On the Non-Homogeneous Quintic Equation with Five Unknowns X 4 -Y 4 =10 p 3 (Z 2 -W 2 )

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1 Journal of Mathematics and Informatics Vol ISSN: (P) (online) Pulished Decemer 07 DOI: Journal of On the Non-Homogeneous Quintic Equation with Five Unknowns X 4 -Y 4 =0 p (Z -W ) M..Gopalan and G.Dhanalakshmi Department of Mathematics SIGC Trich-6000 India mailgopalan@gmail.com Department of Mathematics SIGC Trich-6000 India dhanamselvi9@gmail.com Received 5 Novemer 07; accepted 4 Decemer 07 stract. The quintic non-homogeneous equation with five unknowns represented the Diophantine equation ( 4 ) = 0 ( ) P is analed for its patterns of non-ero distinct integral solutions. Various interesting relations etween the solutions and special numers namel polgonal numers pramidal numers are ehiited. Kewords: Non-homogenous quintic equation quintic with five unknowns integral solutions. MS Mathematics Suject Classification (00): D4. Introduction The theor of Diophantine equations offers a rich variet of fascinating prolems [-]. Particularl in [4-7] quintic equations with three unknowns are studied for their integral solutions. In [89] quintic equations with four unknowns for their non-ero integer solutions are analed. In [0-] anale quintic equations with five unknowns for their non-ero integer solutions. This communication concerns with et another interesting non-homogeneous quintic equation with five unknowns given ( 4 ) = 0 ( ) P is analaed for its infinitel man non-ero distinct integer solutions. Various interesting properties among the values of w P are presented. Notations t mn : polgonal numer of rank n with sie m. n P m : Pramidal numer of rank n with sie m.. Method of analsis The non-homogeneous quintic equation with five unknowns to e solved for its distinct non-ero integral solutions is ( 4 ) = 0 ( ) P ()

2 M..Gopalan and G.Dhanalakshmi Introduction of the linear transformations = u + v = u v = u + v w = u v in () leads to u + v = 0P Different methods of otaining the patterns of integer solutions to () are illustrated elow: () ().. PTTERN P + where a and are non-ero integers. Write 0 as 0 = ( + i)( i) (5) Using (4) (5) in () and appling the method of factoriation define ( u + iv) = ( + i)( a + i (6) u = a + a 9a (7) v = a 9a + a Using (7) and ( ) the values of = a (4) = 4a + a 6a = a a a a = 5a + 5 5a 5a = a a a w a Thus (4) and (8) represent the non-ero integer solutions to (). a) 4 P( [ + P ] + t0 + t. + t4 + t6 = 8 a + w a = 4[ Pa + 5t a P a ] + + w + 6[ 6t P 8P ] = a + [ P a + Pa ] t6 a t8 a = 8a + [ t + t P 8P ] = ) c) ( + d) ) e) ( +.. PTTERN P = a + where a and are non-ero integers. Write 0 as 8 78 (8)

3 On the Non-homogeneous Quintic Equation with Five Unknowns X 4 -Y 4 =0 p (Z -W ) ( + i)( i) 0 = (9) Using (4) (9) in () and appling the method of factoriation define ( u + iv) = ( + i)( a + i (0) u = a + 9a a () v = a a + 9a Using () and ( ) the values of = 4a a + 6a = a + 4 6a a a = 7a a + a = 5a + 5 5a 5a w a Thus (4) and () represent the non-ero integer solutions to (). [ 6 a 0 a 6Pa ] = 8a w + 48 [ P P 0t ] = a) ) + t + t P) ( a a 6 a 09 c) ( a ) + ) + w) 7P) + 6[ t P ] + t = a d) ( + ( ( + P( + t 8P = 9 e) ( ( + 5 P( 6P t = [ ] 0.. PTTERN In () taking u = r + s v = r s () it leads to r + s = 5P (4) P = + (5) where and are non-ero integers. Write 5 as 5 = ( + i)( i) (6) Using (6) (5) in (4) and appling the method of factoriation define ( r + is) = ( + i)( + i (7) r = + 6 s = + 6 (8) From (8) and () the values of u and v are given ()

