SOME NON-EXTENDABLE DIOPHANTINE TRIPLES IN SPECIAL NUMBER PATTERNS

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1 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at 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 (

2 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at 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( (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

3 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at 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,

4 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at 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.

5 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at Vol. (3 July-September, pp.1-7/gopala et al. II: No-extedable D(3 ( Diophatie triple: Let a S 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

6 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at 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 Brow V (1985. Sets i which xy+k is always a square. Mathematics of Computatio Bugeaud Y, Dujella A ad Migotte (007. O the family of Diophatie triples 3 ( k 1, k 1,16k 4k. Glasgow Mathematical Joural Deshpade MN (00. Oe iterestig family of Diophatie Triples. Iteratioal Joural of Mathematical Educatio i Sciece ad Techology Deshpade MN (003. Families of Diophatie Triplets. Bulleti of the Marathawada Mathematical Society Filipi A, Fujita Y ad Migotte M (01. The o extedibility of some parametric families of D(-1- triples. Quarterly Joural of Mathematics Flipi A, Bo He ad Togbe A (01. O a family of two parametric D(4 triples. Glasik Matematicki Ser.III Fujita Y (006. The o-extesibility of D(4k-triples {1,4k(k,4k 1} With ǀkǀ prime. Glasik Matematicki Ser. III Fujita Y (006. The uique represetatio d 4k( k i D(4-quadruples {k-, k+, 4k, d}. Mathematical Commuicatios Fujita Y (008. The extesibility of Diophatie pairs (k-1, k+1. Joural of Number Theory Copyright 014 Cetre for Ifo Bio Techology (CIBTech 6

7 Iteratioal Joural of Iovative Research ad Review ISSN: (Olie A Olie Iteratioal Joural Available at Vol. (3 July-September, pp.1-7/gopala et al. Gopala MA ad Padichelvi V (011. The No Extedibility of the Diophatie Triple (4(m 1,4(m 1,4( m 8(m. Impact Joural of Sciece ad Techology 5( Gopala MA ad Srividhya G (01. Diophatie Quadruple for Fioacci ad Lucas Numbers with property D(4. Diophatus Joural of Mathematics 1( Gopala MA ad Srividhya G (01. Some o extedable P 5 sets. Diophatus Joural of Mathematics 1( Gopala MA ad Srividhya G (01. Two Special Diophatie Triples. Diophatus Joural of Mathematics 1( Gopala MA, Vidhyalakshmi S ad Mallika S (014. Some special o-extedaibility Diophatie Triple. Scholars Joural of Egieerig ad Techology (A Gupta H ad Sigh K (1985. O k-triad Sequeces. Iteratioal Joural of Mathematics ad Mathematical Scieces 8( Srividhya G (009. Diophatie Quadruples for Fibboacci umbers with property D(1. Idia Joural of Mathematics ad Mathematical Sciece 5( Tao Liqu (007. O the property P 1. Electroic Joural of Combiatorial Number Theory 7 #A47. Thamotherampillai V (1980. The set of umbers {1,, 7}. Bulleti of the Calcutta Mathematical Society Yasutsugu Fujita ad Alai Togbe (011. Uiqueess of the extesio of the D(4k -triple ( k 4, k,4k 4. Notes o Number Theory ad Discrete Mathematics 17( Copyright 014 Cetre for Ifo Bio Techology (CIBTech 7

On Non-Extendable Special Dio-3-Tuples

ISSN: 319-8753 (An ISO 397: 007 Certified Organization) Vol. 3, Issue 8, August 014 On Non-Extendable Special Dio-3-Tuples M.A.Gopalan 1 *, S. Vidhyalakshmi, N. Thiruniraiselvi 3 1, Professor, Department