THE NON-HOMOGENEOUS QUINTIC EQUATION WITH FIVE

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1 THE NON-HOMOGENEOUS QUINTIC EQUATION WITH FIVE UNKNOWNS x y (x y )(x y) (z w )p Gopalan M.A. 1, Vidhyalakshmi S. 1, *Premalatha E. and Manjula S. 1 1 Department of Mathematics, Shrimati Indira Gandhi College, Trichy-000 Department of Mathematics, National College, Trichy-0001 *Author for Correspondence ABSTRACT The non-homogeneous Quintic equation with five unknowns given y x y (x y )(x y) (z w )p is considered and analyzed for its non zero distinct integer solutions. A few interesting relations etween the solutions and special numers namely, Polygonal numers, Pyramidal numers, Stella octangular numers, octahedral numers, rhomic dodecagonal numers are presented. Keywords: Non - homogeneous Quintic, Quintic with Five Unknowns, Integral Solutions, Special Numers 010 Mathematics suject classification: 11D1 Notations Special numers Notations Definitions Regular Polygonal Numer tm, n (n )(m ) n Octahedral Numer OH n n(n 1) Stella Octangular Numer SO n(n 1) Copyright 01 Centre for Info Bio Technology (CIBTech) n Rhomic Dodecagonal Numer RD n (n )(n n ) m n(n ) Pyramidal Numer P n ((m )n ( n)) INTRODUCTION The theory of Diophantine equations offers a rich variety of fascinating prolems. In particular, Quintic equations, homogeneous and non-homogeneous have aroused the interest of numerous mathematicians since antiquity (Dickson, 1; Mordell, 1). For illustration, one may refer (Gopalan and Vijayashankar, 010; Gopalan and Vijayashankar, 010; Gopalan et al., 01) for Quintic equation with three unknowns and (Gopalan and Vijayashankar, 011; Gopalan and Vijayashankar, Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01; Vidhyalakshmi et al., 01; Vidhyalakshmi et al., 01) for Quintic equation with five unknowns. This paper concerns with the prolem of the non-homogeneous Quintic equation with five unknowns given y x y (x y )(x y) (z w )p. A few relations among the solutions are presented. Method of Analysis The non-homogeneous quintic equation with unknowns to e solved is given y x y (x y )(x y) (z w )p (1) Assume x u v, y u v, z u v and w u v () Sustituting () in (1), it leads to

2 u v p () () is solved through different approaches and different patterns of solutions of (1) otained are presented elow Pattern-1 Assume p a () where a and are non-zero distinct integers. Write 1 as 1 ( i )( i ) () Using () & () in () and employing the method of factorization, define u i v ( i )(a i () Equating the real and imaginary parts of (), we get u a a a (7) v a a a (8) Sustituting (7) and (8) in (), the integral solutions of (1) are given y x a a 0a 0 () y(a, y a a 0a 0 (10) z 7a 1a 0a (11) w a a 7a (1) along with () Properties y(1, n) w(1, n) z(1, n) OHn SO n 8Pn 8t, n 0(mod ) x(1, n) y(1, n) z(1, n) w(1, n) 7RDn Pn 00t, n (mod 8) x(n,1) RDn 11(mod ) y(a, 0 y (a, z (a, 7w (a, y(a, 0 Note 1 In (), the representations of z and w may e taken as z uv, w uv (1) In this case, the values of z and w are given y z a a 0a 00a 0a 00a 70 (1) w a a 0a 00a 0a 00a 70 (1) Thus (), (10), (1), (1) and () represent a different set of solutions to (1) Note Oserve that z and w in () may also e taken as z uv, w uv (1) For this choice, the corresponding values of z and w are otained as Copyright 01 Centre for Info Bio Technology (CIBTech) z a a a 00a 1a 00a 7 (17) w a a a 00a 1a 00a 7 (18)

3 Thus (), (10), (17), (18) and () represent an another set of integer solutions to (1) Pattern- Instead of (), write 1 as 1 ( i )( i ) Following the procedure presented in pattern-1, the corresponding integer solutions of (1) are y(a, x a a 0a 0 (1) y a a 0a 0 (0) z a a 7a (1) w 7a 1a 0a () along with () Note For the choices of z and w given y (1) and (1), the corresponding two sets (I and II) of values of z and z a w a z a a a a 0a 0a a 00a 00a 00a 0a 0a 1a 00a 00a 00a w a a a 00a 1a 00a 7 Considering (1), (0), () with the aove sets, we have two more choices of integer solutions to (1) Pattern- () can e written as u v p * 1 () Write 1 as 1 ( i )( i ) () (1 i )(1 i ) and 1 as 1 () 81 Using (),() & () in () and employing the method of factorization, define (1 i ) u i v ( i )(a i Equating the real and imaginary parts, we get 1 u ( 7a a a ) 1 v (1a a a 8 ) Replacing a y a and y in the aove set of equations and in () the values of u, v and p ecomes, u a 8a v a a p a 7a 8a 7 Hence in view of (), the integral solutions of (1) are given y () Copyright 01 Centre for Info Bio Technology (CIBTech)

