On rational Diophantine Triples and Quadruples

Transcription

1 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer ISSN On rtionl Diophntine riples nd Qudruples M.A.Gopln *, K.Geeth **, Mnju Somnth *** * Professor,Dept.of Mthemtics,Srimthi Indir Gndhi College,richy-6000,milndu,Indi; ** Asst Professor,Dept.of Mthemtics,Cuvery College for Women,richy , milndu, Indi; *** Assistnt Professor, Dept.of Mthemtics, Ntionl College,richy ,milndu, Indi; Astrct- his pper concerns with the study of constructing strong rtionl Diophntine triples nd qudruples with suitle property. Index erms- Strong rtionl Diophntine triples nd qudruples, Pythgoren eqution 010 MSC Numer: 11D09, 11D99 L et q e non-zero rtionl numer. A set squre of rtionl numer for ll 1 i j m 1,,... m I. INRODUCION of non-zero rtionl is clled rtionl Dq ( ) - mtuple, if i j q. he mthemticin Diophntus of Alexndri considered vriety of prolems on indeterminnt equtions with rtionl or integers solutions. In prticulr, one of the prolems ws to find the sets of distinct positive rtionl numers such tht the product of ny two numers is one less thn rtionl squre [0] nd Diophntus found four ,,, [4,5]. he first set of four positive integers with the sme property, the set is 1,3,8,10 ws positive rtionls found y Fermt. It ws proved in 1969 y Ber nd Dvenport [3] tht fifth positive integer cnnot e dded to this set nd one my refer [6, 7,11] for generliztion. However, Euler discovered tht fifth rtionl numer cn e dded to give the following rtionl Diophntine quintuple ,3,8,10, , Gis [13] found severl exmples of rtionl Diophntine sextuples, eg.,. Rtionl sextuples with two equl elements hve een given in []. In this ,,,,, , ,,,5,, All nown Diophntine qudruples re regulr nd it hs een conjectured tht there re no irregulr Diophntine qudruples [1,13] (this is nown to e true for polynomils with integer co-efficients [8]). If so then there re no Diophntine quintuples ,5,, Mny of these irregulr However there re infinitely mny irregulr rtionl Diophntine qudruples. he smllest is qudruples re exmples of nother common type for which two of the sutriples re regulr i.e.,,, c, d is n irregulr rtionl Diophntine qudruple, while c,, nd,,d re regulr Diophntine triples. hese re nown s semi regulr rtionl Diophntine qudruples. hese re only finitely mny of these for ny given common denomintor l nd they cn redily found. D( m ) Moreover in [1], it hs een proved tht - triple, 1,4 1 D( m ) cnnot e extended to - quintuple. D( ) 1, 1, 4 D( ) In [10], it hs een proved tht - triple cnnot e extended to - qudruple if 5. Also, one my refer [14-0]. hese result motivted us to serch for strong rtionl Diophntine triples nd qudruples with suitle property. Section A II. MEHOD OF ANALYSIS

2 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer 014 ISSN property In this section we generte sequence of strong rtionl Diophntine triples such tht (A,B,C), (B,C,D), (C,D,E), with the D n 1 Cse 1: Consider A nd AB 1 Note tht n B 4n 1 is perfect squre Let C e ny non-zero rtionl integer such tht From (1), we hve 1 AC n 1 BC n C Consider the liner trnsformtions Cse : X X n1 A 4n 1 On sustituting (3) in () nd y using (4) nd (5), we get B C 4n 1 1 n n C Let nd Let D e ny nonzero rtionl integer such tht From (6), we hve 1 BD n 1 CD n D Using the liner trnsformtions n1 B 1 n n n ( 4 1 ) X X 1 n n On sustituting (8) in (7) nd y using (9) nd (10), we get Cse 3: D (1) () (3) (4) (5) (6) (7) (8) (9) (10)

3 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer ISSN Cse 4: C 1 n n Consider Let E e ny non-zero rtionl integer such tht From (11) 1 CE n 1 DE n E n1 C nd D Let us ssume the liner trnsformtions X X 1 n n On sustituting (13) in (1) nd y using (14) nd (15), we get D E 1 E n n 1 Consider nd Let F e ny non-zero rtionl integer such tht From (16) n n 1 DF n 1 EF n F Consider the liner trnsformtion X X n1 D 1 n n On sustituting the vlue of (18) in (17) nd y using (19) nd (0), we get F 3 1 n n From ll the ove cses, (A,B,C), (B,C,D), (D,E,F),. will form sequence of strong rtionl diophntine triples Section B In this section, we serch for distinct rtionl qudruple (A,B,C,D) such tht product of ny two of them dded with 4 is perfect squre. Assume A (1) (11) (1) (13) (14) (15) (16) (17) (18) (19) (0)

4 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer ISSN nd where B (n n3) 1r s 4 rs( n 1) nd rs rs, 0 Let C e ny non zero rtionl integer such tht AC 4 (3) BC 4 From (3) we get, Assume 4 C A X X (n n3) On sustituting (5) in (4) nd y using (6) nd (7), we get () (4) (5) (6) (7) C 1r s 4 rs( n 1) 4 r s (n n 3) 41r s 4 rs( n 1) rsn 1 Let D e ny non zero rtionl integer such tht From (9) we get Assume X 1r s 4 rs( n 1) rs AD 4 (8) BD 4 CD 4 4 D B X (n n3) n 4 (30) in (8) nd y using (3) nd (33), we get D (9) (30) (31) (3) (33) On sustituting 1r s 4 rs( n 1) 16 r s (n n 3) 81r s 4 rs( n 1) rsn 1 1r s 4 rs( n 1) rs Hence (A, B, C, D) is strong rtionl diophntine qudruple in which the product of ny two when dded with 4 is perfect squre. Remr: If we te B different from () we cn generte different qudruples. Some of them re given elow

