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On Non-Homogeneous Sextic Equation with Five Unknowns Asian Journal of Applied Science and Technology (AJAST) Page | 45 Volume 1, Issue 6, Pages 45-47, July 2017 On Non-Homogeneous Sextic Equation with Five Unknowns 2x yx3 y3 39z2 w2 T4 1 2 3 S.Vidhyalakshmi , M.A.Gopalan and S.Aarthy Thangam 1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India. 2Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India. 3Research Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India. Article Received: 27 June 2017 Article Accepted: 10 July 2017 Article Published: 16 July 2017 ABSTRACT The non-homogeneous sextic equation with five unknowns given by 2x yx3 y3 39z2 w2 T4 is considered and analysed for its non-zero distinct integer solutions. Employing the linear transformations x u v,y u v,z 2u v,w 2u v,(u v 0) and applying the method of factorization, three different patterns of non-zero distinct integer solutions are obtained. A few interesting relations between the solutions and special numbers namely Four dimensional numbers, Polygonal numbers, Octahedral numbers, Centered Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers, Kynea numbers and Star numbers are presented. Keywords: Integer solutions, Non-homogeneous sextic equation and Sextic equation with five unknowns. 1. INTRODUCTION Four dimensional Figurate number of rank n whose The theory of Diophantine equations offers a rich variety of generating polygon is a pentagon fascinating problems [1-4] particularly, in [5, 6] Sextic 3n4 10n3 9n2 2n F equations with three unknowns are studied for their integral 4,n,5 4! solutions. [7-12] analyze Sextic equations with four unknowns for their non-zero integer solutions. [13-16] Jacobsthal number of rank n analyze Sextic equations with five unknowns for their 1 n n Jn 2 1 3 non-zero integer solutions. This communication analyzes a Sextic equation with five unknowns given by Jacobsthal-Lucas number of rank n n n 2 x y x3 y3 39 z2 w2 T4 . Infinitely many Quintuples (x, jn 2 1 y, z, w, T) satisfying the above equation is obtained. Various Kynea number of rank n 2 interesting properties among the values of x, y, z, w and T are Ky 2n 1 2 n presented. 3. METHOD OF ANALYSIS 2. NOTATIONS The non-homogeneous sextic equation with five unknowns to Polygonal number of rank n with size m be solved is given by n 1m 2 Tm,n n1 3 3 2 2 4 2 2x yx y 39z w T (1) The substitution of the linear transformations Star number of rank n x u v, y u v, z 2u v, w 2u v, u v 0 (2) S 6nn 11 n in (1) leads to 3u2 v2 39T4 (3) Octahedral number of rank n (3) is solved through different approaches and different 1 2 patterns of solutions thus obtained for (1) are illustrated OHn n2n 1 3 below: Centered Pyramidal number of rank n with size m 3.1 Pattern: 1 mn 1nn 1 6n CP Assume T Ta,b a2 3b2; a,b 0 (4) m,n 6 Write 39 as © 2017 AJAST All rights reserved. www.ajast.net Asian Journal of Applied Science and Technology (AJAST) Page | 46 Volume 1, Issue 6, Pages 45-47, July 2017 39 6 i 36 i 3 (5) Properties 2 Using (4) and (5) in (3) and employing the method of w1,B x1,B 72T4,B 4320CP6,B 24SB 0mod2 factorization and equating positive factors, we get n T2 ,112Jn 4jn 16 4Kyn 4 v i 3u 6 i 3a i 3b 3xA,A yA,A 2zA,Ais a nasty number Equating real and imaginary parts, we have u ua,b a4 24a3b 18a2b2 72ab3 9b4 Remark It is worth to note that 39 in (5) and 1in (7) are also 4 3 2 2 3 4 v va,b 6a 12a b 108a b 36ab 54b represented in the following ways Employing (2), the values of x, y, z, w and T are given by 3 i7 3 3 i7 3 1 i4 3 1 i4 3 39 1 4 3 2 2 3 4 49 x xa,b u v 7a 12a b 126a b 36ab 63b 4 9 i5 3 9 i5 3 11 i4 3 11 i4 3 4 3 2 2 3 4 y ya,b u v 5a 36a b 90a b 108ab 45b 4 169 z za,b 2u v 8a4 36a3b 144a2b2 108ab3 72b4 By introducing the above representations in (5) and (7), one may obtain different patterns of solutions to (1). w wa,b 2u v 4a4 60a3b 72a2b2 180ab3 36b4 2 2 T Ta,b a 3b 3.3 Pattern: 3 Which represent non-zero distinct integer solutions of (1) in Write (3) as two parameters. 