certaly t a urre that t too more tha four hudred year to come wth th ecod Veta le roduct formula from the aearace of frt oe By eemg thee two formula oe ca uect that whether there a more geeral ettg behd thee roducti th aer we gve a elemetary geeralzato for Veta le roduct I artcular th geeralzato we gve erha the ext mlet exreo of Veta le roduct famly ( 5 5 ) ( ) ( ) ( ) ( ) ( ) (3) Secto Let ( ε ) be a equece uch that ε {, } for all e ( ε ) { } ( ) that c [, ] for all { } Defe the equece ( r ( c )), Let the equece r be defed a r + ε + + ε wth r Let ( c ) be ay real equece uch a follow r ( c ) + ε + + ε + c wth r ( c ) + c The oberve that the equece r ( c ) real for all Theorem The equece ( ( )) εε ε r c coverget ad lm r( c) + Frt we rove the followg lemma Lemma () () r + r εε ε ad εε ε εε ε () + + r () For ay, r () r( c) r() or r () r( c) r() Proof of Lemma

3 To how r + εε ε, oberve that, whe the rght had de of the detty ε + ε ε ε ε But + + co ε r 8 Hece whe the reult true (Oberve that we chooe otve quare root ce ε + Therefore value otve) 8 Suoe for the reult true Oberve that + + εε ε εε ε + co + * + εε ε ε + ε + ε + + ε ε + + ε + ε + ε + + ε + + ε + + ε + Hece by ducto reult true + εε ε * Aga we chooe otve quare root ce + εε ε εε ε To how r () + +, oberve that, whe the rght had de of ε ε the detty ( + ε) co ε + r() Hece whe the reult true Suoe for the reult true Oberve that + + εε ε εε ε+ εε ε εε ε+ + + co + + +

4 + εε ε εε ε+ ε + ε + ε + ε + ε + ε ε ε ε + ε + ε + ε + ε + ε + + ε ε + + ε + ε + + r + () Hece the reult Proof of ():- Oberve that ce c, + c ε + c εor ε + c ε + ε + c + ε or + ε + c + ε r() r( c) r() or r r c r () ( ) () Therefore whe the reult true Now uoe for the reult true e r () r( c) r() or r () r( c) r() Oberve that r+ ( c+ ) ca be wrtte a r+ ( c+ ) + r ( c+ ), r ( c ) + ε + + ε + c where But by ducto hyothe r () r ( c+ ) r () or r () r ( c+ ) r () + ε r () + ε r ( c ) + ε r () or + ε r () + ε r ( c ) + ε r () + + r() r+ ( c+ ) r+ () or r r+ c+ r+ Hece the roof of lemma comlete Proof of the Theorem :- () ( ) () Hece by ducto reult true εε ε By Lemma art (), ce the fucto cotuou ad ce the ere abolutely coverget, r () ad r () coverget It clear that εε ε lm r() lm r() +

5 εε ε By lemma art () lm r( c) + Comlete the roof of Theorem Notato:- We ue the otato + ε + ε + to deote the lm r () lm r Theorem Let ( ε ) {,} ad ( ) ε ha the followg roerty + let A {,} be the et of all uch equece Defe ψ : {, } \ A [, ] uch that ψ (( ε) ) lm r, where ( ) radcal equece Theψ - ad o to Proof of Theorem t ε ε & ε +, r the correodg cotued Oberve that by Theorem, ψ well defed To how ψ - uoe there ext two dfferet equece ( ) ε ad ( ), \ A ε { } uch that ( ) ( ) ψ ( ε ) ψ ( ε ) + ε + ε + + ε + ε + Let m { ε ε } The ε + ε+ + ε + ε + +, but ce ε ε ε ε But oberve that for ay( ε ) {,} lm r ε & ε ε+ ε + & ε ε + But ε ε ψ ε A or ε A, cotradcto ψ ( ) ( ) To how ψ o to frt we rove the followg lemma Lemma If < φ < ad co φ the for ay, co φ coφ + ε + + ε + co + φ r( co + φ), where ε co φ

