Diophantine Equation Of The Form. x Dy 2z

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1 IOSR Joual of Mathematics (IOSR-JM) e-issn: p-issn: X. Volume Issue 5 Ve. I (Sep. - Oct.6) PP Diophatie Equatio Of The Fom x D z Nu Asiki Hamda Abdul Latif Samia Nazi Muslim Mala Wold ad Civilizatio (ATMA) Natioal Uivesit of Malasia Abstact: The pupose of this stud is to ivestigate the solutio of Diophatie equatio. This stud will be complete if we kow moe about the pime umbes of Mesee. Besides that this pape will discuss about Diophatie equatio. It is about expeimet with umbes ad to discove pattes. Numbe theo plas a impotat ole i the Diophatie equatio. I this stud we coside Diophatie equatio of the fom: x D z fo a odd umbe D that is pime umbe. Usig coguet method this Diophatie equatio could be solved. Kewods: Diophatie pime umbes pattes umbes odd umbe I. The Histo Of Diophatie Equatio Accodig to Le Veque (969) he defied Diophatie equatio as the equatio with oe o moe vaiables that have the powe of oe two ad so o. If the equatio has a vaiable with the highest powe of oe two o thee the each equatio is kow as liea quadatic o cubic Diophatie equatio. The commo example is Pthagoea Theoem. It ca help to udestad this Diophatie equatio. Pthagoea Theoem states that fo a ight-agled tiagle the squae of the two logest sides is equal to the sum of the squaes of the othe two sides. The logest side is called the hpoteuse. Pthagoea Theoem equatio is x z whee xad z ae the vaiables ad this equatio is called as quadatic Diophatie equatio. Equatio x z ca be descibed as i the figue below: Solutios fo Diophatie equatio must be i itege o atio. Fo this example two of the solutios. Othe solutios ca be foud though simple fomula as ae ( x z) (345) ad ( x z) (53) follow: x a b ab z a b a ad b ae a itege Fo example to obtai solutio (3 4 5) coside a ad b so = 5. Babloias used the Pthagoea equatio to build tigoomet daft schedule. With the stuctue of this daft schedule Pthagoea equatio will be emphasized. Level oe equatio ax b c has appeaed i Geek Aabic ad i Chiese wod puzzles. Howeve the theo eeded to solve equatio ax b c has bee foud i Eleve Euclid. Equatio ax b c is still egaded as Diophatie equatio although it is ot ecoded i Diophatus witig as he might udeestimate it. The wok of Diophatus was ot kow b public i Euope duig the Middle Ages. Howeve the wok has bee discoveed ad taslated twice i the 6 th ad 7 th Cetu. Oe of the secod taslatio copies b Bachet has eached Femat a mathematicia who was detemied to udestad it i depth ad attempted to cotiue the wok of Diophatus. Femat's effots might be descibed as a tuig poit fo the discove of DOI:.979/ Page

2 Diophatie Equatio Of The Fom x z has o o-zeo mode umbe theo. I most of Femat's decisio he stated that the equatio itege solutio. He povided a detailed aalsis of the poblem b showig itege as a sum of squaes of two umbes ad demaded to pove each positive itege is the sum of the squaes of fou umbes. Howeve he was uable to pove it. Afte 5 eas late mathematicias such as Eule Lagage Gauss ad Kumme cotiued the wok of Femat. Thei esults wee limited to eithe the quadatic equatio kow as Pell equatio Femat equatio if is less tha. x z.kumme showed that equatio x z x d c o has o o-zeo itege solutio x z which has bee stated peviousl has o itege solutio if is x ad z. Femat's Theoem itege geate tha o equal to thee also a example of Diophatie equatio with thee vaiables 3 x x x 3 3 is also a Diophatie equatio with two vaiables x ad. This equatio Equatio is called as cubic Diophatie equatio because the highest powe of the vaiable is thee. Thee ae ma examples of Diophatie equatio. This equatio ca be witte i the fom of egula equatio o polomial equatio. Fo x z whee it has ma solutios ( x z ). Fo geate value Femat's Last Theoem states that thee is o solutio fo the positive itege umbes ( x z) that satisf this equatio. Next we will discuss quadatic equatio of the fom the ext chapte. x D z with coditio ( xz ) II. Diophatie Equatio Mesee Pime Numbe Mesee pime umbe is a pime umbe witte i the fom of 5 example 3 is a pime umbe 3. See the table below: povided i a with coditio Table. Example fo a whe =3 3 =7 4 =3.5 5 =3 3 = = = =. 4 = = = = = = = =..7 6 = = = =5.3 7 = = = = = = = =7.3.5 Souce: Joseph H. Silvema 997. A simple obsevatio ca be see easil fom the table above which is if a is a odd umbe the a is the eve umbe. Based o the table above if a is a odd umbe a ca be divided b a. This obsevatio is ve accuate because the statemet ca be pove b usig the fomula fo geometic seies. Fo example: ( )( x x x... x x ) x.( x x... x x ).( x x... x x ) 3 ( x x... x x x) ( x x... x x ) x Based o this geometic fomula assume that x a a ca epeset ma values besides a that is a. Howeve if satisfied. This is show b the table below:. Fo a ca alwas be divided b a. So a fomula is Table. Mesee Pime Numbe Souce: Joseph H. Silvema 997. DOI:.979/ Page

