On Strong Rational Diophantine Quadruples with Equal Members

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1 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 (Print) ISSN (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 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 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 Gibs [] found several examples of rational Diophantine sextuples e.g..rational sextuples with two equal elements have been given in []. In All nown Diophantine quadruples are regular and it has been conjectured that 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: 88

2 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 () d (8) 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: 89

3 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) i 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: 90

4 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 ( ) ( ) i (ABCD) i 9 0i 60 i i i i i 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: 9

5 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 (i 7) 8i 7 i i i i 60 i8 i i d () d (7) d () d () d(8i 7) d( i6 ) ( ) d( ) Available Online: 9

6 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 () ` d (6) Theorem 6 Let A B C (s ) s ( ) and (s ) 8s ( s ) D.Then (ABCD) is a d( ) strong rational Diophantine quadruple. Available Online: 9

7 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: 9 ) Property d( ) d ( 8 ) 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.

8 Gopalan MA et al.; Sch. J. Phys. Math. Stat. 0; Vol-; Issue-(Sep-Nov); pp Dicson LE; History of theory of Numbers. Volume Chelsea New Yor966: -0.. Bashmaova IG; Diophantus of Alexandria Arithmetics and the Boo of Polygonal Numbers (Ed.)NauaMoscow97 (in Russian) A.DujellaThe problem of the extension of a parametric family of Diophantine triples Publ Math Debreen. 997; : Dujella A Petho A; A generalization of a theorem of Baer and Davenport. Quart J Math Oxford Ser. 998; 9(): Dujella A Fuchs C; Complete solutions of the polynomial version of a problem of Diophantus. J Number Theory 00; 06: Dujella A Petricevic V; Strong Diophantine Triples. Experimental Math. 008; 7(): Available from QuASJsHIBw&ved=0CBMQFjAA&usg=AFQjCNFJq-pxbUCuSQjGYVO8tAleN-7qTg 0. Flipin A Fujita Y; The D( )-triple + + with prime. Glas Mat Ser. 0; III 6: -.. Fujita Y; The extensibility of Diophantine pairs +. J Number Theory 008; 8: -.. Fujita Y; Extensions of the D( )-triple ± ±. Period Math Hungarica. 009; 9: Gibbs PE; Computer Bulletin 978; 7:6.. Heath TL; Diophantus of Alexandria: A study in the History of Gree Algebra. Dover Publications Inc. New Yor 96. Available Online: 9

1 Diophantine quintuple conjecture

1 Diophantine quintuple conjecture Conjectures and results on the size and number of Diophantine tuples Andrej Dujella Department of Mathematics, University of Zagreb Bijenička cesta 30, 10000 Zagreb, CROATIA E-mail: duje@math.hr URL :