6 Proof of lemma co φ coφ + co φ + + co φ Oberve that g of the quare root decde by co φ g of co φ, t equal to co φ Hece, whe the reult true co φ Suoe for ome the reult true Oberve that + + co φ + co φ + co φ + co φ Hece by ducto reult true for ay Ed of the roof of lemma To rove ψ o to frt oberve that wheε, the ψ ( ε ) ad whe, the ψ ( ε ) I both cae ( ) ε & ε Now let (, ) ( ) ε A ( ) x The there extφ uch that < φ < ad x coφ Suoe co φ The by above lemma ( co + x r φ ) for ay co φ where ε By Theorem, co φ x r r lm ( co φ) lm () ε ε co φ ψ x Alo oberve that co φ co φ A co φ Now let (, ) x ad uch that co φ Let m{ co φ } co φ The x + ε + + ε, whereε for,, co φ Defe ( ε ) a co φ ε whe ad ε, ε+ & ε + co φ The t clear that ( ε ) A ad ( ) Corollary ( ) ψ ε x Ed of the roof of Theorem

7 Every [, ] r ca be rereeted the form of cotued radcal r + ε + ε + Th rereet uque ule f we allowed ug equece ( ε ) t tal of thee equece cot of two coecutve - follow wth ftely may Proof:- Clear Theorem 3 Let x (, ) &, the ε, ε, ε {, } t x + ε + + ε x co + β, where β odd ad + β Frt we rove the followg lemma Lemma 3 α a odd teger ad + α ε, ε, ε {, } t α ε Proof of lemma 3 Oberve that for ay ε, ε, ε {, }, ε odd ad + ε Let ( ) ε ε Suoe {,, } t ε ε r ( ε ε ) ad ( εr ε r ) r r ' ( εr ε r ) ( ε ε) ( εr ε r ) ( ) Let r max { ε ε } r r r r, cottradcto, ε ε ε ε for,,, Hece the rereetato ε uque the {,, {, } for,, } et ( ) ε ε ε ε ha elemet { α + α ad α odd} ε ε {, } for,, Hece the lemma Proof of theorem 3:-

8 Let x + ε + + ε The,, ε ε ε ca be exteded to the equece ( ε ) { }, \ A t frt term of the equece ε, ε, ε ad ε +, ε +, ε + 3 The oberve that x + ε + Therefore by Theorem, εε ε δ δ x co Oberve that, where δ εε ε δ δ εε ε δ x co co + +, where δ εε ε for,, x co δ Let α δ The by α β Lemma 3 + α ad α odd x co co K+ + β where β ad β odd Now uoe that β odd ad x co + The oberve β α α that by ettg, α t + α,α odd ad x co The by lemma 3, α δ But the we ca elect,, {, } ductvely whe ε, ε εr already elected, elect x co εε ε δ ε ε ε t ε δ, ε ad ε δ ε r εε r εr + ε + + ε Hece the theorem Note:- Oberve that ay fte eted radcal rereetato of the form + ε + + ε Ca be exteded to cotued radcal rereetato two way, by uttg etherε+, ε+, ε+ 3 ε+ or ε+, ε+, ε+ 3 ε+ Defto Coder the cotued radcal + ε + The th cotued radcal(or the equece( ε ) ) ad to be evetually erodc ff There ext ad t ε++ m ε+ m {} ad {,, } The above cotued radcal ad to be totally erodc (or erodc) Iff

9 ε ++ m + ε m {} ad {,, } cotued radcal (or the equece( ε ) ) I both cae called a erod of the called the mmum erod of the cotued radcal f a erod of cotued radcal the Theorem Coder the cotued radcal r + ε + the r totally erodc r co q for ome q, ad q ha a odd deomator (We remove the trval cae ε, ) Furthermore, mmal erod of r m{ d d ± mod, the deomator of q} Proof:- Suoe ( ) ε totally erodc Let be a erod of( ε ) Oberve that f εε ε the a erod of ( δ ), (Whereδ εε ε) If εε ε, the ot a erod of ( δ) Hece wthout lo of geeralty let be a erod of ( δ) ad δ but a erod By ug ummato of geometrc ere, δ δ δ δ δ ( ( δ+ δ + + δ + ) Oberve that ( δ+ δ + + δ + ) mod Hece the deomator of δ q odd q get t maxmum whe δ, ad get t mmum whe t δ,, hece < q < Now uoe r co q, q (, ), q, odd & (, t ) t t Hece co + ε +, where ε, (ce odd, co ) co t co t ϕ( ) ϕ( ) + + mϕ( ) t + t Sce (, t ), mod t t mod co co ε ad {,, ϕ( ) } ( ) totally erodc ad ϕ() a + + mϕ ( ) ε + m {} erod (Need ot be a mmal erod) ε

