Abstract We obtain infinitely many non-zero integer sextuples ( x, y, z, w, p, T )

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1 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY O the No-Homogeeous Equatio of the Eighth Degree with Six Ukows x 5 -y 5 +(x 3 -y 3 xy = p(z -w T 3 S.Vidhyalakshmi *1, K.Lakshmi, M.A.Gopala 3 *1,,3 Departmet of Mathematics,Shrimati Idira Gadhi College,Trichy, Idia mayilgopala@gmail.com Abstract We obtai ifiitely may o-zero iteger sextuples ( x, y, z, w, p, T satisfyig the o-homogeeous equatio of degree eight with six ukows give by x y + ( x y xy = p( z w T. Various iterestig relatios betwee the solutios ad special umbers, amely, polygoal umbers, Pyramidal umbers, Star umbers, Stella Octagular umbers, Octahedral umbers, Proic umber, Jacobsthal umber, Jacobsthal- Lucas umber, keyea umber, Cetered pyramidal umbers are exhibited. Keywords: Replica No-homogeeous equatio, itegral solutios, polygoal umbers, Pyramidal umbers, Cetered pyramidal umbers. MSC 000 Mathematics subject classificatio: 11D41. NOTATIONS: T m, -Polygoal umber of rak with size m m P - Pyramidal umber of rak with size m SO -Stella octagular umber of rak S -Star umber of rak PR - Proic umber of rak OH - Octahedral umber of rak J -Jacobsthal umber of rak of j - Jacobsthal-Lucas umber of rak KY -keyea umber of rak CP,3 - Cetered Triagular pyramidal umber of rak CP,6 - Cetered hexagoal pyramidal umber of rak CP,7 - Cetered heptagoal pyramidal umber of rak Itroductio The theory of diophatie equatios offers a rich variety of fasciatig problems. I particular, homogeeous ad o-homogeeous equatios of higher degree have aroused the iterest of umerous Mathematicias sice atiquity [1-3].Particularly i [4,5] special equatios of sixth degree with four ad five ukows are studied. I [6-8] heptic equatios with three ad five ukows are aalysed. This paper cocers with the problem of determiig o-trivial itegral solutio of the o- homogeeous equatio of eighth degree (C Iteratioal Joural of Egieerig Scieces & Research Techology

2 with six ukows give by x 5 y 5 ( x 3 y 3 xy p( z w T 3 ad the special umbers are preseted. + =. A few relatios betwee the solutios Method of Aalysis The Diophatie equatio represetig the o- homogeeous equatio of degree eight is give by x y + ( x y xy = p( z w T (1 Itroductio of the trasformatios x = u + v, y = u v, z = u + 1, w = u 1, p = v, v > 1 ( i (1 leads to 3 u + v = T (3 The above equatio (3 is solved through differet approaches ad thus, oe obtais differet sets of solutios to (1 Approach1: T a b = + (4 Let Substitutig (4 i (3 ad usig the method of factorisatio, defie ( u iv ( a ib 3 + = + (5 Equatig real ad imagiary parts i (5 we get u a ab = 3 3 v = 3a b b I view of (, (4 ad (6, the correspodig values of x, y, z, w, p, T are represeted by 3 3 x = a + 3a b 3ab b 3 3 y = a 3a b 3ab + b z = a 3ab + 1 w = a 3ab 1 3 p = 3a b b T = a + b The above values of x, y, z, w, p ad T satisfies the followig properties: x( a,1 + y( a,1 + T ( a,1 6 pa + 6T3, a + 1T4, a 8T5, a pa + CPa,6 = 1 4. z( a,1 + w( a,1 + T ( a,1 6 pa + Sa 6T3, a 19T4, a + 16PRa = 0 3. The followig are asty umbers: a x(, + y(, + z(, + w(, + p(, + j6 b 30( p(, + T (, 3J6+ 1 KY + 4 j 4. 9( z w zw + xy + p is a cubic iteger [ z w + ( x + y( z + w(1 p + p x y ] is a biquadratic iteger 6. ( x y ( z + w 8 p( zw + 1 = 0 7. zw( z + w = ( x + y( xy + p 1 (6 (7 (C Iteratioal Joural of Egieerig Scieces & Research Techology