4 M..Gopalan and G.Dhanalakshmi u = v = + 9 Using (9) and ( ) the values of + 9 ( = ( = = 7 + = w Thus (0) and (5) represent the non-ero integer solutions to (). (9) (0) + = a) ( ) P( ) + 6[ 7t 5P ] ( P( + 8P t ( mod ) + 8 c) ( ) ( ) + P( ) 6P 0t = 0 d) ( + ( ( + 5 6P P( + t e) ( ) + 4( ) + w( ) + 5 P( ) 4P + t.4. PTTERN P = + where and are non-ero integers. + s [ ] = 0 [ + t ] = Write 5 as 5 = ( + i)( i) () Using () (5) in (4) and appling the method of factoriation define ( r + is) = ( + i)( + i () r = + 6 () s = 6 + From () and () the values of u and v are given u = + 9 v = (4) Using (4) and ( ) the values of 4

5 On the Non-homogeneous Quintic Equation with Five Unknowns X 4 -Y 4 =0 p (Z -W ) ( = ( = = = 7 + w Thus (5) and (5) represent the non-ero integer solutions to (). a) ( ) 4P( ) = 4[ P 5t ] ( ( + [ 6P P( + t6 ] = 4 c) ( ) + [ P( ) P + 4t ] = 6 8 d) ( ) + ( ) + P( ) + 0w( ) 46P + 454t = e) ( ) ( ) + 7P( ) + [ P t t t ] =. Conclusion In this paper we have made an attempt to determine different patterns of nonero distinct integer solutions to the non-homogeneous quintic equations with five unknowns. s the quintic equations are rich in variet one ma search for other forms of quintic equation with variales greater than or equal to five and otain their corresponding properties. REFERENCES. L.E.Dickson Histor of Theor of Numers Chelsea Pulishing compan New York (95).. L.J.Mordell Diophantine equations cademic Press London (969).. S.G.Telang Numer theor Tata Mc Graw Hill pulishing compan New Delhi Carmichael R.D. (996) The theor of numers and Diophantine nalsis Dover pulications New York (959). 4. M..Gopalan and.vijsashankar Integral solutions of ternar quintic Diophantine 5 equation + ( k + ) = International Journal of Mathematical and Sciences 9(-) (00) M..Gopalan G. Sumathi and S. Vidhalakshmi Integral solutions of nonhomogeneous ternar quintic equation in terms of pell equation JMS 6() (0) M..Gopalan S.Vidhalakshmi and K.Lakshmi Integral solutions of the nonhomogeneous equation ternar quadratic Diophantine equation a + = ( a + 5 a > > 0 rchimedes Journal of Mathematics () (0) M..Gopalan and G.Sangeetha Integral solutions of ternar quadratic Diophantine equation + = ( k + ) 5 ulletin of Pure and pplied Sciences 9() (00) -8. (5) 5

6 M..Gopalan and G.Dhanalakshmi 8. M..Gopalan S. Vidhalakshmi and. Kavitha Oservations on the quintic equation with four unknowns 4 4 ( + )( + + ) + ( + )( + ) w = ( ) Impact J. Sci. Tech. 7() (0) M..Gopalan S.Vidhalakshmi and.kavitha On the quintic equation with four unknowns = ( k + l ) w essel J. Math. () (0) M..Gopalan S.Vidhalakshmi and J.Shanthi The non-homogeneous quintic equation with five unknown 4 + k( )( + k) = ( a + )( ) P Open Journal of pplied and Theoretical Mathematics () (06) 8-.. M..Gopalan S.Vidhalakshmi.Kavitha and E.Premalatha On the quintic 5 equation with five unknowns = w + 6t International Journal of Current Research 5(6) (0) M..Gopalan S.Vidhalakshmi and. Kavitha On the quintic equation with five unknowns ( )( + ) = 9 ( ) P International Journal of Engineering Research Online () (0) M..Gopalan V.Krithika and.vijaasankar Integral solutions of nonhomogeneous quintic equation with five unknowns 4 = 6 P 5(8) (07) 5-. 6