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JETIR Research Journal © 2018 JETIR March 2018, Volume 5, Issue 3 www.jetir.org (ISSN-2349-5162) ON THE NON-HOMOGENEOUS QUINTIC EQUATION WITH THREE UNKNOWNS 5(x2 y2) 9xy 2(x y) 4 (k2 19s2)n z5 Dr.G.Sumathi M.Sc.,M.Phil.,B.Ed.,Ph.D.,SET Assistant Professorr,Dept.of Mathematics, Shrimati Indira Gandhi College, Trichy-620002,Tamilnadu,India; Abstract : The non-homogeneous quintic equation with five unknowns represented by the diophantine equation 2 2 2 2 n 5 5(x y ) 9xy 2(x y) 4 (k 19s ) z is analyzed for its non-zero distinct integral solutions. Introducing the transformations x u v, y u v and employing the method factorization,three different patterns of non-trival distinct integer solutions to the quintic equation under consideration are obtained. A few interesting properties between the solutions and special numbers namely,Polygonal numbers,,Centered Pyramidal numbers , Thabith-ibn-kurrah number, Gnomic number, Jacobsthal Lucas number , Jacobsthal number and five dimensional numbers are exhibited. Keywords: Integral solutions, lattice points,non- homogeneous quintic equation with five unknowns. 2010 Mathematics Subject Classification: 11D41 NOTATIONS: t : Polygonal number of rank n with size m m,n m Pn : Pyramidal number of rank n with size m j : Jacobsthal Lucas number of rank n n J : Jacobsthal number of rank n GNO : Gnomic number of rank n Tk : Thabith-ibn-kurrah number of rank n Ct : Centered Polygonal number of rank with size m,n Cf3,n.30 : Centered Tricontagonal Pyramidal number of rank F5,n,7 : Fifth Dimensional Figurate Heptagonal number of rank GFn (k, s) : Generalized Fibonacci Sequences of rank GLn (k,s) : Generalized Lucas Sequences of rank ________________________________________________________________________________________________________ I. INTRODUCTION The theory of diophantine equations offers a rich variety of fascinating problems. In particular,quintic equations, homogeneous and non- homogeneous have aroused the interest of numerous mathematicians since antiquity[1,2,9].For illustration, one may refer [3-5] for quintic equation with three unknowns ,[6] for quintic equation with four unknowns and [7,8] for quintic equation with five unknowns. This communication concerns with yet another interesting a non-homogeneous sextic equation with 5 unknowns given by 2 2 2 2 n 5 5(x y ) 9xy 2(x y) 4 (k 19s ) z for determining its infinitely many non-zero integer quintuples (x, y, z,w,t) .Three different methods are illustrated. In mehod1,the solutions are obtained through the method of factorization. In mehod2,the binomial expansion is introduced to obtain the integral solutions. In method3,the integral solutions are expressed in terms of Generalized Fibonacci and Lucas sequences along with a few properties in terms of the above integer sequences.Also a few interesting properties among the values of x, y and z are presented. 2. METHOD OF ANALYSIS The diophantine equation representing a non-homogeneous quitic equation with five unknowns is 2 2 2 2 n 5 5(x y ) 9xy 2(x y) 4 (k 19s ) z (1) JETIR1803209 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 1101 © 2018 JETIR March 2018, Volume 5, Issue 3 www.jetir.org (ISSN-2349-5162) Introducing the linear transformations (2) x u v, y u v in (1), it leads to 2 2 2 2 2n 5 (u 2) 19v (k 19s ) z (3) The above equation (3) is solved through three different methods and thus, one obtains three distinct sets of solutions to (1) 2.1:Method: 1 Let z a 2 19b2 (4) Substituting (4) in (3) and using the method of factorization,define n 5 (u 2 i 19v) (k i 19s) (a i 19b) (5) n 5 r exp(in)(a i 19b) 2 2 19s where r k 19s , 1 (6) tan k Equating real and imaginary parts in (5) we get u r n cos n(a5 190a3b2 1805ab4 ) 19 sin n(5a4b 190a2b3 3611b5 ) 2 n 4 2 3 5 sin n 5 3 2 4 v r cos n(5a b 190a b 361b ) (a 190a b 1805ab ) 19 Substituting the values of u and v in (2), the corresponding values of x, y and z are represented by cos n (a5 190a3b2 1805ab4 5a4b 190a2b3 361b5 ) x(a,b,k) r