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On Homogeneous Ternary Quadratic Diophantine Equation ( )( R.Anbuselvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 11, ( Part -4) November 2017, pp.22-25 RESEARCH ARTICLE OPEN ACCESS On Homogeneous Ternary Quadratic Diophantine Equation 2 2 2 z 45x y R.Anbuselvi1, S.A.Shanmugavadivu2 1. Associate Professor of Mathematics, ADM College for women (Autonomous), Nagapattinam, Tamilnadu, India 2. Assistant Professor of Mathematics, Thiru.Vi.Ka. Govt. Arts College, Thiruvarur, Tamilnadu, India. Abstract The ternary quadratic homogeneous equation representing homogenous cone given by z 2 45x 2 y 2 is analyzed for its non-zero distinct integer points on it. Four different patterns of integer points satisfying the cone under consideration are obtained. A few interesting relations between the solutions and special number patterns namely Polygonal number, Pyramidal number, Octahedral number, Stella Octangular number and Oblong number are presented. Also knowing an integer solution satisfying the given cone,two triples of integers generated from the given solution are exhibited. Keywords: Ternary homogeneous quadratic,integral solutions. ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: 01-09-2017 Date of acceptance: 17-11-2017 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. INTRODUCTION we have different pattern of solutions of (1) which The Ternary quadratic Diophantine are illustrated below. equations offer an unlimited field for research Pattern-I because of their variety [1-5]. For an extensive view Consider (1) as of various problem one may refer [6-20].This z 2 1 45 x 2 y 2 communication concerns with yet another (2) interestingfor ternary quadratic equation Assume 2 2 z 2 45x 2 y 2 for determining its infinitely z a 45b (3) many non-zero integral solutions. Also a few Write 1 as 22 2n 2n2 i 45 2n 122 2n 2n2 i 45 2n 1 interesting relations among the solutions have been 1 2 presented. 23 2n 2n2 Notations used 푇푚,푛 -Polygonal number of rank n with size m. Substituting (3) and (4) in (2) and employing the m CPn - Centered Pyramidal number of method of factorization, define 2 rank n with size m. 22 2n 2n2 i 45 2n 1a i 45 b y i 45x 2 푃푟푛 - Pronic number of rank n. 23 2n 2n 푆푂푛 -Stella Octangular number of rank n. 푂푏푙푛 -Oblong number of rank n. Equating the real and imaginary parts in the above 푂퐻푛 - Octahedral number of rank n. equation, we get 푃푡푛 -Pentatope number of rank n. 푃푃푛 - Pentagonal Pyramidal number of rank n 22 2n 2n 2 2ab 2n 1a 2 45b 2 Method of analysis x 2 The ternary quadratic equation under consideration 23 2n 2n is z 2 45x 2 y 2 (1) www.ijera.com DOI: 10.9790/9622-0711042225 22 | P a g e R.Anbuselvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 11, ( Part -4) November 2017, pp.22-25 2 2 2 2 22 2n 2n a 45b 2n 190ab z 23 2n 2n2 A2 45B2 y 2 23 2n 2n For simplicity and clear understanding, taking n=1 in the above equations, the corresponding integer solutions of(1) are given by Replacing a by (23-2n+2n2)A,b by(23-2n+2n2)B in the above equation the corresponding integer x 23 A2 1035 B 2 1012 AB solutions of (1) are given by y 506 A2 22720 B 2 2070 AB 2 2 2 2 x 23 2n 2n 22 2n 2n 2AB2n 1A 45B z 529 A2 23805 B 2 y 23 2n 2n2 22 2n 2n2 A2 45B 2 90AB2n 1 Properties 1) 23x A,7A2 4 z A,7A2 4 69828CP 14 1058Pr 0 mod1058 A A 2) x B 1, B y B 1, B z B 1, B1058 Pr T 0mod 47609 B 95222,B 2 2 3) 23xA,2A 1 z A,2A 11058obl A 23276 SOA 0mod1058 4) 22x A, A A 1 y A, A A 1 48668PP 0 A 5) xA,11035obl T 1035mod1034 A 2062,A 6) z 1, B 203805 Pr 0 mod23805 B Pattern-II It is worth to note that 1 in (2) may also be represented as 45 4n2 i 45 4n45 4n2 i 45 4n 1 2 45 4n2 Following the analysis presented above, the corresponding integer solution to (1) are found to be x 45 4n2 45 4n2 2AB 4n A2 45B 2 y 45 4n2 45 4n2 A2 45B 2 360 ABn 2 z 46 4n2 A2 45B2 For the sake of simplicity, taking n=1 in the above equations, the correspondinginteger solutions of (1) are given by 2 2 x 196A 8820B 4018AB y 2009A2 90405B 2 17640AB 2 2 z 2401 A 108045B Properties 1) 41xA, AA 1 4 yA, AA 1 141120PP 0 A 2 2 2) 2`009 xA,2A 1196 y A,2A 111529602 SOA 0 www.ijera.com DOI: 10.9790/9622-0711042225 23 | P a g