On Sextic Equation With Five Unknowns

Transcription

1 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut 017 ISSN On Setic Equation With Five Unknown ( )( ) = 8( w ) S.Vidhalakhmi 1, S. Aath Thangam, G. Dhanalakhmi 1 ofeo, Depatment of Mathematic, Shimati India Gandhi College, Tich-60 00, Tamil Nadu, India. vidhaigc@gmail.com Reeach Schola, Depatment of Mathematic, Shimati India Gandhi College, Tich-60 00, Tamil Nadu, India. aaththangam@gmail.com M.hil Schola, Depatment of Mathematic, Shimati India Gandhi College, Tich-60 00, Tamil Nadu, India. dhanamelvi9@gmail.com ABSTRACT The non-homogeneou etic equation with five unknown epeented b the Diophantine equation i ( )( ) = 8( w ) analed fo it patten of non-eo ditinct integal olution ae illutated. Vaiou inteeting elation between the olution and pecial numbe namel, polgonal numbe, pamidal numbe ae ehibited. KEYWORDS Integal olution, Non-homogenou equation, Setic equation. 1. INTRODUCTION The theo of Diophantine equation offe a ich vaiet of facinating poblem (Dickon, 195; Camichael, 1959; Modell, 1969; Telang, 1996). aticulal, in (Gopalan et al., 007; Gopalan and Sangeetha, 010; Gopalan et al., 010), etic equation with thee unknown ae tudied fo thei integal olution. (Gopalan and VijaaShanka, 010; Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01) anale etic equation with fou unknown fo thei non-eo intege olution (Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 01; Gopalan et al., 015) anale etic equation with five unknown fo thei non-eo intege olution. Thi communication concen with et anothe inteeting non-homogeneou etic equation with five unknown ( )( ) = 8( w ) given b tuple i analed fo it infinitel man non-eo ditinct intege olution (,,, w, ) atifing the above equation ae obtained. Vaiou inteeting popetie among the value of,,, w, ae peented. NOTATIONS t m,n : olgonal numbe of ank n with ie m. n m : amidal numbe of ank n with ie m.. METHOD OF ANALYSIS The non-homogeneou etic equation with five unknown to be olved fo it ditinct non-eo integal olution i

2 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut 017 ISSN = 8 w ( )( ) ( ) (1) Intoduction of the linea tanfomation = u v, = u v, = u v, w = u v, u v 0 () in (1) lead to u v = 8 () Diffeent method of obtaining the patten of intege olution to (1) ae illutated below:.1 ATTERN: 1 Let = a b () whee a and b ae non-eo intege. Wite 8 a ( 9 i )( 9 i ) 8 = (5) Uing (), (5) in () and appling the method of factoiation, define Fom which, we have ( u i v) = ( 9 i )( a i b) (6) u = 9a v = a 81b 9b 16a b 18a b 6a 1a b 6ab b 108ab (7) Uing (7) in ( ), the value of,, and w ae given b ( a, b) = 10a 90b 180a b a b 7ab ( a, b) = 8a 7b 1a b 8a b 1ab ( ) ( ) a, b = 19a 171b a b 1a b 6ab, b = 17a 15b 06a b 60a b 180ab w a (8) Thu () and (8) epeent the non-eo intege olution to (1). ( 1,b ) ( 1,b ) 18 ( 1,b ) [ t t t t t 18 ] 696b 8,b 10,b 1,b 16,b 18,b b =

3 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut 017 ISSN ( a,1) ( a,1) w( a,1) 10( a,1) 1[ t8,a t1,a t1,a t16,a 1a ] 0( mod6) ( a,1) w( a,1) ( a,1) 8[ ( a,1) 6a ] 0( mod 8) ( 1,b ) w( 1,b ) 6[ 16 ( 1,b ) 6[ t t t ] 0( mod ) b w( a,1) ( a,1) ( a,1) 15( a,1) 61 [ t t t ] 0( mod) 10,b a 1,b 1,a 18, b 16,a 18, a. ATTERN: Conide (6) a Wite (9) in the fom of atio a u ( ) 81 = v (9) u 9 v ( v) α =, ( β 0) = u 9 β which i equivalent to the following two equation βu αv βv αu ( 9β α ) = 0 ( 9α β ) = 0 On emploing the method of co multiplication, we get u = 9α 6αβ 7β v = β α 18αβ (10) β α = (11) which i atified b α = β = Subtituting the value of α and β in (10) and (11), we get u = 81 v = Subtituting the value of u and v in (), the non-eo ditinct integal value of and ae given b