3 Although this table shows ol a few values but it ca be explaied that ca be divided b 3= Diophatie Equatio Of The Fom Whe is a eve umbe Whe ca be divided b 3 ca be divided b 7= 3 Whe ca be divided b 5 ca be divided b 3= 5 So it ca be cocluded that if ca be divided bm the ca be divided b m. Fom mk m k the obsevatio it is clea that this statemet is tue. Coside mk the ( ). Geometic seies fomula with x k will be used to obtai m k m m k ( ) ( ) ( ) m k m m ( )... ( ) ( ) This suggests that if is a plual umbe the is also a plual umbe. Plual umbes ae umbes othe tha pime umbes. This ca be explaied b the facts below. Facts If a is pime umbe fo some umbes with a ad the a ad must be pime umbes. This meas that the pime umbe is i the fom of a so it is ecessa to coside the case a ad must be pime umbes. Pime umbes ae i the fom of p is kow as Mesee Pime Numbe Amog Mesee Pime Numbes ae Not all umbes ae pime umbes. Amog them ae: Mesee Pime Numbes ae amed accodig to the ame of Fathe Mai Mesee ( ). I 944 he p is pime umbe fo the followig umbes: claimed that p = The umbes above ae pime umbes less tha 58 that satisf the fomula p i ode to become pime umbe. Howeve it is ukow how Fathe Maie Mesee discoveed that fact especiall afte he stated that his pevious eseach was ot tue. Fiall he submitted complete pime umbes p less tha which satisfied the fomula p which is p = III. Quadatic Diophatie Equatio Sstem Vaious esults ae obtaied fom the equatio whe fuctios f ( x ) ad f ( ) x ae quadatic polomial. This fact was stated b Baes (953) Goldbeg (954) ad Mills (954). Thee ae ifiite umbes of tivial solutios of the equatio. Apat fom that thee ae ol ifiite values possible fo z. Fo example quadatic equatio with x ad values ae eithe ifiit o ca be obtaied fom the Pell Equatio o equivalet alkhawaizmi ecusive. The equatio x x xz is equivalet to x ad x which has a positive itege ( x ) ( u u ) whe u is of a sequece of whee 3 solutio z. This sequece also has a alteate fom of a Fiboacci sequece Equatio x x xz with x ad has solutio ( x ) two cosecutive tems of u 5u u. equatio x x xz with the sequece with x ad ol has solutio if x. Modell (969) showed that equatio ax b c xz whee x. This solutio ca be povided i the fom of ab ad c ae iteges has ifiite solutios fo a DOI:.979/ Page

4 Diophatie Equatio Of The Fom polomial i obtaied. ab ad c. I this stud all itege solutios fo the equatio x D z will be x D z The Solutio Of Diophatie Equatio Of The Fom Next will be discussed o Diophatie equatio of the fom (Edwad L. Cohe 98) x Q z ( xz ) () whee Q is Mesee pime umbe as descibed above. Pime umbes ae a umbe of the p fomq. I this case p is a pime umbe. All Mesee pime umbes ae i the fom of Q except Q 3. Ou objective is to solve all geeal Diophatie equatio solutios of the fom x D z ( xz ) () whee D is a pime umbe of the fom D. Theefoe fo the ext solutio we coside all pime umbes except umbe 3. The method used to fid the solutio is b chagig equatio () to a paametic fom so that the equatio obtaied is much simple. The equatio is i the fom of Fo each itege umbe D obtaied though ax b cz. ote that x z ae solutios to (). Thus we obtai the same idetit equatio which is ( ) ( ) ( ) solutio is obtaied b assumig ( x z ) as itege solutios fo equatio (). The value of xz equatio () we get that If summaized the esult obtaied is. Paametic x ( ) x ( ) z ( ) z ( ) x ( ) z. Hece the solutios ae as below. Fom z z ( ) ( ) x ( ) x() ( ) ( ) ( ) ( ) x K z K x z z( ) K x() So ( ) ( ) ad K x ( ) z ( ) x K z K is costat x ( ) K z ( ) K z ( ) x ( ) The iitial solutios obtaied ae x ( ) K( ) Kz (i) Kx K( ) ( ) z (ii) DOI:.979/ Page