n 2 sin n 5 3 2 4 4 2 3 5 (a 190a b 1805ab 95a b 3610a b 6859b ) 19 cos n (a5 190a3b2 1805ab4 5a4b 190a2b3 361b5 ) y(a,b,k) r n 2 sin n 5 3 2 4 4 2 3 5 (a 190a b 1805ab 95a b 3610a b 6859b ) 19 z(a,b) a2 19b2 2.2:Method :2 n Using the binomial expansion of (k i 19s) in (5) and equating real and imaginary parts, we have u f ()(a5 190a3b2 1805ab4 ) 19g()(5a4b 190a2b3 361b5) 2 v f ()(5a4b 190a2b3 361b5) g()(a5 190a3b2 1805ab4 ) where n 2 f () (1)r (19)r (s)2r n k n2r c2r r0 (7) n1 2 r1 r1 2r2 n2r1 g() () (19) (s) nC k 2r1 r1 In view of (2) and (7) the corresponding integer solution to (1) is obtained as x f () g()(a5 190a3b2 1805ab4 ) f () 19g()(5a4b 190a2b3 361b5 ) 2 5 3 2 4 4 2 3 5 y f () g()(a 190a b 1805ab ) f () 19g()(5a b 190a b 361b ) 2 z a2 19b2 2.3:Method :3 Taking n 0 and u 2 U in (3), we have, U 2 19v2 z5 (8) Substituting (4) in (8), we get 2 2 2 2 5 U 19v (a 19b ) (9) whose solution is given by JETIR1803209 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 1102 © 2018 JETIR March 2018, Volume 5, Issue 3 www.jetir.org (ISSN-2349-5162) 5 3 2 4 U0 (a 190a b 1805ab ) 4 2 3 5 v0 (5a b 190a b 361b ) Again taking n 1 in (3), we have, 2 2 2 2 2 2 5 U 19v (k 19s )(a 19b ) (10) whose solution is represented by U1 kU0 19sv0 v1 sU0 kv0 The general form of integral solutions to (1) is given by An i 19Bn Un U0 Bn ,n 1,2,3..... vn i An v0 19 (k i 19s)n (k i 19s)n where A n 2 (k i 19s)n (k i 19s)n B n 2 Thus, in view of (2), the values of xn , yn and z as follows: 5 3 2 4 4 2 3 5 i xn {(a 190a b 1805ab 5a b 190a b 361b )As Bs 19 4 2 3 5 5 3 2 4 (95a b 3610a b 685b a 190a b 1805ab )} 2 5 3 2 4 4 2 3 5 i yn {(a 190a b 1805ab 5a b 190a b 361b )As Bs 19 4 2 3 5 5 3 2 4 (95a b 3610a b 685b a 190a b 1805ab )} 2 2 2 z a 11b Thus, in view of (2), the following of integers interms of Generalized Lucas and fiboanacci sequence satisfy (1) are as follows: xn , yn 1 GL (2k,k 2 19s)(a5 190a3b2 1805ab4 5a4b 190a2b3 361b5 ) 2 n xn 2 2 4 2 3 5 5 3 2 4 GFn (2k,k 19s)(95a b 3610a b 685b a 190a b 1805ab ) 1 2 5 3 2 4 4 2 3 5 GLn (2k,k 19s)(a 190a b 1805ab 5a b 190a b 361b ) 2 yn 2 2 4 2 3 5 5 3 2 4 GFn (2k,k 19s)(95a b 3610a b 685b a 190a b 1805ab ) The above values of xn , yn satisfy the following recurrence relations respectively 2 2 xn2 2kxn1 (k 19s )xn 0 2 2 yn2 2kyn1 (k 19s )yn 0 2.3.1:Properties a a 1. z(2 ,2 ) 20( j2a Tk2a 9J2a ) 2. z(a,a) 108Ct2,a 10Gna 80t3,a Bn 3. xn (1,a,k) yn (1,a,k) 2i (Ct372,a 180t 2 93t3,a 190t4,a ) 19 4,a 2A (361F 792Cf 190P3 190t 2166t 5t 1198t ) n 5,a,7 3,a,30 a 8,a 4,a2 4,a 3,a 2A (38784F 1616t 32P5 ) n 5,a,7 4,a2 a 4.xn (a, a, k) yn (a, a, k) 4 i38Bn (4224F5,a,7 1056t 2 176Cf3,a,30 2112t3,a 176t12,a 880t4,a ) 19 4,a JETIR1803209 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 1103 © 2018 JETIR March 2018, Volume 5, Issue 3 www.jetir.org (ISSN-2349-5162) 3. CONCLUSION To conclude, one may search for other pattern of solutions and their corresponding properties. 4. REFERENCES [1] R.D.Carmichael.,The theory of numbers and Diophantine Analysis,Dover Publications, New York (1959) [2] L.E.Dickson, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York(1952) [3] M.A.Gopalan & A.Vijayashankar, An Interesting Dio.problem x3 y 32 z 5 , Advances in Mathematics, Scientific Developments and Engineering Application, Narosa Publishing House, Pp 1-6, 2010. [4] M.A.Gopalan & A.Vijayashankar, Integrated solutions of ternary quintic Diophantine equation x2(2 k 1) y 2 z 5 ,International Journal of Mathematical Sciences 19(1-2), 165-169,(jan-june 2010) [5] M.A.Gopalan., Sumathi.G., and Vidhyalakshmi.S., Integral solutions of the non- homogeneous ternary quintic equation in terms of pell sequence equation x3 y3 xy(x y) 2z5 ,International Journal of Applied Mathematical Sciences,Vol6,No.1,59-62,2013. [6] M.A.Gopalan.,Vidhyalakshmi.S., and Sumathi.G., Integral solutions of the non-homogeneous quintic equation with four unknowns 5 5 4 4 4 2 x y (x y )z 52w z 4z(1 7w ) , Bessel J.Math,Vol 3(1),175-180,2013.
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