10 d ϕ ( ) Let m{ d ± mod } the, ce d mod The ± mod + + md t + t t ± t mod co( ) co( ) + + md t + t ( ) ( ) co co ε + + dm ε + m { } ad {,,,( d ) } d a erod Now uoe d d ad md ( ) ( ) d a erod ε ε { } & {,,,( d ) } + + md + m + + t + t + + md t + t co co, + md ± ( t ± t ) md ( ) ± ± By mmalty of d mle d t md mod d Hece the mmal erod of r m{ d ± mod } Corollary r + ε + evetually erodc r co q d for ome (, ) q δ Proof:- Suoe r evetually erodc, the r co ad t δ+ r+ m δ+ r for r {,,,( )} for m Ug ummato of geometrc ere, δ δ + Hece δ q Now coder the fte eted radcal rereetato of r co q, q (, ) If the β deomator of q ha o odd factor other tha, the q or q, +, β ad + β odd Hece ote to the theorem 3 cotued radcal rereetato of r evetually erodc Now uoe the deomator of q of the form, the umerator of q, the t odd The by lemma, ad a odd umber greater tha 3 Let t be co t co r + ε + + ε + ε, ε co t, where {,,,( ) } t ε ca be choe a to et rereetato Hece r evetually erodc < t < By theorem, co t ha totally erodc cotued radcal

11 Followg defto facltatg to develo a mle ad ractcal method to fd cotued radcal rereetato of co q where q ratoal Defto If a odd umber the em order of modulo m { d d mod } ± We gve three te to fd cotued radcal rereetato of co q, where q ratoal t Ste:- Wrte dow q a, q, { } geeralty we ca aume that t < <, ad ad t are relatvely rme umber Wthout lo of Ste :- If, the co q ha the form co q or co q + ε + ε, co q where ε to fd ε followg ractcal method ca be ued co q Chec whether the agle q frt, ecod, thrd or fourth quadrat If t frt or fourth quadrat ε, f t ecod or thrd quadratε Ste 3:- If 3, the fd the em order of modulo, let t Th the mmal erod of relevat cotued radcal To fd frt coeffcet ε, ε, ε ue the method te Now co q co t + ε + ε + ε +, where ε + ca be elected a to et < t < Aga, 3 ue the method te to fd frt coeffcet of relevat cotued radcal of co t Now th totally erodc wth erodc bloc ( ε, ε 3,, ε ) Examle:- Fd the cotued radcal rereetato of co 36 m d ± mod7 d Oberve that the em order of modulo 7, ce { } co 3 t 7 quadrat, co 3 d 7 quadrat, co rd 7 quadrat ε, ε, ε 3 co + + co + + ε co, whereε Now calculate frt four coeffcet of co 7

12 co 7 t quadrat, co 7 d quadrat, co 7 t quadrat, co 3 7 th quadrat, co 7 th quadrat, ε, ε, ε, ε Hece co ha reeated bloc ( +++,,, ) Coequetly rereetato of co +,, +++, +++, +++, 36 7 Secto Theorem 5 Let ( r ) be a totally erodc cotued radcal equece of mmal erod, wth the correodg ε ε ε Let m +, where, m { } The, bloc (,, ) ( ) σ σ σ m m ( ) ( ) ( ) ( ) m m m m + ( + ) co +, wheδ r rm+ σ σ σ co +, wheδ δ Where σ, δ, δ εε ε Proof:- By lemma, εε ε δ co co r Oberve that δ δ δ δ m m + m, but δm ( εε ε )( εε ε )( εε ε ) ( δ) mbloc m m m m δ ( ) ( ) δ δ δ δ m σ σm σ m Alo oberve that, + + δ δ δ δ δ δ m δ δ δ σm ( ) + δ δ ( m ) + ( m ) + m σ m σ σ m +, wheδ ( ) m σ ( ) σ δ m + ( + ) whe Hece the theorem

