Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. SOME NON-ETENDABLE DIOPHANTINE TRIPLES IN SPECIAL NUMBER PATTERNS *M.A.Gopala, S. Vidhyalakshmi ad N. Thiruiraiselvi Departmet of Mathematics, SIGC, Trichy-6000, Tamil Nadu *Author for Correspodece ABSTRACT I this paper, we preset three o-extedable Diophatie triples whose members are special umbers, amely, Triagular umber, Jacobsthal umber, Jacobsthal-Lucas umber, Kyea umber ad Star umber with suitable property. Keywords: Diophatie Triples, Iteger Sequeces 010 Mathematical subject classificatio; 11D99 Notatios t - Polygoal umber of rak with size m. m, ky -Kyea umber of rak S -Star umber of rak J -Jacobsthal umber of rak j -Jacobsthal-Lucas umber of rak INTRODUCTION A Set of positive itegers ( a 1, a, a3,... am is said to have the property D(, z 0, if a i a j is a perfect square for all 1 i j m ad such a set is called a Diophatie m-tuple with property D(. May mathematicias cosidered the problem of the existece of Diophatie quadruples with the property D( for ay arbitrary iteger [1] ad also for ay liear polyomial i.i this cotext, oe may refer (Thamotherampillai, 1980; Brow, 1985; Gupta ad Sigh, 1985; Beardo ad Deshpade, 00; Deshpade, 00; Deshpade, 003; Bugeaud et al., 007; Liqu, 007; Fujita, 008; Filipi et al., 01; Gopala ad Padichelvi, 011; Fujita ad Togbe, 011; Gopala ad Srividhya, 01; Gopala ad Srividhya, 01; Filipi, 005; Gopala et al., 014; Gopala et al., 014; Padichelvi, 011 for a extesive review of various problem o Diophatie triples. These results motivated us to search for o-extedable Diophatie triples with elemets represeted by special umbers, amely, polygoal umbers ad other special umber patters. I this paper, we exhibit three o-extedable Diophatie triples whose members are Triagular umber, Jacobsthal umber, Jacobsthal-Lucas umber, Kyea umber ad Star umber with suitable property. METHOD OF ANALYSIS 1 I: No-extedable D(4t 1 3, Diophatie triple: Let a ky ( ad b j be two itegers such that ab+1 is a perfect square Let c be ay o-zero iteger such that 1 ( (c (4t 1 3, (1 1 (( (c (4t 3, Elimiatig c from (1 ad (, we obtai Copyright 014 Cetre for Ifo Bio Techology (CIBTech 1 (
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. (( 1 ( ( ( [(( ](4t (3 Usig the liear trasformatios ( T (4 (( T (5 i (3, it leads to the pell equatio (( ( T (4t 3, 1 Let T 0 1 ad 0 ( ( be the iitial solutio of (6. Thus (4 yields 0 ( 1 Ad usig (1, we get c 8t 1 3, Hece (a, b, c=( ky, j 1, 8t 1 is the Diophatie triple with property 1 D(4t 1 3, 3, Copyright 014 Cetre for Ifo Bio Techology (CIBTech 3, Some umerical examples are preseted below (a, b, c with property 1 D(4t 1 3, 1 (7, 4, 5 with property D(1 (3, 16, 81with property D(73 3 (79, 64, 89with property D(73 4 (87, 56, 1089with property D(1057 5 (1087, 104, 45with property D(4161 We show that the above triple caot be exteded to quadruple Let d be ay o-zero iteger such that 1 (( (d (4t 1 p 3, 3, 1 ( (d (4t q 1 ( 4( 4( (d (4t r 3, Elimiatig d from (8 ad (9, we obtai (4( 4( (q ( Usig the liear trasformatios (r [4( 4( ( ]((4t 3, 1 q (4( 4( T (11 r ( T (1 i (10, it leads to the pell equatio (4( 4( ( T (4t 3, 1 (6 (7 (8 (9 (10 (13
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. Let T 0 1 ad 0 ( ( 1 be the iitial solutio of (13. Thus (11 yields q0 3( 1 Ad usig (8, we get d 9( 6( 3 Verify Quadruple: Substitutig the above value of d i L.H.S of (7, we have 1 ad (4t 1 [3( 4( ] ( 1 (5( 1 3, Note that the R.H.S is ot a perfect square 1 Hece the triple ( ky, j 1,8t with property D(4t 1 3, 3, quadruple. caot be exteded to a Note: The triple ( ky, j 1,8t is a strog Diophatie triple ad the quadruple 3, (ky, j 1,8t 1,9(j 3(( is almost strog Diophatie quadruple. 3, II: No-extedable D( Diophatie triple: Let a 8t3, 4 4 ad Let c be ay o-zero iteger such that b j be two itegers such that ab+1 is a perfect square ( c (14 ( 4 4(c (15 Elimiatig c from (14 ad (15, we obtai (4 4( ( ( [((4 4 ] (16 Usig the liear trasformatios ( T (17 (4 4T (18 i (16, it leads to the pell equatio (4 4( T ( (19 Let T 0 1 ad 0 ( ( be the iitial solutio of (19. Thus (17 yields 0 ( ( Ad usig (14, we get c 8t (j ( (3J j Copyright 014 Cetre for Ifo Bio Techology (CIBTech 3 3, Hece (a, b, c=( t, j 1,8t (j ( (3J j is the Diophatie triple with property D( 8 3, 3,
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. Some umerical examples are preseted below (a, b, c with property D( 1 (8, 4, 4 with property D(4 (4, 16, 80 with property D(16 3 (48, 64, 4 with property D(64 4 (80, 56, 64 with property D(56 5 (10, 104, 1848 with property D(104 We show that the above triple caot be exteded to quadruple Let d be ay o-zero iteger such that ( 4 4(d p (0 ( (d q (1 ( 4 4 ( (4 (d r ( Elimiatig d from (1 ad (, we obtai (4 4 (4 (q ( (r [4 4 (4 ]( (3 Usig the liear trasformatios q ( T (4 r (4 4 (4 T (5 i (3, it leads to the pell equatio (4 4 (4 T (6 Let T 0 1 ad 0 ( ( be the iitial solutio of (6. Thus (4 yields q0 ( ( Ad usig (1, we get d 8t 4[(j ( (3J j ] 3, Verify Quadruple: Substitutig the above value of d i L.H.S of (0, we have ad [4 4( Note that the R.H.S is ot a perfect square ] 8 8 Copyright 014 Cetre for Ifo Bio Techology (CIBTech 4 8 Hece the triple ( t, j 1,8t (j ( (3J j with property D( caot be exteded to a quadruple. 8 3, 3, Note: The triple ( t, j 1,8t (j ( (3J j is a strog Diophatie triple ad the 8 3, 3, quadruple ( 8t 3,, j 1,8t 3, (j ( (3J j,8t 3, 4[(j ( (3J j ] is almost strog Diophatie quadruple.
