Formation of Triples Consist Some Special Numbers with Interesting Property

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 04 Issue: 07 | July -2017 www.irjet.net p-ISSN: 2395-0072

FORMATION OF TRIPLES CONSIST SOME SPECIAL NUMBERS WITH INTERESTING PROPERTY

V. Pandichelvi1, P. Sivakamasundari2

1Assistant Professor, Department of Mathematics, UDC, Trichy. 2Guest Lecturer, Department of Mathematics, BDUCC,Lalgudi, Trichy. ------***------Abstract- In this communication, we discover the triple them is a perfect square is given in the following two a,b,c involving some figurate numbers such that the sum sections. of any two of them is a perfect square. Also, we find some interesting relations among the triples. Section- A

Key words Diophantine m -tuples, polygonal numbers. : Let an  2CP2n1 bn Dec Gno INTRODUCTION: Let n be an integer. A set of positive 2n2 2n2 which are equivalent to the following two equations integers a1,a2,a3,...... amis said to have the an  20n2  30n 12,bn 16n2  30n 13 the property D(n) if ai a j  n is a perfect square for all Now, we assume that 1  i  j  m such a sets is called a Diophantine m-tuple. such an bn   2 sets were studied by Diophantus [1].The set of numbers 1,2,7is Diophantine triple with property D(2).For an Let cn be any non-zero integer such that extensive review of various articles one may refer [2-12]. In b(n)  cn   2 (1) this communication, we find the triple a,b,c involving centered pentagonal numbers, decagonal numbers, a(n)  cn   2 (2) Gnomonic numbers, Kynea numbers and Jacobsthal-lucas Subtracting (2) from (1), we get numbers such that the sum of any two of them is a perfect  2  2  b(n)  an (3) square. Also, a few interesting relations among the triples are presented. Put   A 1,  A in (3), we get   A  2n2 (4) Notations: Substituting (4) in (2), the values of c are represented by 4 2 5n2  5n  2 cn  4n  20n  30n 12 (5) Let CP  be a centered pentagonal number of n 2 Hence, rank . 20n2  30n 12,16n2  30n 13,4n4  20n2  30n 12 is a 2 t10,n  4n  3n be a decagonal number of rank n . triple in which the sum of any two of them is a perfect square. Gno  2n 1 be a Gnomonic number of rank n . n 2n n1 Kn  2  2 1 be a Kynea number of rank n . Table-1: Some numerical examples are illustrated n n below: jn  2  1 be a Jacobshthal-lucas number of rank n .

n an bn cn a(n)  bn a(n)  cn b(n)  cn Method of analysis: 1 62 59 58 112 22 12 The procedure for finding the triple involving 152 137 88 2 2 2 some interesting numbers such that the sum of any two of 2 17 8 7

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3 282 247 42 232 182 172 Table-2: Some numerical examples are illustrated below: 4 452 389 572 292 322 312 n an bn cn a(n)  bn a(n)  cn b(n)  cn 5 662 563 1838 352 502 492

1 23 58 266 92 172 182 A few interesting relations among the numbers are given 287 154 4202 2 2 2 below: 2 21 67 66 4223 538 3392426 2 2 2 1. anbn 36 Pron 12Gnon1 13 69 1843 1842 66047 2074 572 2 2 2 2. (1/ 4)an cn is a bi-quadratic integer 261 31987 31986

1050623 8218 3. (1/ 4)[bncn36Pron 12Gnon 37] is a 2.7165151661011 10292 5212032 5212022

bi-quadratic integer

4. (1/ 4)[ancn 40Pron 10Gnon 34]is a CONCLUSION:

bi-quadratic integer In this communication, we discover the triple involving SECTION-B various special numbers in such a way that the sum of any two of them is a perfect square. In this manner, one may seek out other triples and quadruples satisfying some other Let an  K 2n properties. bn  8j2n  j1  j4 which are equivalent to the following two equations REFERENCES: an  24n  2.22n 1,bn  8.22n  26 [1]. Baker A.,Davenport H., The equations 3x2  2  y 2 Now, we assume that 2 2 an bn   2 and 8x 7  z , J.Math.Oxford Ser. 20 (1969),129-137. Let cn be any non-zero integer such that [2]. I.G Bashmakova(ed), Diophantus of Alexandria, Arithmetic and the Book of polygonal numbers, Nauka, b(n)  cn   2 (6) Moscow, 1974. a(n)  cn   2 (7) [3]. Dickson L.E; History of theory of numbers ,Vol.2, Chelsea , New York 1966; 513-520. Subtracting (7) from (6), we get  2  2  b(n)  an (8) [4]. Thamotherampillai N; The set of numbers {1, 2, 7}.Bull of Calcutta math soc.,1980; 72:195-197. The choices   A 1,  A lead (8) to [5]. Brown E; sets in which xy  k is always a square, Math   A  3.22n  24n1 13 (9) comp., 1985; 45: 613-620. Substituting (9) in (7), the values of c are represented by 4n 8n2 6n1 2n 4n1 [6]. Gupta H, singh K; On K-triad sequences. Internet J Math cn  8.2  2 6.2  76.2  26.2 170 Sci., 1985; 5: 799 - 804. Hence, [7]. Dujella A., Complete solution of a family of simultaneous 24n  2.22n 1,8.22n  26,8.24n  28n2  6.26n1  76.22n  26.24n1 170 pellian equations, Acta Math. Inform. Univ. Ostraviensis is a triple in which the sum of any two of them is a perfect 6 (1998), 59-67. square. [8]. Beardon AF, Deshpande MN; Diophantine triples. The athematical Gazette, 2002, 86:258-260 [9]. Deshpande MN; Families of Diophantine triplets,Bulletin of the Marathwada Mathematical Society, 2003; 4: 19-21.

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[10].Bugeaud Y, Dujella A, Mignotte M; On the family of Diophantine triples { k 1,k 1,16k 3  4k }. Glasgo Math J. 49(2007),333-334. [11].Gopalan M.A., Pandichelvi V; On the extendibility of the Diophantine triple involving Jacobsthal numbers

J2n1, J2n1  3,2J2n  J2n1  J2n1  3, International journal of Mathematics and Applications, 2009;2(1):1-3. [12]Pandichelvi V, Construction of the Diophantine triple involving polygonal numbers. Impact J SciTech., 2011; 5(1): 7-11.