5 Solvig equatios (i) ad (ii) b emovig z => x ( ) K( ) Kz K Kx K( ) ( ) z => x ( ) K( ) Kz = K x ( ) ( ) K Kz => K x K K K ( ) [ ( ) ( ) ( ) ( ) ] ( K ) x [ ( ) K( ) K ( ) K( ) ] = => => => ( ) [( ) 4 ( ) ( )] K x K K x ( ) 4 K( ) K ( ) K Next gettig the value of z b elimiatig x => Diophatie Equatio Of The Fom K x ( ) K( ) Kz (iii) Kx K( ) ( ) z (iv) Kx ( ) K( ) K K z => Kx K( ) ( ) z => [ ( ) K K ( ) K( ) ( )] ( K ) z => [ K( ) K ( ) ( )] K z => [ K( ) K ( ) ( )] K z => = z K( ) K ( ) ( ) K Coside => => s K the geatest commo facto (FSTB) fo ( s t) t t x = = s s s s s ( ) 4( ) ( ) ( ) ( ) ( ) t t t t t xt t zt ( ) s 4 st( ) t ( ) t s ( ) s st( ) t ( ) x z foms obtaied ae e f g z e s st t ( ) 4 ( ) ( ) f t s (3) g s st t ( ) ( ) ( ) (FSTB) the geatest commo facto of d fo the deomiato e f g ae divided util DOI:.979/ Page

6 Diophatie Equatio Of The Fom ( ) e ( ) g which ae Note also d divides divide ( ) f t s ad FSTB ( )( s t ) ( s t t s ) o. Theefoe it satisfies that d ca D. So d = D o D. The ew equatio will be obtaied which is c x s st t d c ( t s ) d c z s st t d ( ) 4( ) ( ) ( ) ( ) ( ) (4) whee c is a itege. Next values equied will be obtaied though equatio (4) b takig ito accout s K whee t. The values ( x z) ae the solutio fo equatio (). Theefoe equatio () is t equivalet to equatio (4). Note the followig:. Whe s is a odd umbe the FSTB ( e f g ) = o D. Whe s is a eve umbe the FSTB ( e f g ) = o D if it is odd it ca be cocluded that H = D. similal if it is eve the H= D. Theoem : H = FSTB( e f g ) if ad ol if s t(mod D) Pove b equatio (3) ( ) s 4 st( ) t ( ) (mod D) t s (mod D) So to solvig these two equatios ae though deletig method ( ) s 4( ) st t ( ) (mod D) (5) (i) ( ) s t ( ) (mod D) (ii) The followig equatios ae obtaied ( ) s ( ) st (mod D) ( ) s ( ) t (mod D) (6) Multipl equatio (6) with ( ) will poduce s t (mod D). To pove this statemet is tue the it must show that equatio (5) ad (6) ae satisfied. Geeall it is kow ( )( ) (mod D) so equatio (5) is satisfied whe s t is multipl b. Fo equatio (6) whe s t (mod D) it shows that s 4 t (mod D). Fiall sice 4 (mod D) the s t (mod D). Coolla: H = FSTB( e f g ) if ad ol if t s(mod D). Theoem Fo equatios that have specific solutios thee ae ma ifiite solutios. Each solutio ( x z) of equatio s (4) is obtaied fom fou atio umbes K which is the highest commo facto (FSTB). t DOI:.979/ Page

7 Diophatie Equatio Of The Fom Theoem 3 had s x x had z t z Theoem 4 Whe D7( ) the pioit of this solutio ae x z that have bee metioed ealie. It is also equal to FSTB fo fou umbes that ae 7 4. Note the table below: Table.3 Example of FSTB whe D7( ) FSTB S T Souce: Edwad L. Cohe 98. It has bee stated that t so we defiitel will ot use equatio (3) ad (4). The Similaities Of Liea Diophatie Liea Coguet Ad Matix THEOREM Coside that is the last divide i Eukledea alkhawaizmi fo the pupose of fidig FSTB fo two positive qi iteges which ae a ad b whee ab. Coside that Q i ad Q Qi i whee i qii i fo i a. So Q a b the Q b. So Q ad a b the a b. Next c c k fo all k costat. So c ( k) a ( k) b the x k ad k ae specific solutios fo this liea Diophatie equatio (T. Kosh 996). THEOREM Coside i qii i fo i Coside qi Qi ad ( ) Qi b ( ) a Q i the Eukledea algoithm to get FSTB ( ab ) whee a i Qi Veificatio Geeall it has bee idetified that the top ow of of i. The (T. Kosh. 996) b. Q i ae [ ] so we just have to focus o the secod lie (T. Kosh 996). Q. This ca be pove b usig iductio method which icludes equatio a q q. DOI:.979/ Page

