ONTERNARY QUADRATIC DIOPHANTINE EQUATION x + 3y = 4z M.A.Gopala 1, K.Geetha ad Maju Somaath 3 1 Professor,Dept.of Mathematics,Shrimati Idira Gadhi College,Trichy- 0 Tamiladu, Asst Professor,Dept.of Mathematics,Cauvery College for Wome,Trichy-18, Tamiladu, 3 Assistat Professor, Dept.of Mathematics, Natioal College,Trichy- 01,Tamiladu, Idia Abstract-The terary quadratic diophatie equatiox + 3y = 4z is aalyzed for its o-zero distict itegral poits o it. A few iterestig properties amog the solutios are preseted. Keywords:Itegral poits, Terary quadratic, Polygoal umbers, Pyramidal umbers ad Special umbers. Mathematics subject classificatio umber: 11D09 Notatio: t = Polygoal umber of rak with sides m m, ct m, = Cetered Polygoal umber of rak with sides m g = Gomoic umber wo = Woodhal umber mer = Mersee umber I. INTRODUCTION Diophatie equatios is a iterestig cocept, as it ca be see from [1-]. For a extesive review of various problems oe may refer[3-11]. I this cotext oe may also see [1-3]. This commuicatio cocers with yet aother iterestig terary quadratic diophatie equatiox + 3y = 4z for determiig its ifiitely may o-zero itegral solutios. Also a few iterestig properties amog the solutios are preseted. II. METHOD OF ANALYSIS The terary quadratic equatio to be solved for its o-zero itegral solutio is x + 3y = 4z (1) x = X, y =Y () O substitutig () i (1), we get X + 3Y = z (3) Let X = α+3β, Y = α-β, z = 5γ (4) Substitutig (4) i (3), we get α + 6β = γ (5) Where α, β ad ϒ are o-zero itegers. @IJMTER-016, All rights Reserved 644
Differet patters of solutio for (1) are give below Iteratioal Joural of Moder Treds i Egieerig ad Research (IJMTER) Volume 03, Issue 0, [February 016] ISSN (Olie):349 9745; ISSN (Prit):393-8161 PATTERN:1 Equatio (5) ca be writte as α + 6β = γ *1 (6) Write 1 as 1 = γ = a + 6b (8) Where a, b are o-zero distict itegers. Substitutig (7) & (8) i(6), we get α + 6β = (a + 6b ) O employig the method of factorizatio ad o equatig real ad imagiary parts, we get + 6 6 = (+ 6 ) O comparig the positive ad egative factors, we get ( 6 ) + 6 = + 6 (9) 6 = 6 (10) O comparig the ratioal ad irratioal parts from the above equatio, we get α = (a 6b -4ab) (11) β = [ (a 6b +ab)] (1) As our iterest is to fid oly iteger solutio, it is see that α, β are itegers for suitable choices of a ad b. Let us assume a = 5A ad b = 5B i (11) ad (1), we get α = 5A 30B -10AB (13) β = 10A 60B + 10AB (14) Substitutig (13) ad (14) i (), theo-zero distict itegral solutios of (1), we get x = x(a, B) = (35A 10B -90AB) y = y(a, B) = (-15A + 90B -140AB) z = z(a, B) = 5 5 +150 1. x( A,1 ) + y ( A,1) t 4t 40( mod 45) 4, A 1, A. z ( A,1) = 50( ct + 55ct ) 5( 3437g + A + 1663) PATTERN: Write (5) as, 5, A 5, A γ - α = 6β A (7) @IJMTER-016, All rights Reserved 645
Case 1: Iteratioal Joural of Moder Treds i Egieerig ad Research (IJMTER) Volume 03, Issue 0, [February 016] ISSN (Olie):349 9745; ISSN (Prit):393-8161 (γ+ α) (γ α) = 6β (15) = =! This is equivalet to the followig two equatios qα 6pβ + qγ = 0 pα +qβ - pγ = 0 Applyig the method of cross multiplicatio, we get α = α(p, q) = 6p q β = β(p, q) = pq (17) γ = γ(p, q) = 6p + q Substitutig the values of (17) i (), the o-zero distict itegral values of x,y ad z satisfyig (1) are give by x = x(p, q) = (6p q + 6pq) y = y(p, q) = (6p " - 4pq) z = z(p, q) = 5 (36p 4 + q 4 +1p q ) 1. ( ) ( ) ( ) 11, x 1, q y 1, q + z 1, q = ct + t + 37g + 16 10, q x, + 1 y, + 1 41 = t. ( ) ( ) 54, Case : Equatio (15) ca be rewritte as = =! O followig the procedure as i case (1) the o-zero distict solutios of (1) are give by x = x(p, q) = (p 6q + 6pq) y = y(p, q) = (p 6q - 4pq) z = z(p, q) = 5 (p 4 + 36q 4 +1p q ) 1. (, ) x + Mer + 6wo + 7 = Nasty umber. ( ) ( ) ( ) PATTERN:3 Rewrite (5) as, Write 1 as,,, = 10 l + + 0 x y car ky q q (16) # - 6β = *1 (18) 1 = 5+ 65 6 (19) α = a 6b (0) Usig (19) ad (0) i (18), we get @IJMTER-016, All rights Reserved 646