4 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN (, ) = (, ) = (, ) = (, ) = ( ), = w (1) Thu (1) epeent the non-eo intege olution to (1). w ( 1, ) ( 1, ) ( 1, ) ( 1, ) 6[ t, 6 6t, 1 ] 0( mod 6) (, 1) (,1) (,1) (,1) 18[ 6 ( t10, t1, t16, )] 0( mod ) (, ) ( 1, ) 7( 1, ) 6[ 8 ( t t t )] 18( mod ) 1 1, 16, 18, (,1) (,1) 7(,1) 6[ t t t t t 0t 6 ] 1( mod18) 1, 1, 16, 18, w ( 1, ) ( 1, ) ( 1, ) 6[ t 1 ] 0( mod1) 16, 0,,. ATTERN: Wite (9) in the fom of atio a u 9 ( v) ( v) α =,( β 0) = u 9 which i equivalent to the following two equation βu αv βv αu β ( 9β α ) ( 9α β ) = 0 = 0 On emploing the method of co multiplication, we get u = 7α 6αβ 9β v = α β 18αβ (1) α β = (1) which i atified b α = β = Subtituting the value of α and β in (1) and (1), we get

5 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN u = 81 v = Subtituting the value of u and v in (), the non-eo ditinct integal value of and ae given b (, ) = (, ) = (, ) = (, ) = ( ), = w (15) Thu (15) epeent the non-eo intege olution to (1). ( 1, ) 10 ( 1, ) = [ 6 t8, t10, ] (, 1) 8[ (,1) 108 6t1, ] = 58 (, ) ( 1, ) 7[ ( 1, ) 7 p 6( t t )] 0( mod) 1 10, 1, ( 1) (,1) 7(,1) 6[ 6 t t ] 6( mod18), 10, 1, 1, 1, 1, 1, 118 8, 10, = 0 ( ) ( ) w( ) ( ) [ t t ] ATTERN: Wite (9) in the fom of atio a u 9 ( v) ( v) α =,( β 0) = u 9 which i equivalent to the following two equation βu αv βv αu β ( 9β α ) = ( 9α β ) = 0 On emploing the method of co multiplication, we get 0 u = 7α 6αβ 9β v = α β 18αβ (16) = α β (17) which i atified b

6 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN α = β = Subtituting the value of α and β in (16) and (17), we get u = 81 v = Subtituting the value of u and v in (), the non-eo ditinct integal value of and ae given b (, ) = (, ) = (, ) = (, ) = ( ), = w (18) Thu (18) epeent the non-eo intege olution to (1). ( 1, ) ( 1, ) ( 1, ) ( 1, ) 181 [ 6t, t6, ] 0( mod6) ( 1) w(,1) 6[ (,1) 6[ 6 ( t t t t ) ] 0( mod), 8, 10, 1, 16, (, ) 8( 1, ) [ 7 9t 1t ] 0( mod8) 1, 10, (, 1) w(,1) (,1) 5(,1) 6[ 00t, 96 t8, ] 6( mod0) (, ) w( 1, ) ( 1, ) ( 1, ) [ ( 1, ) 6( t t t )] 0( mod96).5 ATTERN: 5 Wite (9) in the fom of atio a 1 8, 6,, 1 u 9 ( v) ( v) α =, ( β 0) = u 9 which i equivalent to the following two equation βu αv βv αu β ( 9β α ) = 0 ( 9α β ) = 0 On emploing the method of co multiplication, we get

7 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN u = 9α 6αβ 7β v = α β 18αβ (19) = (0) β α which i atified b α = β = Subtituting the value of α and β in (19) and (0), we get u = 81 v = Subtituting the value of u and v in (), the non-eo ditinct integal value of and ae given b (, ) = (, ) = (, ) = (, ) = ( ), = w (1) Thu (1) epeent the non-eo intege olution to (1). (, 1) (,1) w(,1) (,1) 18[ 6 t10, t1, t16, ] 0( mod ) ( 1, ) 8[ 6 ( 1, )] 0( mod 8) ( 1, ) ( 1, ) ( 1, ) ( 1, ) 18[ t8, t10, 1 ] 0( mod6) (, 1) 6[ 7 5(,1) t, t6, ] 0( mod ) (,1) w(,1) 11t [ 9(,1) 7 ( t t t t t t t )]. CONCLUSION 8( mod7), In thi pape, we have made an attempt to detemine diffeent patten of non-eo ditinct intege olution to the non-homogeneou etic equation with five unknown. A the etic equation ae ich in vaiet, one ma each fo othe fom of etic equation with vaiable geate than o equal to five and obtain thei coeponding popetie.. REFERENCES 8, 10, 1, 1, 16, 18, 0,