e R.Anbuselvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 11, ( Part -4) November 2017, pp.22-25 2 2 3) 2009xA,2A 1196 y A,2A 1 34588806 OH A 0 4) xA,1196PrA 8820mod3822 5) 41xA,A 1A 2A 3 4yA,A 1A 2A 31693440 PtA 0 Pattern-III Equation (1) can be written as z y x P 45x z y Q (5) From equation (5),we get two equations 45P xQ y Q z 0 Q x P y P z 0 We get the integer solutions are x xP,Q 2PQ y yP,Q 45P 2 Q2 2 2 z zP,Q 45P Q Properties 1) y a,a z a,a 90Pra 0mod90 2) x a,a 2 2CP 6 0 a 2 2 2 3) xP, 2P 1 yP,2P 1 zP,2P 1 290PrP T402,P 2SOP 0mod489 4) xP,1 yP,1 45oblP 1mod47 [2] Mordell L.J., “Diophantine Equations” II. GENERATION OF INTEGER Academic Press, new York, 1970 SOLUTIONS [3] RD. Carmicheal, “The Theory of Numbers Let (x0, y0, z0) be any given integer solution and Diophantine Analysis”, Dover of (1). Then, each of the following triples of integers Publications, New York 1959 satisfies (1): [4] M.A. Gopalan, ManjuSomnath and Triple 1: (x1, y1, z1) N.Vanitha, “On ternary cubic Diophantine n equation x2+y2 = 2z3 Advances in x1 3 x0 Theoretical and Applied Mathematics”, Vol. 1 n n n n y1 83 2 3 y0 43 4 3 z0 1, no.3, 227-231, 2006. 6 [5] M.A. Gopalan, ManjuSomnath and N. Vanitha, “Ternary cubic Diophantine equation 1 z 43n 4 3n y 23n 8 3n z x2 – y2 = z3”, ActaCienciaIndica”, 1 6 0 0 Vol.XXXIIIM, No. 3, 705-707, 2007. [6] M.A. Gopalan and R.Anbuselvi, “Integral Solutions of ternary cubic Diophantine III. CONCLUSION equation x2 + y2 + 4N = zxy”, Pure and In this paper, we have presented four different Applied Mathematics Sciences Vol.LXVII, integer No. 1-2, 107-111, March 2008. patterns of infinitely many non-zero distinct [7] M.A. Gopalan, ManjuSomnath and N. solution of the homogeneous cone given by Vanitha, “Ternary cubic Diophantine equation 2 2 2 z 45 x y .To conclude, one may search for 22a-1 (x2+y2) = z3, actaCienciaIndia”, other patterns of solutions and their corresponding Vol.XXXIVM, No. 3, 1135-1137, 2008. properties. [8] M.A. Gopalan, J.Kaligarani Integral solutions 3 3 3 x + y + 8k(x+y) = 2k+1)z Bulletin of Pure REFERENCES and Applied Sciences, vol 29E, No.1, 95- [1] Dickson, L.E. “History of Theory of Numbers 99,2010 and Diophantine Analysis”, Vol.2, Dover [9] M.A.Gopalan, J.Kaligarani Integral solutions Publications, New York 2005. x3 + y3 + 8k(x+y) = (2k+1)z3 Bulletin of www.ijera.com DOI: 10.9790/9622-0711042225 24 | P a g e R.Anbuselvi. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 7, Issue 11, ( Part -4) November 2017, pp.22-25 Pure and Applied Sciences, Vol. 29E, No.1, [17] M.A.Gopalan, B.Sivakami Integral solutions 95-99, 2010 of the ternary cubic equation 4x2 – 4xy + 6y2 [10] M.A.Gopalan, S.Premalatha On the ternary = ((k+1)2+5)w3 Impact J. Sci. Tech., Vol 6 cubic equation x3 + x2 + y3 – y2 = 4(z3 + z2) No. 1, 15-22, 2012 Cauvery Research Journal Vol. 4, iss 1&2 87- [18] M.A.Gopalan, B.Sivakami on the ternary 89, July 2010-2011. cubic diophantine equation 2xz = y2(x+z) [11] M.A.Gopalan, V.Pandichelvi, observations on Bessel. J.Math 2(3), 171-177, 2012. the ternary cubic diophantine equation x3 + y3 [19] S.Vidhyalakshmi, T.R.Usha Rani and + x2 - y3= 4(z3 + z2 ) Archimedes J.Math 1(1), M.A.Gopalan Integral solutions of non- 31-37, 2011 homogenous ternary cubit equation ax2+by2 = [12] M.A.Gopalan, G.Srividhya Integral solutions (a+b)z3DiophantusJ.Math, 2(1), 31-38, 2013 of ternary cubic diophantine equation x3 + y3 [20] M.A.Gopalan, K.Geetha On the ternary cubic = z2 ActaCienciaIndica, Vol XXXVII No.4, diophantine equation x2 + y2 – xy = z3 Bessel 805-808, 2011 J.Math., 3(2), 119-123, 2013 [13] M.A.Gopalan, A.Vijayashankar, [21] M.A.Gopalan, S.Vidhyalakshmi and S.Vidhyalakshmi Integral solutions of ternary A.Kavitha Observations on the ternary cubic cubic equation x2+ y2– xy + 2(x + y + z) = equation x2 + y2 + xy = 12z3Antartica (k2+ 3)z3 Archimedes J.Math 1(1), 59-65, J.Math.10(5), 453-460, 2013 2011 [22] M.A.Gopalan, S.Vidhyalakshmi and [14] M.A.Gopalan, G.Sangeetha on the ternary K.Lakshmi Lattice points on the non cubic diophantine equation y2=Dx2+z3 homogeneous ternary cubic equation x3 + y3 + Archimedes J.Math 1(1), 7-14, 2011 z3 (x+y+z) = 0 Impact J.
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