2 4 4 2 3u T 36T v (10) Factorizing (10) we have Properties 2 2 2 2 3u T u T 6T v6T v (11) 234OH z(a,1) 2w(a,1) 0mod2 a This equation is written in the form of ratio as 504F4,b,5 x1,b 246CP6,b 105T8,b 1mod2 3 u T2 6T2 v a , b 0 2T1,b1is a nasty number 6T2 v u T2 b (12) Which is equivalent to the system of double equations 3.2 Pattern: 2 3bu av 6a 3b T2 0 (13) One may write (3) as 2 au bv a 6bT 0 (14) v2 3u2 39T4 *1 (6) Applying the method of cross multiplication, we get Also, write 1 as 2 2 u a 3b 12ab (15) 1 i 31 i 3 1 (7) v 6a2 18b2 6ab 4 (16) 2 2 2 Substituting (4), (5) and (7) in (6) and employing the method T a 3b (17) of factorization and equating positive factors we get Now, the solution for (17) is 2 2 2 2 6 i 31 i 3 4 a 3p q , b 2pq, T 3p q (18) v i 3u a i 3b 2 Using (18) in (15) and (16), we get Equating real and imaginary parts, we have u up,q 9p4 72p3q 18p2q2 24pq3 q4 1 4 3 2 2 3 4 u ua,b 7a 12a b 126a b 36ab 63b v vp,q 54p4 36p3q 108p2q2 12pq3 6q4 2 (8) 1 In view of (2), the integer values of x, y, z, w, T are given by v va,b 3a4 84a3b 54a2b2 252ab3 27b4 2 (9) x xp,q u v 45p4 108p3q 90p2q2 36pq3 5q4 The choices a=2A and b=2B in (8), (9) lead to y yp,q u v 63p4 36p3q 126p2q2 12pq3 7q4 u uA,B 56A4 96A3B1008A2B2 288AB3 504B4 z zp,q 2u v 36p4 180p3q 72p2q2 60pq3 4q4 v vA,B 24A4 672A3B 432A2B2 2016AB3 216B4 w wp,q 2u v 72p4 108p3q 144p2q2 36pq3 8q4 In view of (2), the integer values of x, y, z, w and T are given 2 2 by T Tp,q 3p q x xA,B 80A4 576A3B1440A2B2 1728AB3 720B4 Which represent non-zero distinct integer solutions of (1) in two parameters. y yA,B 32A4 768A3B576A2B2 2304AB3 288B4 z zA,B136A4 480A3B 2448A2B2 1440AB3 1224B4 Properties w wA,B 88A4 864A3B1584A2B2 2592AB3 792B4 2z1,q w1,q156CP6,q 0mod3 T TA,B 4A2 12B2 n 2, if n is odd Which represent non-zero distinct integer solutions of (1) in T2 ,1 3ky n 6jn 10 , if n is even two parameters. © 2017 AJAST All rights reserved. www.ajast.net Asian Journal of Applied Science and Technology (AJAST) Page | 47 Volume 1, Issue 6, Pages 45-47, July 2017 468CP6,p 2xp,1 2yp,1 zp,1 0mod2 [11] Gopalan, M.A., Sumathi, G., Vidhyalakshmi, S., “Integral Solutions of Non- homogeneous Sextic Equation with four unknowns x4 y4 16z4 32w6 ”, Antarctica J.Math, 4. CONCLUSION First of all, it is worth to mention here that in (2), the values of 10(6), 623-629, 2013. z and w may also be represented by z 2uv 1,w 2uv 1 and z uv 2,w uv 2 and thus, will obtain other choices of [12] Gopalan, M.A., Sumathi, G., Vidhyalakshmi, S., solutions to (1). In conclusion, one may consider other forms “Gaussian Integer Solutions of Sextic Equations with four of Sextic equation with five unknowns and search for their unknowns x6 y6 4z(x4 y4 w4) ”, Archimedes, J.Math, integer solutions. 3(3), 263-266, 2013. REFERENCES [13] Gopalan, M.A., Vidyalakshmi, S., Lakshmi, K., [1] Dickson, L.E., History of theory of numbers, vol.2, “Integral Solutions of Sextic Equation with Five unknowns Chelsea publishing company, Newyork (1952). x3 y3 z3 w3 3(x y)t5 ” , IJERST, 1(10), 562-564, 2012. [2] Carmichael, R.D., The theory of numbers and Diophantine analysis, Dover publications, Newyork (1959). [14] Gopalan, M.A., Sumathi, G., Vidyalakshmi, S., “Integral Solutions of Sextic Non-Homogeneous Equation [3] Mordell, L.J., Diophantine equations, Academic press, with Five unknowns (x3 y3) z3 w3 6(x y)t5 ”, London (1969). International Journal of Engineering Research, Vol.1, Issue.2, 146-150, 2013. [4] Telang, S.G., Number Theory, Tata MC Graw Hill Publishing Company, New Delhi (1996). [15] Gopalan, M.A., Aarthy Thangam, S., Kavitha, A., “On Non-homogeneous Sextic equation with five unknowns [5] Gopalan, M.A., Manju Somanath and Vanitha, N., 2x yx3 y3 28z2 w2 T4 ”, Jamal Academic Research “Parametric Solutions of x2 y6 z2 ”, Acta Ciencia Indica Journal (JARJ) , special Issue, 291-295, ICOMAC- 2015. XXXIII, vol.3, 1083-1085, 2007. [16] Meena, K., Vidhyalakshmi, S., Aarthy Thangam, S., [6] Gopalan, M.A., Sangeetha, G., “On the Sextic Equations “On Non-homogeneous Sextic equation with five with three unknowns x2 xy y2 (k2 3)n z6 ”, Impact unknowns x yx3 y3 26z2 w2 T4 ”, Bulletin of J.Sci.tech, Vol.4, No: 4, 89-93, 2010.
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