IJESMR International Journal OF Engineering Sciences & Management Research

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1 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research CONSTRUCTION OF DIOPHANTINE QUADRUPLES WITH PROPERTY D(A PERFECT SQUARE) V.Geetha *, M.A.Gopala * Assistat Professor, Departmet of Mathematics, Cauvery Collage For wome, Trichy ,Tamiladu, Idia DProfessor, Departmet of Mathematics, Shrimathi Idira Gadhi Collage, Trichy , Tamiladu, Idia Keywords: Diophatie quadruples, System of equatios. MSC 000 Mathematics Subject Classificatio:11D99 ABSTRACT This paper cocers with the study of costructio of Diophatie quadruples such that the product of ay two elemets of the set added by a perfect square is a perfect square. INTRODUCTION Let q be a o-zero umber. A set a1, a,... a m of o-zero ratioal is called a - mtuple, if is a square for all 1 i j m. The mathematicia Diophatus of Alexadria cosidered a variety of problems Dq ( ) o idetermiat equatios with ratioal or itegers solutios. I particular, oe of the problems was to fid the sets of distict positive ratioal umbers such that the product of ay two umbers is oe less tha a ratioal square [1] ad Diophatus foud four positive ratioals,,, [,5]. The first set of four positive itegers with the same property, the set 1,3,8,10 was foud by Fermat. It was proved i 1969 by Baer ad Daveport [3] that a fifth positive iteger caot be added to this set ad oe may refer [6, 7,11] for geeralizatio. However, Euler discovered that a fifth ratioal umber ca be added to give the followig ratioal Diophatie quituple 1,3,8,10,. Ratioal sextuples with two equal elemets have bee give i []. I this 1999, Gibs [13] foud several examples of ratioal Diophatie sextuples, eg., ,,,,, , ,,,5,, All ow Diophatie quadruples are regular ad it has bee cojectured that there are o irregular Diophatie quadruples [1,13] (this is ow to be true for polyomials with iteger co-efficiets [8]). If so the there are o Diophatie quituples. However there are ifiitely may irregular ratioal Diophatie quadruples. The smallest is,5,,. May of these irregular quadruples are examples of aother commo type for which two of the subtriples are regular i.e., a, b, c, d is a irregular ratioal Diophatie quadruple, while abc,, ad,,d ab a a i j q are regular Diophatie triples. These are ow as semi regular ratioal Diophatie quadruples. These are oly fiitely may of these for ay give commo deomiator l ad they ca readily foud. Moreover i [1], it has bee proved that D( ) - triple, 1, 1 caot be exteded to a D( ) - quituple. I [10], it has bee proved that to a D( ) - quadruple if 5. - triple 1, 1, D( ) caot be exteded http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[10]

2 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research METHOD OF ANALYSIS SECTION I: I this sectio we search the diophatie quadruple (,,, ) a b c d such that product of ay two of them added with is a perfect square. Also,the fourth tuple is either iteger or ratioal umber. Cosider a (6 ) 6,& b (6 ) 6 ab Note that is a perfect square. Let c be ay o-zero iteger such that ac (1) bc () From (1), we have c a (3) Assume X ((6 ) 6) T () X ((6 ) 6) T (5) O substitutig the value of (3) i () ad by usig () ad (5), we get X ((6 ) 6)((6 ) 6) T whose iitial solutio is T0 1, X0 6( 1) Thus 6( 1) (6 ) 6 6( 1) (6 ) 6 Therefore from (3) c( 1) Let d be ay o-zero iteger such that ad A (6) bd B (7) cd C (8) Solvig (6), (7) ad (8) we get the value of d ( 1) d 1( 1) Substitutig the value of d i (6),(7) & (8) the 1( 1) 1 ( 1) ad 1( 1) 1 ( 1) bd 88( 1) cd Therefore ( a, b, c, d ) is a mixed diophatie quadruple with property D( ) as the fourth tuple may ot always be iteger. I what follows, a few examples of diophatie quadruple iteger are preseted. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[11]

3 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research Table ( a, b, c, d) (,8,,80) 3 (9,15,8,30) (1,,7,5760) 6 (,36,10,11880) 8 (3,50,168,1835) 1 (5,78,6,31680) (,10,,19) 3 (6,18,8,70) (10,6,7,1386) 6 (18,,10,880) 1 (,90,6,77) (0,1,,7) (6,30,7,576) 8 (18,66,168,1890) (-,1,,30) 3 (0,,8,1) Sectio II: I this sectio we search the diophatie quadruple ( a, b, c, d ) such that product of ay two of them added with is a perfect square. Assume a ad 1 b. ab is a perfect square. Let c be ay o-zero iteger such that ac (9) bc (10) From (9), we have Assume c (11) a X T (1) 1 X (. ) T (13) O substitutig the value of (11) i (10) ad by usig (1) ad (13), we get Whose iitial solutio is T0 1, Thus Therefore from (11) X (. ) T 1 0 X ( ) ( ) 1 ( ) (. ) c ( 1) ( 1) Let d be ay o-zero iteger such that ad A (1) bd B (15) http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[1]