Iteratioal Joural of Moder Treds i Egieerig ad Research (IJMTER) Volume 03, Issue 0, [February 016] ISSN (Olie):349 9745; ISSN (Prit):393-8161 ( a b ) ( )( ) γ 6β = 6 5 + 6 5 6 O employig the method of factorizatio ad equatig the positive ad egative factors, we get α = a 6b β = a + 1b + 10ab (1) γ = 5a + 30b + 4ab Substitutig the values of (1) i (), the o-zero distict itegral values of x,y ad z satisfyig (1) are give by x = x(a, b) = (7a + 30b + 30ab) y = y(a, b) = (-3a 30b -0ab) z = z(a, b) = 5 (5a +30b + 4ab) 1. x( a + 1,1) + y ( a + 1,1) ( t8, a + t1, a + 1ga ) = 49. y ( + 3, ) = ( t68, t6, t4, + 11g 16) III. CONCLUSION Oe may search for other patters of solutio ad their correspodig properties. REFERENCES [1] Dickso L.E., History of theory of umbers, Vol., Chelsea publishig compay, New York, 195. [] Mordell L.J., Diophatie Equatios, Academic press, New York, 1969. [3] Gopala M.A., ad Padichelvi V., Itegral solutio of terary quadratic equatio ( ) 4 Idica, Vol. XXXIVM, No.3, Pp.1353-1358, 008. z x + y = xy, Acta Ciecia y = DX + Z, Impact J.Sci. [4] Gopala M.A., ad Kaliga Rai J., Observatio o the Diophatie equatio Tech., Vol., No., Pp. 91-95, 008. [5] Gopala M.A., ad Padichelvi V., O terary quadratic equatio X + Y = Z + 1, Impact J.Sci.Tech., Vol., No., Pp.55-58, 008. [6] Gopala M.A., Maju Somaath ad Vaith N., Itegral solutios of terary quadratic Diophatie equatio ( 1) x + y = k + z, Impact J.Sci. Tech.,Vol.,No.4, Pp.175-178, 008. [7] Gopala M.A., ad Maju Somaath Itegral solutio of terary quadratic Diophatie equatio xy + yz = zx, Atartica J.Math., Vol.5, o.1, Pp.1-5, 008. [8] Gopala M.A., ad Gaam A., Pythagorea triagles ad special polygoal umbers, Iteratioal J.Math. Sci., Vol.9, No.1-, Pp. 11-15, Ja-Ju 010. [9] Gopala M.A., ad Padichelvi V., Itegral solutio of terary quadratic equatio ( ) 4 z x y = xy, Impact J.Sci. Tech., Vol. 5, No.1, Pp. 1-6, 011. [10] Gopala M.A., ad Vijaya Sakar, Observatio o a Pythagorea problem, Acta Ciecia Idica, Vol. XXXVIM, No.4, Pp.517-50, 010. [11] Gopala M.A., ad Kaliga Rai J., O terary quadratic equatio X + Y = Z + 8, Impact J.Sci. Tech., Vol. 5, No.1, Pp. 39-43, 011. [1] Gopala M.A., ad Geetha D., Lattice poits o the Hyperboloid of two sheets x 6xy + y + 6x y + 5 = z + 4, Impact J.Sci. Tech., Vol. 4, No.1, Pp. 3-3, 011. @IJMTER-016, All rights Reserved 647
Iteratioal Joural of Moder Treds i Egieerig ad Research (IJMTER) Volume 03, Issue 0, [February 016] ISSN (Olie):349 9745; ISSN (Prit):393-8161 [13] Gopala M.A., Vidyalakshmi S., ad Kavitha A., Itegral poits o the homogeeous coe z = x 7y, Diophatus J.Math., Vol. 1, No.5, Pp. 17-136, 01. [14] Gopala M.A., Vidyalakshmi S., ad Sumati G., Lattice poits o the hyperboloid of oe sheet 4z = x + 3y 4, Diophatus J.Math., Vol. 1, No., Pp. 109-115, 01. [15] Gopala M.A., Vidyalakshmi S., ad Lakshmi K., Lattice poits o the hyperboloid of two sheets 3y = 7x z + 1, Diophatus J.Math., Vol. 1, No., Pp. 99-107, 01. [16] Gopala M.A., Vidyalakshmi S., Usha Rai T.R., ad Malika S., Observatios o 6z = x 3y, Impact J.Sci. Tech., Vol.6, No.1, Pp. 7-13, 01. [17] Gopala M.A., Vidyalakshmi S., ad Usha Rai T.R, Itegral poits o the o-homogeeous coe z + 4xy + 8x 4z + = 0, Global J.Math. Sci., Vol., No.1, Pp.61-67, 01. [18] Gopala M.A., ad Geetha.K, Itegral poits o the Homogeeous coe Research Joural of Multidiscipliary, Vol.1,Issue 4, Pp. 6.71, 01. x = 6z 4y, Asia Academic [19] Gopala M.A., ad Geetha.K, Itegral solutio of terary quadratic Diophatie equatio z = a ( x + y + bxy),idia Joural of Sciece,Vol.,No.4, Pp.8-85, 013. [0] Gopala M.A., ad Geetha.K,Observatios o the hyperbola y = 18x + 1, RETELL, Vol.13, No.1, PP.81-83, Nov 01 [1] Gopala M.A., Geetha.K ad Maju Somaath,Itegral solutios of quadratic equatio with four ukows xy + z( x + y) = w, Impact Joural of sciece ad Techology, Vol.7, No.1, PP.1-8, Ja-Mar 013. [] Gopala M.A., Geetha.K ad Maju Somaath, A terary quadratic Diophatie equatio ( ) ( ) 8 x + y 15xy + x + y + 1 = 3z, Proceedigs of the iteratioal coferece o Mathematical methods ad computatio, Feb 13 th ad 14 th 014, 46-51. [3] Gopala M.A., Geetha.K ad Maju Somaath, A terary quadratic Diophatie equatio Bulleti of Mathematics ad statistics research, Vol., issue.1, Pp.1-8, 014. 7x + 9y = z, @IJMTER-016, All rights Reserved 648