3 a 3, a a a,6 5 a 3, a a a a,6 x( a, a + y( a, a + z( a, a + w( a, a + p( a, a + T ( a, a + 36 p 40T + 16 p 8CP = 0 y( a, a + z( a, a w( a, a + p( a, a + T ( a, a SO 4T + 6( OH p + 6CP = 0(mod 10. x y zw p Approach: Now, rewrite (3 as, + = 3 u + v = T 1 (8 Also 1 ca be writte as 1 = ( i ( i (9 Substitutig (4 ad (9 i (8 ad usig the method of factorisatio, defie, ( u iv i ( a ib 3 + = + (10 Equatig real ad imagiary parts i (10 we get π π 3 u = cos ( a 3 ab si (3 a b b π 3 π v = cos (3 a b b + si ( a 3 ab I view of (, (4 ad (11, the correspodig values of x, y, z, w, p, T are represeted π 3 3 π 3 3 x = cos ( a 3ab + 3 a b b + si ( a 3ab 3 a b + b π 3 3 π 3 3 y = cos ( a 3ab 3 a b + b si ( a 3ab + 3 a b b π π 3 z = cos ( a 3 ab si (3 a b b + 1 π π 3 w = cos ( a 3 ab si (3 a b b 1 π 3 π p = cos (3 a b b + si ( a 3ab T = a + b Approach3: 1 ca also be writte as (( m + i m(( m i m 1 = ( m + Followig the same procedure as above we get the itegral solutio of (1 as x = m + m a ab + a b b + m a ab a b + b ( [( ( 3 3 ( 3 3 ] y = m + m a ab a b + b m a ab + a b b ( [( ( 3 3 ( 3 3 ] z = m + m a ab m a b b ( [( ( 3 (3 ] 1 w = m + m a ab m a b b 3 3 ( [( ( 3 (3 ] 1 p = m + m a b b + m a ab 3 ( [( (3 ( 3 ] = ( + ( + T m a b (11 (13 (1 (14 (C Iteratioal Joural of Egieerig Scieces & Research Techology

4 Approach4: Writig 1 as (m + i( m 1 = ( m (m i( m + Followig the same procedure as above we get the itegral solutio of (1 as x = m + m a ab + a b b + m a ab a b + b ( [ ( 3 3 ( ( 3 3 ] y = m + m a ab a b + b m a ab + a b b ( [ ( 3 3 ( ( 3 3 ] z = m + m a ab m a b b ( [ ( 3 ( (3 ] 1 w = m + m a ab m a b b 3 3 ( [ ( 3 ( (3 ] 1 p = m + m a b b + m a ab 3 ( [ (3 ( ( 3 ] T = ( m + ( a + b Approach5: The solutio of (3 ca also be obtaied as u = m( m +, v = ( m +, T = ( m + (16 I view of (16 ad (, the itegral solutios of (1 is obtaied as x = ( m + ( m + y = ( m + ( m z = m( m (17 w = m( m + 1 p = ( m + T = ( m + The above values of x, y, z, w, p ad T satisfies the followig properties: 1. z( a,1 + w( a,1 + p( a,1 4CPa,3 T4, a = 1. 6( p(, + T (, + z(, w(, j6+ 1 j4+ 1 is a asty umber x( a,1 + y( a,1 + p( a,1 4Pa + T3, a 5T4, a + T7, a = ( ( +, + ( +, + p + T J6+ 3 KY j+ 1 is a cubic iteger. 5. The followig are biquadratic itegers: a 3 8( p( a,1 T( a,1 48F4, a,4cpa, T4, a α = r + s, u = r s, v = 4 b 8 [ x( a, a + y( a, a + z( a, a w( a, a + p( a, a 18pa + 9GNa 1T3, a + 6 T4, a] Approach6: Assumig T = α (18 i (3, we have u + v = ( α rs, r > s > 0 which is i the form of Pythagorea equatio, whose solutio is, (15 (C Iteratioal Joural of Egieerig Scieces & Research Techology

