Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 Scholars Journal of Physics Mathematics and Statistics Sch. J. Phys. Math. Stat. 0; ():88-9 Scholars Academic and Scientific Publishers (SAS Publishers) (An International Publisher for Academic and Scientific Resources) ISSN 9-806 (Print) ISSN 9-806 (Online) On Strong Rational Diophantine Quadruples with Equal Members M. A. Gopalan * S. Vidhyalashmi N. Thiruniraiselvi Professor Department of Mathematics SIGC Trichy-6000 Tamil Nadu India Research Scholar Department of Mathematics SIGC Trichy-6000 Tamil Nadu India *Corresponding Author: M.A. Gopalan Email: Abstract: This paper concerns with the study of constructing strong rational Diophantine quadruples from two given equal members. Keywords: Diophantine quadruples strong/regular rational Diophantine quadruples. 00 MSC Number: D09 D99 INTRODUCTION Let q be a non-zero rational number. A set a a a m of non-zero rational is called a rational d (q) m-tuple if a i a j + q is a square of a rational number for all i j m.the Gree mathematician Diophantus of Alexandria considered a variety of problems on in determinant equations with rational or integer solutions. In particular one of the problems was to find the sets of distinct positive rational numbers such that the product of any two numbers is one less than a rational square [] and Diophantus found four positive rationales 7 0 []. The first set of 6 6 6 four positive integers with the same property the set {80} was found by Fermat. It was proved in 969 by Baer and Davenport [] that a fifth positive integer cannot be added to this set and one may refer [6 7 ] for generalization. However Euler discovered that a fifth rational number can be added to give the following rational Diophantine quintuple 80 77780 8886 999Gibs [] found several examples of rational Diophantine sextuples e.g..rational sextuples with two equal elements have been given in []. In 8087 9 9 7 7 8 6 7 6 60. All nown Diophantine quadruples are regular and it has been conjectured that 8 8 8 7 79 there are no irregular Diophantine quadruples [] (this is nown to be true for polynomials with integer co-efficient [8]).If so then there are no Diophantine quintuples. However there are infinitely many irregular rational Diophantine quadruples. The smallest is 0. Many of these irregular quadruples are examples of another common type for which two of the sub-triples are regular i.e. a b c d is an irregular rational Diophantine quadruple while a b c and a b d are regular Diophantine triples. These are nown as semi-regular rational Diophantine quadruples. There are only finitely many of these for any given common denominator l and they can be readily found. Moreover in [] it has been proved that the D( )- triple ± ± cannot be extended to a D( )- quintuple. In [0] it has been proved that D( )- triple { + + } cannot be extended to a D( )- quadruple if. These results motivated us to search for strong rational Diophantine quadruples generated from two equal members. STRONG RATIONAL DIOPAHANTINE QUADRUPLES WITH EQUAL MEMBERS Theorem 9 Let A B C and D.Then (A B C D) is a d( ) strong rational Diophantine quadruple. Available Online: http://saspjournals.com/sjpms 88
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 AB ( ) ( ) AC ( ) ( ) BC ( ) 9 AD ( ) ( ) ( ) BD ( ) 9 CD ( ) ( ( i i ) ) (ABCD) Property d( d () 8 8 d (6) d () 6 7 6 6 d (8) 99 09 0 0 d (7) 6i 6i 8i d i i 8i 8i i 7 i8 i8 7 ( i ) ( i ) i i d(i ) ( ) (6 ) i d( ) ( ) 8i ) Available Online: http://saspjournals.com/sjpms 89
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 Theorem Let A B C and D ( ).Then (ABCD) is a d( ) strong rational Diophantine quadruple. AB ( ) ( ) AC ( ) ( ) BC ( ) AD ( ) ( ) ( ) ( ) BD ( ) CD ( ) ( )[( ) ] ( ) ( ) (ABCD) Property d( ) d () 6 ( i) 6 70 8 86 7 6 7 8i 7i ( i) ( i) i d (6) d () d (8) d (7) d(6 ) d( i) ( i8 i ) i i i i d(9 i6 ) Available Online: http://saspjournals.com/sjpms 90
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 Theorem 0 9 Let A B C and D.Then (ABCD) is a d( ) strong rational Diophantine quadruple. AB ( ) ( ) AC ( ) ( ) BC ( ) 0 9 AD ( ) ( ) BD ( ) CD ( ) 6 0 9 ( ) i (ABCD) 9 9 89 8 8 89 09 9 9 6 6 0 09 0 0 7i 9 0i 60 i i i i 7 7 69 0 8i 9 i80 ( i ) i i 8 i 8 i Property d( ) d( ) d () d (7) d () d () d(6i 0) d(9 6 d (i ) ) Available Online: http://saspjournals.com/sjpms 9
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 Theorem Let A B C and D.Then (ABCD) is a d( ) strong rational Diophantine quadruple. AB ( ) ( ) AC ( ) ( ) ( ) ( ) BC ( ) AD ( ) ( ) BD ( ) CD ( ) ( ) ( ) ( ) (ABCD) Property Property d( ) d( ) 0 i i 06 7 7 86 6 (i 7) 8i 7 i i i i 60 i8 i i d () d (7) d () d () d(8i 7) d( i6 ) ( ) 0 9 6 d( ) Available Online: http://saspjournals.com/sjpms 9
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 Theorem Let A B C ( ) and ( ) 8 D (ABCD) is a d( ) strong rational Diophantine quadruple. AB ( ) ( ) AC ( ) ( ) ( ) ( ) BC ( ) AD ( ) ( ) 8 BD ( CD ( ) {( ) } ( ) ).Then ( ) 8 ( ( ) (ABCD) ) Property d( ) 7 d () 0 07 0 9 8 9 9 ` 6 6 00 d (6) 0 0 0 Theorem 6 Let A B C (s ) s ( ) and (s ) 8s ( s ) D.Then (ABCD) is a d( ) strong rational Diophantine quadruple. Available Online: http://saspjournals.com/sjpms 9
Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp-88-9 AB ( ) ( ) AC ( ) (s ) s ( ) ( ) ( s) BC ( ) (s ) 8s ( s ) AD ( ) s BD ( ) CD ( ) {(s ) s ( )} ( ) (s ) 8s ( s ) ( (s ) s ( ) s S (ABCD) Available Online: http://saspjournals.com/sjpms 9 ) Property d( ) d ( 8 ) 0 07 0 9 8 9 9 869 7 6 CONCLUSION In this paper we have presented six strong rational Diophantine quadruples. To conclude one may search for other families of strong and almost strong Diophantine quadruples with suitable property. ACKNOWLEDGEMENT The finicial support from the UCG New Delhi (F-MRP-/(SERO/UCG) dated march 0) for a part of this wor is gratefully acnowledged. REFERENCES. Arin J Hoggatt VE Straus EG; On Euler s solution of a problem of Diophantus Fibonacci Quart. 979; 7 : -9.. Arin J Arney DC Giordano FR Kolb RA Bergaum GE; An extension of an old classical Diophantine problem. Applications of Fibonacci numbers Kluwer Acad Publ Dordrecht. 99; : -8.. Baer A Davenport H; The equations x = y and 8x 7 = z. Quart J Math Oxford Ser. 969; 0 (): 9-7.
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