13 Corollary 5 lm σ ( ) σ ( ) co, wheδ r co, wheδ Proof :- Tae the lmt of both de the detty of theorem 5 Theorem 6 Let ( r ) be a totally erodc cotued radcal equece of Let ( u ) ( ) u r r, where r lm r Cae I: - δ the ( ) If for,,( ) Cae II:- δ the ( ) If for,, ( ) Proof of the theorem:- Let be the equece defed a, σ σ u ha dtct lmt ot whch are, ( σ ) ( ) δ, the equece ( ) σ σ u ha dtct lmt ot whch are, ( σ ) ( + + ) r ca be arttoed to ubequece accordg to ( r m + ) { },,( ), r, m Sce the equece( r ) coverge, each of thee o σ ubequece coverget to r co( ) arttoed to ubequece a( m ) Coder the ubequece ( m ) Coequetly the equece ( ) u +,,( ), m { } u + for ome,, ( ) & m { } m+ + From theorem 5, { ( ) ( ) } co σ co σ σ σ u m + m m ( ) + u ca be Now ug the coe addto formula co A co B, ( A+ B) ( BA)

14 ( ) ( ( ) ) σ σ σ σ σ m m m+ m+ m+ + u m ( ) ( 8 ( ) ) σ σ σ σ σ m m+ m+ lm u lm lm + m+ + m+ m m m σ σ σ 8 ( ) σ 8 m + σ lm m σ σ 8 m σ σ ( σ ) ( ) α, {ug lm α } Dfferet ubequece gve dfferet lmt for,,( ), ce σ σ for Partcularly, the ub equece δ, artto the equece ( ) Now let for,,( ), r, m { } σ ubequece coverget to r co( + ) arttoed to ubequece a( m ) o σ σ um coverge to ( σ ) ( ) r to dfferet ubequece accordace to ( r m+ ) Sce the equece( r ) coverge, each of thee Coequetly the equece ( ) u +,,( ), m { } From theorem 5, ubttutg m tead of m, we ca get the detty, ( m m ( ) ) σ σ σ m+ co + + r +, for,, ( ) Coder the ubequece ( m ) u + for ome,, ( ) & m { } m+ + From theorem 5, { co( ) co( ( ) ) } σ σ σ σ u m+ + m m By mlar calculato a cae I, lm u m + m u ca be ( ) ( ( ) ) m m m+ m σ σ ( σ ) ( + + ) m+ + σ σ σ σ σ 8

15 Oberve that σ σ for,, {,,( ) } lmt for,, ( ) Hece the theorem Corollary 6 Lmt ot of the equece u ( r r ) algebrac umber Hece Dfferet ubequece gve dfferet,where r lm r, have the formα, where α a Proof:- If q ratoal the q algebrac[] Hece each lmt ot of the formα, where α a algebrac umber Reult of Theorem 6 Let ( r ) be a totally erodc cotued radcal equece of wth mmal erod 3 wth erodc bloc (,, ) ε, ε, ε δ, δ, δ (& δ ) δ δ 3 δ σ, σ, σ +,(& σ ) 3 3 r r ( 3 ) lm co co Hece by theorem 6 (cae I) the equece σ co 7 + ha three lmt ot, whch are, root 3 σ3 ( 3 σ ), ( 3 σ ) 3 σ3 ( 3 σ ) σ , 7 Partcularly lm co + 3 root Let ( r ) be a totally erodc cotued radcal equece of wth mmal erod wth reeated g ( ) e r ε, ε δ, δ, (& δ ) root