5 3 α = r + s, u = rs, v = r s, r > s > 0 (Or (19 3 α = r + s, u = r s, v = rs r > s > 0 (0 Solvig the first equatio of (19 we have two choices of solutios, amely, r m m s m α m m = ( +, = ( +, = +, > > 0 (1 = 3, = 3, α = +, > > 0 ( 3 3 r m m s m m m I view of (18, (19, ad (1 ad (, we get the itegral solutio of (1 as x = m + m + m ( ( ( y m m m = ( + ( ( z = m( m w m m = ( + 1 p = ( m + ( m T m = ( + I view of (18, (19, ( ad (, we get a differet itegral solutio of (1 as x = m m m + m + m m ( 3 (3 15 ( y m m m m m m = ( 3 ( ( 3 3 z = ( m 3 m (3 m + 1 w m m m 3 3 = ( 3 ( p = m + 15 m ( m T m = ( + Similarly takig (0, istead of (19 ad performig the same procedure we will get two more patters. Approach7: Assumig u = UT, v = VT (5 i (3,we get, U + V = T (6 T = w, w = a + b (7 Assume i (6, Ad Write (6 as U + V = w 1 (8 Also Write 1 as m m 1 = ( i ( i (9 Substitutig (7 ad (9 i (8 ad usig the method of factorisatio defie m ( + = ( + (30 U iv i a ib π i( m + θ -1 b = r e, where r = a + b, θ = ta a Equatig the real ad imagiary parts, we have, π U = r cos( m + θ π V = r si( m + θ (3 (31 (4 (C Iteratioal Joural of Egieerig Scieces & Research Techology

6 I view of (31, (5 ad (, we get mπ mπ x = r [cos( + θ + si( + θ ]( a + b mπ mπ y = r [cos( + θ si( + θ ]( a + b mπ z = r [cos( + θ ( a + b ] + 1 mπ w = r [cos( + θ ( a + b ] 1 mπ p = r si( + θ ]( a + b T = ( a + b (3 Coclusio I coclusio, oe may search for differet patters of solutios to (1 ad their correspodig properties. Refereces [1] L.E.Dickso, History of Theory of Numbers, Vol.11, Chelsea Publishig compay,new York (195. [] L.J.Mordell, Diophatie equatios, Academic Press, Lodo(1969 [3] Carmichael,R.D.,The theory of umbers ad Diophatie Aalysis,Dover Publicatios, New York (1959 [4] M.A.Gopala,S.Vidhyalakshmi ad K.Lakshmi, O the o-homogeeous sextic equatio x + ( x + w x y + y = z,ijama,4(, ,nov.01 [5] M.A.Gopala,S.Vidhyalakshmi ad K.Lakshmi, Itegral Solutios of the sextic equatio with five ukows x + y = z + w + 3( x + y T, IJESRT,50-504, Dec.01 [6] M.A.Gopala ad sageetha.g, parametric itegral solutios of the heptic equatio with 5ukows x y + ( x + y ( x y = ( X Y z,bessel Joural of Mathematics 1(1, 17-, 011. [7] M.A.Gopala ad sageetha.g, O the heptic diophatie equatios with 5 ukows x y = ( X Y z,atarctica Joural of Mathematics, 9( , 01 [8] Majusomath, G.sageetha ad M.A.Gopala, O the o-homogeeous heptic equatios with 3 3 p 5 7 ukows x + ( 1 y = z,diophatie joural of Mathematics, 1(, , 01 (C Iteratioal Joural of Egieerig Scieces & Research Techology