8 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN [1] L.E. Dickon, Hito of theo of numbe, vol.11, Chelea publihing compan, Newok (195). [] R.D. Camichael, The theo of numbe and Diophantine anali, Dove publication, Newok (1959). [] L.J. Modell, Diophantine equation, Academic pe, London (1969). [] S.G. Telang, Numbe Theo, Tata MC Gaw Hill ublihing Compan, New Delhi (1996). [5] M.A. Gopalan, Manju Somanath and N. Vanitha, aametic Solution of vol., 007, [6] M.A. Gopalan, G. Sangeetha, On the Setic Equation with thee unknown J.Sci.tech, Vol., No:, (010), [7] M.A. Gopalan, R. Sikanth and Uha janaki, aametic integal olution of Vol., No:, 010, = [8] M.A. Gopalan and A. VijaaShanka, Integal Solution of the etic equation of Mathematic and Mathematical cience, Vol.6., No:, 010, 1-5. = (k =, ActaCiencia Indica XXXIII, 6 = ) n 6, Impact, Impact J. Sci. Tech., 6 w, Indian Jounal [9] M.A. Gopalan, S. Vidhalakhmi and A. VijaaShanka, Integal Solution of Non-homogeneou etic equation = 6 6 w, Impact J. Sci. Tech., Vol.6, No:1, 01, 7-5. [10] M.A. Gopalan, S. Vidhalakhmi and K. Lakhmi, On the non-homogeneou etic equation ( w) =, IJAMA, (), Dec 01, [11] M.A. Gopalan, G. Sumathi and S. Vidhalakhmi, Gauian Intege Solution of etic equation with fou unknown ( ) 6 6 = w, Achimede J. Math., (), 01, ( ) 6 6 [1] M.A. Gopalan, G. Sumathi and S. Vidhalakhmi, Integal Solution of = ( w ) intem of Genealied Fibonacci and Luca Sequence, Diophantu J. Math., (), 01, [1] M.A. Gopalan, G. Sumathi and S. Vidhalakhmi, Integal Solution of Non-homogeneou etic equation with fou unknown = 6 16 w, Antactica J. Math., 10(6), 01, [1] M.A. Gopalan, S. Vidhalakhmi and A. Kavitha, Obevation on the non-homogeneou etic equation with fou 5 unknown = ( K ) w IJIRSET, Vol., Iue. 5, 01, Ma-01. [15] M.A. Gopalan, S. Vidhalakhmi and K. Lakhmi, Integal Solution of Non-homogeneou etic equation with five = w T IJESRT, 1(10), 01, unknown ( ) 5 [16] M.A. Gopalan, G. Sumathi and S. Vidhalakhmi, Integal Solution of Non-homogeneou etic equation with five = w 6 t Vol.1, iue., 01, unknown ( ) 5 [17] M.A. Gopalan, S. Vidhalakhmi and K. Lakhmi, Integal Solution of the Setic equation with five unknown ( ) 6 = ( w) 6 w Jul-01 6 T Intenational Jounal of Scientific and eeach ublication, Vol., iue.7, [18] M.A. Gopalan, S. Vidhalakhmi and A. Kavitha, Integal Solution of the Setic equation with thee unknown ( k 1)( ) ( k ) = ( k 1 ) 6 No:7, Jul-015, -8. AUTHORS :, Intenational Jounal of Innovation Science and Reeach, Vol., Fit Autho - S. Vidhalakhmi, ofeo, Depatment of Mathematic, Shimati India Gandhi College, Tich-60 00, Tamil Nadu, India. vidhaigc@gmail.com

9 Intenational Jounal of Scientific and Reeach ublication, Volume 7, Iue 8, Augut ISSN Second Autho S. Aath Thangam, Reeach Schola, Depatment of Mathematic, Shimati India Gandhi College, Tich , Tamil Nadu, India. aaththangam@gmail.com Thid Autho G. Dhanalakhmi, M.hil Schola, Depatment of Mathematic, Shimati India Gandhi College, Tich-60 00, Tamil Nadu, India. dhanamelvi9@gmail.com.