5 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer ISSN ) If B (n 4 n) D(4), then the qudruple is 1r s 4 rs( n ) rs, (n 4 n), rs 1r s 4 rs( n ) r s rs n r s rs r s rs n n 1 4 ( ) 4 n(n 4) ( ), 1r s 4 rs( n ) rs 1r s 4 rs( n ) 16r s n(n 4) 8rs 1r s 4 rs( n ) n where, ) If B n 1 1r s 4 rs( n ) rs D( n ), then the qudruple is (n1) (n 1) n, n1 (1 n) ( n 1),, n 1 rs n3r s rs( n ) n n nd B p npq D( nq ) 3) If p npq where q( p n) n q, p 3r s 4rs 1 qn B n q 4) If nd rs nq, p npq, then the qudruple is p nq p npq n q D( p npq), then the qudruple is q n ( p qn) 4q n 4 ( p qn), nq,, n provided q B npq D( p n q ) 5) If, then the qudruple is pqn ( p qn) 8npq 4 ( p qn), npq,, 4p n 10 pq 4q provided 4p n 6 pq 4q or III. CONCLUSION o conclude, one my serch for other fmilies of strong rtionl diophntine triples nd qudruples.

6 Interntionl Journl of Scientific nd Reserch Pulictions, Volume 4, Issue 9, Septemer ISSN REFERENCES [1] J.Arin., V.E.Hoggtt nd E.G.Strus, On Euler s solution of prolem of Diophntus, Fioncci Qurt. 17(1979), [] J.Arin, D.C.Arney, F.R.Giordno, R.A.Kol nd G.E.Bergum, An extension of n old clssicl Diophntine prolem prolem, Applictions of Fioncci Numers, Kluwer Acd. Pul. Dorrecht, (1993), [3] A.Ber nd H.Dvenport, he equtions nd, Qurt. J.Mth. Oxford Ser., 0()(1969), [4] L.E.Dicson, history of theory of numers, Vol., Cheles, New Yor, 1966, [5] Diophntus of Alexndri, Arithmetics nd the oo of polygonl numers, (I.G.Bshmov, Ed.), Nu, Moscow, 1974 (in Russi) , 3. [6] A.Dujell, he prolem of the extension of prmetric fmily of Diophntine triples, Pul. Pul.Mth.Dereen, 51(1997), [7] A.Dujell nd A.Petho, A generliztion of theorem of Ber nd Dvenport, Qurt.J.Mth.Oxford Ser. 49()(1998), [8] A.Dujell nd C.Fuchs, Complete solutions of the polynomil version of prolem of Diophntus, J.Numer theory, 106(004), [9] A.Dujell nd Vino Petricevie, Strong Diophntine triples, Experimentl Mth., 17, (1), (008). [10] A.Flipin nd Fujit.Y., he triple with prime, Gls. Mt.Ser. III46(011), [11] Y.Fujit, he extensiility of Diophntine pirs, J.Numer theory, 18, (008), [1] Y.Fujit, Exension of the - triple, Period.Mth. Hungric, 59(009), [13] P.E.Gis, Computer Bulletin, 17 (1978), 16. M.A.Gopln nd G.Srividhy (009), Diophntine Qudrpules for Fioncci numers with property D(1), Indin Journl of Mthemtics nd mthemticl Sciences, 5 (), [14] M.A.Gopln nd G.Srividhy (010), Diophntine Qudrpules for Pell numers with property D(1), Antrctic Journl of Mthemtics, 7(3), M.A.Gopln nd V.Pndichelvi (011), Construction of the Diophntine triple involving Polygonl numers, Impct J.Sci.ech., 5(1), M.A.Gopln nd G.Srividhy (01), wo specil Diophntine riples, Diophntus J.Mth.,1(1), [15] M.A.Gopln, K.Geeth nd Mnju Somnth (014), On specil Diophntine riple, Archimedes Journl of Mthemtics, 4(1), [16] M.A.Gopln, V.Sngeeth nd Mnju Somnth (014), Construction of the Diophntine polygonl numers, Sch. J.Eng. ech., [17] M.A.Gopln, K.Geeth nd Mnju Somnth (014), Specil Dio 3 tuples, Bulletin of society of Mthemticl services nd stndrds, 3(), [18].L.Heth, Diophntus of Alexndri: A study in the History of Gree Alger, Dover Pulictions, Inc., New Yor, AUHORS First Author M.A.Gopln, Professor,Dept.of Mthemtics,Srimthi Indir Gndhi College,richy- 6000, milndu, Indi; e- mil: myilgopln@gmil.com Second Author K.Geeth, Asst Professor,Dept.of Mthemtics,Cuvery College for Women,richy ,milndu, Indi, geeth_othn@yhoo.co.in hird Author Mnju Somnth, Assistnt Professor, Dept.of Mthemtics, Ntionl College,richy ,milndu, Indi, mnjujil@yhoo.com Correspondence Author K.Geeth, Asst Professor,Dept.of Mthemtics,Cuvery College for Women,richy ,milndu, Indi, geeth_othn@yhoo.co.in