16 δ σ lm r r co( + ) co 3 σ equece root σ ( σ ) σ ad + 3 ( + σ) 3 Hece by theorem 6(cae II) the ha two lmt ot, whch are, Partcularly the equece 3 3 coverge to 3 ad the equece root coverge to 3 root Let ( r ) be a totally erodc cotued radcal equece of wth erodc bloc ( +, ) ε, ε δ, δ, δ, δ (& δ ) 3 3 δ δ δ σ, σ, σ (& σ ) 3 r r ( + ) lm co co Hece by theorem 6 (cae II) the equece ( ) co σ 5 σ 5 r ha four lmt ot, whch are ( σ ) 5 5 +, σ ( σ ), σ ( σ ) 5, + 5 ( σ ) 3 σ Partcularly lm co + root Theorem t Let q, uch that < q < wth deomator of q odd, e q for ome ad + t {,,3 } Let be the mmal erod of correodg fte eted radcal { } of co q e m d d mod ( ) Defe the equece ( ) a follow ± +

17 + for { } ad {,, 3,, }, + ε + +, co q where ε for,,3, co q The, co q co q + co q + + ( q) q Proof of theorem 7:- Let u ( co q r ), where r the th terato of the cotued radcal of co q, q a of the theorem The from theorem 6, otced that the equece u coverge to ( q) q σ σ (Smle ubttuto q ( ) or q ( + ) gve th) By the defto of the equece ( ) we ca have the detty, for, co q ( co q + )( co q + + ) ( co q + + ) + co q ε co q + ( + ) co q ε co q ( + ) Now ce ε co q co q, oberve that co + q ad ε + have ame g Therefore, co ε q + ( co q + ) ( co q + + ) + +, for Now oberve that by teratg above detty for,,, we ca derve the detty co q ( + ) otce that r + ( δ ) ( co q ) co q + ( + ) Now by the defto of equece ( r ) ad ( ) for all By theorem 6, lm u( + ) ( q) q, where,

18 ( + ) ( ( ) ) ( ( + ) ) ( ) u co q r ( + ) + + co q + ( δ ) ( q ) ( co q + + ) ( + ) co + ( δ ) ( co q ) co q + + Now f δ, ce the equece u ( + ) coverge the roduct tag lmt both de, ad wth ome maulato gve the reult Ifδ, ubttutg, tead of, we ca have the followg detty ( + ) co ( q ( + ) ) ( co q ) co q + + d But oberve that, ce mod( + ), deomator of q Hece co q co mod( ) co q + + coverge ad + +, where + the q for,,, Th mle, + co q + + co q + + Now a earler, tag the lmt both de ad co q + co q + co q + + wth ome maulato gve lm ( q) q Tae w co q + +, ow ce w coverge, lm ww + But oberve that the defto of the equece ( ) + mle lm + lm ( + ) for,,, + + lm w lm w + Hece w coverge ad lm w lm w Th gve the reult, co q co q + co q + + ( q) q Hece the theorem

19 Reult of theorem7 let q The by theorem7, (Aaro Lev reult) Let q, The by theorem7, ( 55 ) ( ) ( ) ( ) ( ) ( ) Referece George Polya, Gabor Szego, Problem ad Theorem Aaly I, Srger 978, - Herchfeld,A, O fte radcal, Amer Math Mothly (935) 3 Berdt, BC, Ramaua ote boo, Srger-Verlarg, Newyor,985 Hugh C Wllam, Edouard Luca ad Prmalty Tetg, A Wley-Itercece Publcato (998), - 5 Cota J Efthmou, A cla of erodc cotued radcal, Amer Math Mothly 9 (), MA Nyblom, More eted quare root of, Amer Math Mothly ( 5), LD Serv,Neted quare root of, Amer Math Mothly (3), Seth Zmmerma, Chugwu Ho, O ftely eted radcal, Mathematc Magaze, 8 (8),3-5 9 Lev, Aaro, A ew cla of fte roduct geeralzg Vete roduct formula for,the Ramaua Joural (5),35-3 Aaro Lev, A Geometrc Iterretato of a fte roduct for the Lemcate Cotat Amer Math Mothly, 3 (6) 5-5 Iva Nve, Irratoal Number, The Mathematcal Aocato of Amerca 956, 37-38