4 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research cd C (16) Solvig (1), (15) ad (16) we get the value of d ( 1) ( ) 8 ( 1) 8 (3.. d ) Substitutig the value of d i (1),(15) & (16) the ( ) (. 1) ad (. 1) (3 ) ( 1) bd (. 3) (3.. 1) ( 1) cd Therefore ( a, b, c, d) is a mixed diophatie quadruple with property D ( ) as the fourth tuple may ot always be iteger, a few umerical examples of diophatie quadruple iteger are preseted i the followig table. Table. Sectio III: I this sectio we search the diophatie quadruple ( a, b, c, d ) such that product of ay two of them added with 1 3 ( a, b, c, d ) 1 (1,8,15,58) (1,1,1,30) 3 (1,16,7,80) (1,0,33,73) 6 (1,8,5,88) 1 (,7,110,1709) (,80,10,38808) (,96,10,1390) 5 (,10,150,10500) 8 (,18,180,638) 10 (,1,00,530) 1 (9,59,77,159760) (9,608,765,1888) 3 (9,6,783,195700) (9,60,801,115630) 6 (9,67,837,565500) 8 (9,70,873,38880) 9 (9,70,891,88360) 1 (16,18,658, ) (16,160,69, ). is a perfect square. Assume a Carl 1 1 ad ab. is a perfect square. b Ky 1 1 http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[13]

5 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research Let c be ay o-zero iteger such that ac. (17) bc. (18) From (17), we have. c (19) a 1 Assume X ( 1) T (0) 1 X ( 1) T (1) O substitutig the value of (19) i (18) ad by usig (0) ad (1), we get Thus X whose iitial solutio is Therefore from (19) ( 1)( 1) T. 1 1 T0 1, 0 1 X ( 1). 1. c ( 1) Let d be ay o-zero iteger such that ad A () bd B (3) cd C () Solvig (), (3) ad () we get the value of d d Substitutig the value of d i (),(3) & () the 3 ad bd cd Remar: It is see that the fourth tuple d is iteger oly whe 1 (-1,7,1,15) with the property D ( ). ad the correspodig quadruple is CONCLUSION To coclude oe may costruct a Diophatie quadruples with suitable properties. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[1]

6 [Geetha, (10): October, 015] ISSN Impact Factor (PIF):.3 Iteratioal Joural OF Egieerig Scieces & Maagemet Research REFERENCES [1]..G.Bashmaova (ed), Diophatus of Alexadria, Arithmetics ad the Boo of Polygoal Numbers, Naua, Moscow,197. [].A.F.Beardo ad M.N.Deshpade, Diophatie triples, The mathematical Gazette, 86, 00, [3].Bo He, A.Togbe, O the family of Diophatie triples 1,,9 3,Period Math Hugar,58,009, , A A,( A 1) ( A 1) with two []. Bo He, A.Togbe, O the family of Diophatie triples, parameters, Acta Math, Huger, 1,009, [5]. Bo He, A.Togbe, O the family of Diophatie triples,, A A,( A 1) ( A 1) with two parameters, Period Math Huger, 6,01, [6].Y.Bugeaud, A.Dujella ad M.migotte, O the family of Diophatie triples 1, 1,16 3, Glassgow Math J, 9, 007, [7].M.N.Deshpade ad E.Brow, Diophatie triples ad the Pell sequece, Fibaacci Quart, 39,001,-9. [8]. M.N.Deshpade, Oe iterestig family of Diophatie triples, Iterat. J.Math Ed.Sci.Tech., 33,00, [9]. M.N.Deshpade, Families of Diophatie triplets, Bulleti of the Marathwada mathematical society,,003,19-1. [10].A.Dujella ad C.Fuchs, Complete solutio of the polyomial versio of a problem of Diophatus, J.NumberTheory,106,00,36-3. [11].A.Dujella ad F.Luca, O a problem of Dioiphatus with polyomials, Rocy Moutai J. Math, 37,007, [1].A.Dujella ad V.Petricevic, Strog Diophatie triples, Experimet Math 17,008, F F, F, Fibaacci Quart, 8,01,19-7., 6 [13]. A.Filipi, Bo He, A.Togbe, O the D()-triple [1]. A.Filipi, Bo He, A.Togbe, O the family of two parameters D()-triples, Glas. Mat.ser.III,7,01, [15]. A.Filipi, No-extedability of D(-1) triples of the form {1,10,c}, Iterat J Math. Sci., 35,005,17-6. [16].M.A.Gopala ad G.Srividhya, Two special Diophatie Triples Diophatus J Math., 1(1) 01,3-7. [17].M.A.Gopala,V.Sageetha admaju Somaath, Costructio of the Diophatie Triple ivolvig polygoal umbers, Sch. J.Eg.Tech., (1),01,19- [ [18].M.A.Gopala, S.vidhyalashmi, S.Mallia, Special family of Diophatie Triples,Sch. J.Eg.Tech., (A), 01, [19].V.Padichelvi, Costructio of the Diophatie Triple ivolvig Polygoal umbers Impact J.Sci.Tech., 5(1), 011,7-11. http: // Iteratioal Joural of Egieerig Scieces & Maagemet Research[15]