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Special Dio-Quadruples Comprising of Centered Square Numbers with Property D(2)

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Special Dio-Quadruples Comprising of Centered Square Numbers with Property D(2) ADALYA JOURNAL ISSN NO: 1301-2746 SPECIAL DIO-QUADRUPLES COMPRISING OF CENTERED SQUARE NUMBERS WITH PROPERTY D(2) G.Janaki 1 and C.Saranya 2 Associate Professor1, Assistant Professor2, PG & Research Department of Mathematics, Cauvery College for Women(Autonomous),(Affiliated to Bharathidasan University), Trichy-18. E-mail: [email protected] & [email protected] Corresponding Author- C.Saranya, [email protected] Abstract: In this communication, we accomplish special Dio-quadruples comprising of centered square numbers such that the product of any two members of the set subtracted by their sum and increased by 2 is a perfect square. Keywords: Centered Square number, Diophantine quadruples, Pell equation. Notation CSn Centered square number of rank n. Introduction: Numbers are as unbounded as human understanding is confined, so number theory and its differing subfields will keep enthralling the brains of mathematicians for a very long time [1-4]. While individual equations present a kind of riddle and have been considered since the start, the meaning of general theories of Diophantine equations was an achievement of the twentieth century [5-6]. Many mathematicians considered the problem of the presence of Diophantine triples with the property D(n) for any integer n, furthermore for any linear polynomial in n [7-9]. In this specific situation, one may refer for a comprehensive review of various problems on special dio quadruples [10-11]. In this communication, we construct special Dio-quadruples a,b, c0 , c1 involving centered square numbers such that the product of any two members of the set subtracted by their sum and increased by 2 is a perfect square. Method of Analysis: 2 2 2 2 Let a CSn n (n -1) and b CSn1 (n 1) (n - 2) be Centered Square numbers of rank n and n 1respectively such that ab a b 2 is a perfect square. Let ck be any non-zero integer such that 2 ack a ck 2 uk (1) 2 bck b ck 2 vk (2) Volome 8, Issue 8, August 2019 602 http://adalyajournal.com/ ADALYA JOURNAL ISSN NO: 1301-2746 Eliminating ck between (1) and (2), we get 2 2 b a b 1uk a 1vk (3) Considering, the linear transformations uk xk a 1yk & vk xk b 1yk (4) in (3), it leads to the pell equation 2 2 2 2 xk 2n 2n2n 6n 5yk 1 (5) whose general solution is given by k1 k1 1 2 4 3 2 2 4 3 2 xk 2n 4n 1 4n 16n 22n 10n 2n 4n 1 4n 16n 22n 10n 2 k1 2 4 3 2 (6) 1 2n 4n 1 4n 16n 22n 10n y k 4 3 2 k1 2 4n 16n 22n 10n 2 4 3 2 2n 4n 1 4n 16n 22n 10n Put k 0 in (6), and substituting the values of x0 and y0 in (4), we get 2 u0 4n 6n 1 substituting u0 in (1), we get 2 c0 8n 16n 7 2(CSn CS n1 ) 5 put k 1 and applying the same procedure, we get 6 5 4 3 2 c1 128n 768n 1760n 1920n 1016n 240n 21 3 2 CS 3 2 80n 246n 202n 21 8n 24n 19n Thus, we attain CS , CS , 2(CS CS ) 5,CS 80n 3 246n 2 202n 21 as n n-1 n n1 8n3 24n2 19n a special Dio-quadruple with the property D(2). Some of the numerical examples of the above special Dio-quadruples are presented below. Table 1 n a,b,c0 ,c1 1 (1,1,-1,-3) 2 (5,1,7,21) 3 (13,5,31,5853) 4 (25,13,71,80853) 5 (41,25,127,484221) Volome 8, Issue 8, August 2019 603 http://adalyajournal.com/ ADALYA JOURNAL ISSN NO: 1301-2746 Remarkable Observation: For other choices of centered square numbers of different ranks, we obtain special Dio- quadruples with the property D(2) and are given in the following table. Table 2 a b c0 c1 3 2 2(CS CS ) 5 CS 3 16n 6n 10n 4 CSn CSn1 n n1 8n 5n 3 2 2(CS CS ) 5 CS 4 CS 3 2 240n 614n 414n 28 CSn n n1 3n2 24n 56n 33n 3 2 CS 2(CS CS ) 5 CS 4 CS 3 2 2928n 10790n 11966n 1980 n-1 n n1 3n3 24n 88n 97n Conclusion: In this paper, we construct the special Dio-quadruples involving centered square numbers with the property D (2). One may search for other special Dio-quadruples made up of different numbers with suitable properties. REFERENCES: [1] R.D.Carmichael, “History of Theory of numbers and Diophantine Analysis”, Dover Publication, New york, 1959. [2] L.J.Mordell, “Diophantine equations”, Academic press, London, 1969. [3] T.Nagell, “Introduction to Number theory”, Chelsea publishing company, New york, 1981. [4] L.K.Hua, “Introduction to the Theory of Numbers”, Springer-Verlag, Berlin-New york, 1982. [5] Y.Fujita , “The extendability of Diphantine pairs k 1, k 1”, Journal of Number Theory, 128, 322-353, 2008. [6] M.A. Gopalan and V.Pandichelvi , “On the extendability of the Diophantine triple involving Jacobsthal numbers J 2n1, J 2n1 3,2J 2n J 2n1 J 2n1 3 ”, International Journal of Mathematics & Applications, 2(1), 1-3, 2009. [7] G.Janaki and C.Saranya, “Special Dio 3-tuples for pentatope number”, Journal of Mathematics and Informatics, vol.11, Special issue, 119-123, Dec 2017. [8] G.Janaki and C.Saranya, “Construction of the Diophantine Triple involving Pentatope Number”, International Journal for Research in Applied Science & Engineering Technology, vol.6, issue III, 2317-2319, March 2018. [9] C.Saranya and G.Janaki , “Integer triples comprising of Jarasandha numbers in Arithmetic progression & Geometric progression”, International Journal of Research and Analytical Reviews, vol.6, issue 1, 751-752, Jan 2019. [10] A.Vijayasankar, M.A.Gopalan and V.Krithika, “Special Dio-quadruples with property D(2)”, International Journal of Applied Research, 2(9), 719-720, 2016. [11] M.A.Gopalan, S.Vidhyalakshmi and J.Shanthi, “On special Dio-quadruple with property D( S 2 1)”, World Journal of Engineering Research and Technology, 2(5), 142-145, 2016. Volome 8, Issue 8, August 2019 604 http://adalyajournal.com/.
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