4 y(a, x a 78a 80a 0 (7) y 0a a 0a 70 (8) z a a a 0 () w 1a 017a 11a (0) along with () Properties y(a, 0 y(a, 0 y(a, 0 x (a, 7y (a, 8w (a, 8 y(a, 0 y(a, 0 7x (a, y (a, 8z (a, 8 y(a, 0 Note For the choices of z and w given y (1) and (1), the corresponding two sets (I and II) of values of z and z 78a 00a 80a 0000a 1700a 7000a 70 w 78a 00a 80a 0000a 1700a 7000a 70 z 8a 01a 17a 000a 787a 700a 8 w 8a 01a 17a 000a 787a 700a 8 Considering (), (7), (8) with the aove sets, we have two more choices of integer solutions to (1) Pattern- Instead of (17), write 1 as ( i )( i ) 1 Following the procedure presented in pattern-, the corresponding integer solutions of (1) are y(a, x a 8a 810a 080 (1) y a 70a 0a 170 () z a a a 0 ` () w 7a 77a 0a () along with () Note For the choices of z and w given y (1) and (1), the corresponding two sets (I and II) of values of z and z 810a 0a 070a 0000a 070a 000a 010 w 810a 0a 070a 0000a 070a 000a 010 Copyright 01 Centre for Info Bio Technology (CIBTech) 7

5 z 0a 01a 07a 000a 187a 700a 0 w 0a 01a 07a 000a 187a 700a 0 Considering (), (1), () with the aove sets, we have two more choices of integer solutions to (1) Pattern- Also (17) can e written as ( i )( i ) 1 Following the procedure as presented aove, the values of x,y,z,w and p are given y x 8a 08a 70a 80 () y(a, y 80a 7a 700a 170 () z a 7a 1a 1 (7) w 1a 87a 11a 7 (8) p(a, p a () Note For the choices of z and w given y (1) and (1), the corresponding two sets (I and II) of values of z and z 78a 787a 80a 8100a 17870a a 70 Copyright 01 Centre for Info Bio Technology (CIBTech) 8 w 78a 787a 80a 8100a 17870a a 70 z 7a 777a 787a 000a 8171a 00a 717 w 7a 777a 787a 000a 8171a 00a 717 Considering (), (), () with the aove sets, we have two more choices of integer solutions to (1). CONCLUSION In this paper, we have illustrated different methods of otaining non-zero integer solutions to the Quintic equation with unknowns given y x y (x y )(x y) (z w )p. As the Quintic Diophantine equation are rich in variety one may consider other forms of Quintic equation with variale and search for their corresponding integer solutions along with the corresponding properties. REFERENCES Dickson LE (1). History of Theory of Numers (Chelsea Pulishing Company) New York 11. Gopalan MA and Vijayashankar A (010). An Interesting Diophantine prolem x y z, Advances in Mathematics, Scientific Developments and Engineering Application (Narosa Pulishing House) 1-.

6 Gopalan MA and Vijayashankar A (010). Integral solutions of ternary quintic Diophantine equation x (k ) y z. International Journal of Mathematical Sciences 1(1-) 1-1. Gopalan MA, Sumathi G and Vidhyalakshmi S (01). Integral solutions of non-homogeneous ternary quintic equation in terms of pells sequencex y xy( x y) Z. JAMS (1) -. Gopalan MA and Vijayashankar A (011). Integral solutions of non-homogeneous quintic equation with five unknowns xy zw R. Bessel Journal of Mathematics 1(1) -0. Gopalan MA and Vijayashankar A (No Date). Solutions of quintic equation with five unknowns x y ( z w ) P. Accepted for Pulication in International Review of Pure and Applied Mathematics. Gopalan MA, Sumathi G and Vidhyalakshmi S (01). On the non-homogenous quintic equation with five unknowns x y z w T. IJMRA () Gopalan MA, Vidhyalakshmi S, Kavitha A and Premalatha E (01). On The Quintic Equation with five unknows x y z w t. International Journal of Current Research () Gopalan MA, Vidhyalakshmi S and Kavitha A (01). On The Quintic Equation with five unknows ( x y)( x y ) ( z w ) P. International Journal of Engineering Research 1(). Mordell LJ (1). Diophantine Equations (Academic Press) London. Vidhyalakshmi S, Lakshmi K and Gopalan MA (01). Oservations on the homogeneous quintic equation with four unknowns x y z ( x y)( x y ) w. IJMRA () 0-. Vidhyalakshmi S, Mallika S and Gopalan MA (01). Oservations on the nonhomogeneous Quintic equation with five unknowns. International Journal of Innovative Research in Science Engineering and Technology () Copyright 01 Centre for Info Bio Technology (CIBTech)