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. II: No-extedable D(3 ( Diophatie triple: Let a S 6 6 1 ad b j be two itegers such that ab+1 is a perfect square Let c be ay o-zero iteger such that ( c (3 ( (7 ( 6 6 (c (3 ( (8 Elimiatig c from (7 ad (8, we obtai (6 6 ( ( ( [((6 6 (3 ( ] (9 Usig the liear trasformatios ( T (30 (6 6 T (31 i (9, it leads to the pell equatio (6 6 ( T (3 ( (3 Let T 0 1 ad 0 (3 ( be the iitial solutio of (3. Thus (30 yields 0 ( (3 Ad usig (7, we get c S (j (3 (3J j Hece (a, b, c=( S, j 1,S (j (3 (3J j is the Diophatie triple with property D(3 ( Some umerical examples are preseted below (a, b, c with property D(3 ( 1 (1, 4, 13 with property D(1 (13, 16, 69 with property D(19 3 (37, 64, 9 with property D(178 4 (73, 56, 681 with property D(188 5 (11, 104, 041 with property D(76800 We show that the above triple caot be exteded to quadruple Let d be ay o-zero iteger such that ( 6 6 (d (3 ( p (33 ( (d (3 ( q (34 ( 6 6 (6 ( ( (d (3 ( r (35 Elimiatig d from (34 ad (35, we obtai (6 6 (6 ( ( Usig the liear trasformatios (q ( (r Copyright 014 Cetre for Ifo Bio Techology (CIBTech 5 [(6 6 (6 ( ](3 ( (36 q ( T (37 r (6 6 (6 ( ( T (38
Iteratioal Joural of Iovative Research ad Review ISSN: 347 444 (Olie A Olie Iteratioal Joural Available at http://www.cibtech.org/jirr.htm 014 Vol. (3 July-September, pp.1-7/gopala et al. i (36, it leads to the pell equatio ( (6 6 (6 ( ( T (3 ( (39 Let T 0 1 ad 0 (3 ( be the iitial solutio of (39. Thus (37 yields q0 ( (3 Ad usig (34, we get d S 4(j (3 (3J j ] Verify Quadruple: Substitutig the above value of d i L.H.S of (33, we have ad (3 ( [(S (3 (3J j Note that the R.H.S is ot a perfect square ] 9 Hece the triple ( S, j 1,S (j (3 (3J j with property D(3 ( be exteded to a quadruple. caot Note: The triple ( S, j 1,S (j (3 (3J j is a strog Diophatie triple ad the quadruple { S, j 1,S (j (3 (3J j,s 4(j (3 (3J j } is almost strog Diophatie quadruple. ACKNOWLEDGEMENT *The fiacial support from the UCG, New Delhi (F-MRP-51/14(SERO/UCG dated march 014 for a part of this work is gratefully ackowledged. REFERENCES Bashmakova IG (1974. Diophatus of Alexadria. Arithmetics ad the Book of Polygoal Numbers (Nauka, Moscow. Beardo AF ad Deshpade MN (00. Diophatie triples. The Mathematical Gazette 86 53-60. Brow V (1985. Sets i which xy+k is always a square. Mathematics of Computatio 45 613-60. Bugeaud Y, Dujella A ad Migotte (007. O the family of Diophatie triples 3 ( k 1, k 1,16k 4k. Glasgow Mathematical Joural 49 333-344. Deshpade MN (00. Oe iterestig family of Diophatie Triples. Iteratioal Joural of Mathematical Educatio i Sciece ad Techology 33 53-56. Deshpade MN (003. Families of Diophatie Triplets. Bulleti of the Marathawada Mathematical Society 4 19-1. Filipi A, Fujita Y ad Migotte M (01. The o extedibility of some parametric families of D(-1- triples. Quarterly Joural of Mathematics 63 605-61. Flipi A, Bo He ad Togbe A (01. O a family of two parametric D(4 triples. Glasik Matematicki Ser.III 47 31-51. Fujita Y (006. The o-extesibility of D(4k-triples {1,4k(k,4k 1} With ǀkǀ prime. Glasik Matematicki Ser. III 41 05-16. Fujita Y (006. The uique represetatio d 4k( k i D(4-quadruples {k-, k+, 4k, d}. Mathematical Commuicatios 11 69-81. Fujita Y (008. The extesibility of Diophatie pairs (k-1, k+1. Joural of Number Theory 18 3-353. Copyright 014 Cetre for Ifo Bio Techology (CIBTech 6
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