8 Diophatie Equatio Of The Fom.. q q Whe this alkhawaizmi has the equatio of a q ad Q q b a = whee b q. Theefoe Q. So the solutio is tue whe. Now suppose that the solutio of the equatio is satisfied fo k. Take the value of k. The alkhawaizmi ivolves equatio k. Put it o top of the equatio. The equatio k. These ( ). equatios will fom alkhawaizmi to fid FSTB Coside k Q' Qi ' whe Qi' Qi?? i Q ' k k ivolved. So ( ) ( ) ' k k Q Q k k ( ) ( ) q k k =???? k k q ( ) ( ) k k?? k b k a ( ) ( ) k k =. B usig hpothesis whee? states that the solutio fo the top of the matix is ot ( ) B usig iductio method this solutio is satisfied fo each itege. It has bee stated peviousl that a b a a b ( ) ( ) b = Q b a DOI:.979/ Page

9 COROLLARY : Suppose that Q b a Diophatie Equatio Of The Fom is the fial divide i alkhawaizmi Eukledea to fid FSTB ( ) ( ) ( ) ( ) b c is deived fom x x t ad ab ad. The solutio fo this liea Diophatie equatio is ax b c whee ( ) a t. Fom coolla show above the matix method might be iflueced b calculatio fom calculato like TI-85. It has fou mai beefits which ae:. It calculates costat ad ol i the liea equatio a b. Whe the value of ad ae kow specific solutios fo Diophatie liea equatio will be c moe eas which ae: x k k ad k. 3. A value i the secod ow is costat fo paamete t i geeal solutio of this Diophatie liea equatio. 4. It does ot show how difficult each calculatio is. Whe liea coguet is fom this liea Diophatie equatio coolla is satisfied. It is said that usig the matix solutio is suitable fo solvig liea coguet. COROLLARY : Liea coguet equatio ax c(mod b) ca be solved if ad ol if c ad the solutio is ( ) b x x t whee FSTB( a b). IV. Coclusio This stud shows that oe of the mathematical solutios is though liea Diophatie equatio o Diophatie equatio of the fom x D z. Actuall thee ae ma foms of Diophatie equatio. Oe of the iteestig topics i umbe theo is Diophatie equatio which states that "a liea Diophatie equatio ax b c whe a b ad c ae whole umbe have the whole umbe solutio if ad ol if FSTB ab divide c totall". It ca be cocluded that Diophatie equatio i geeal is a equatio which states that its vaiables ae fom the elemets of a whole umbe. Actuall the solutios fo Diophatie equatio ae ma. The moe the umbe of vaiables used the loge the calculatio. Liea Diophatie equatio ca be solved i diffeet was. If we pefe itege solutio we ca use this Diophatie equatio. Diophatie equatio also ca be used if we pefe esults that ae ot egative. To futhe facilitate the Diophatie equatio Eukledea alkhawaizmi ca be used. Oe thig to keep i mid i solvig Diophatie equatio is oe should be caeful i solvig eve poblem especiall whe witig the iitial aswes obtaied fom the paametes used because duig the ed of the pocess oe should e-ete each aswe that is peviousl obtaied ito the oigial equatio. []. P. Novikov A ew solutio of the idetemiate equatio []. Doklad Akad Nauk SSSR (N.S.) 6:5-6. [3]. Baes E. S O the Diophatie equatio [4]. Soc. 8:4-44. [5]. Geogikopoulos O the equatio [6]. Edwad L. Cohe. 98. The Mathematics Studet Vol. 5:6-. DOI:.979/ Page Refeeces ax b cz. x c xz. J. Lodo Math. ax b cz. Bul. Soc. Math. Gece 4:-5.

10 Diophatie Equatio Of The Fom [7]. Goldbeg K. Newma M. Staus E. G. & Swift J. D The epesetatio of iteges b bia quadatic atioal fom. Ach. Math. 5:-8. [8]. Joseph H. Silvema Mesee pime. A fiedl itoductio to umbe theo [9]. K. H. Rose Elemeta umbe theo ad its applicatios 3 d editio []. Addiso-Wesle Readig Massachusetts. []. Le Veqeu W. J A bief suve of diophatie equatio. MAA studies i Mathematic. 6:4-3. []. L. J. Modell Diophatie Equatio. Pue ad Applied Mathematics Vol 3 Academic Pess. [3]. Mills W. H A method fo solvig cetai Diophatie equatio. Poc. Ame. Math. Soc. 5: [4]. T. N. Siha da Vekatamaiah The equatio x P z : P a Mesee pime. Idia J. Mech. Math. 6:45-47.T. Kosh. Novembe 996. The Euclidea Algoithm via matices ad a calculatomath. Gaz. 8: [5]. Wikipedia the fee ecclopedia Diophatie equatio (dalam talia). [3 Mac ]. [6]. Wikipedia the fee ecclopedia Diophatie equatio of secod powes (dalam talia). [ Mac ]. DOI:.979/ Page