OBSERVATIONS ON TERNARY QUADRATIC EQUATION z 82 x

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1 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: OBSERVATIONS ON TERNARY UADRATIC EUATION z 8 x y G. Jnki 1 nd S. Vidhy 1,Assistnt rofessor, Deprtment of Mthemtics, Cuvery College for Women, Trichy-18, Tmilndu, Indi Abstrct-The udrtic Diophntine Eqution with three unknowns z = 8x + y is nlyzed for its non-zero distinct integer solutions. A few interesting reltions mong the solutions nd some specil polygonl, Octhedrl nd Four dimensionl figurte numbers re presented. Keywords udrtic eqution with three unknowns, Integrl solutions, olygonl Numbers, Four Dimensionl Figurte Numbers, Octhedrl number. 1. Introduction Ternry qudrtic equtions re rich in vriety. For n extensive review of sizble literture nd vrious problems one my refer [1-7]. In [8], the ternry qudrtic Diophntine eqution of the form kxy m x y) z hs been studied for its non-trivil integrl solutions. In [9,10], A specil ythgoren tringle problem hve been discussed for its integrl solutions. In [11], two prmetric non-trivil integrl solutions of the ternry qudrtic homogeneous Diophntine eqution X XY Y Z, where is non-zero constnt hve been presented. In [1], the ternry qudrtic homogeneous eqution k x y ) bxy 4k z, k,,b 0) hs been studied for its non-trivil integrl solutions. In [13], the ternry qudrtic Diophntine eqution of the form ngles nd their prmetric representtions re obtined. x y) x z) y 0, is nlyzed for its integrl solutions t different In this communiction, we consider yet nother interesting ternry qudrtic eqution z = 8x + y nd obtin different ptterns of non-trivil integrl solutions. Also, few interesting reltions mong the solutions nd specil polygonl, Octhedrl nd Four dimensionl numbers re presented.. Nottions Denoting the rnks of some specil polygonl, Octhedrl nd Four dimensionl figurte numbers. n 1m m, n 1 = olygonl Number with rnk n nd sides m. T n O n 3 n n) = Octhedrl Number of rnk n. 3 n n 1) 4DF n = Four dimensionl figurte Number whose generting polygon is squre. 1 3.Method of Anlysis The ternry qudrtic Diophntine eqution to be solved for its non-zero integrl solution is 017, IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 139

2 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: z = 8x + y. 1) Assuming z = z,b) = + 8b. ) where nd b re non-zero integers. ttern : 1 Eqution 1) cn be written s z 8x = y 3) Assuming y = y,b) = 8b 4) we get 8b 8x y Using the fctoriztion method, we hve Compring rtionl nd irrtionl fctors, 8 8 z 8x b 5) z 8x b 6) x = x,b) = b The corresponding non-zero distinct integer solutions re Observtions x = x,b) = b z = z,b) = + 8b y = y,b) = 8b z = z,b) = + 8b 1. If = b nd is even, then z is divisible by.. For ll the vlues of nd b, x + y + z nd x y + z is divisible by. 3. If is even nd b is odd, then z is divisible by. 4. y,) x,) z,) T 0mod 4) 8, 5. y,1) z,1) 4T 0mod ) 3, 6. x,3) y,3) z,) 4T 0mod 1) 7, 7. x,1). y,1) 49O 0mod 330 ). 8. Ech of the following expressions i) x,) + y,) + z,) n 017, IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 140

3 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: ii) 3 x,1) T14, T6, represents Nsty number. ttern : Eqution 1) cn be written s 8x + y = z * 1 7) Assuming z = z,b) = + 8b. nd write 9 i8 8 )9 i8 8 ) 1 8) 539 Using fctoriztion method, eqution 7) cn be written s 9 i8 8 9 i y i 8x) y i 8x) i 8b) i8 8b We get Compring rel nd imginry prts, we get 1 y b 73 ) i 8x) 9 i8 8) i 8 ) 9) 1 y b 73 i 8x) 9 i8 8) i 8 ) 10) x = [8 656b + 18b] y = [9 738b 131b] Since our interest is on finding integer solutions, we hve choose nd b suitbly so tht xy, nd z re integers. Let us tke = 73A nd b = 73B The integer solutions re x = 584A 47888B AB y = 657A 53874B 95776AB z = 539A B 017, IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 141

4 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: Observtions 1. If A = B, then x + y + z is divisible by.. y A,1) x A,1) T mod 73). 18, A 3. x A,1) z A,1) T mod 73). 1, A ttern 3: Eqution 1) cn be written s z y = 8x nd we get z+y)z-y) = 8x.x 11) Cse 1: Eqution 11) cn be written s z 8 y x x z y 1) From eqution 1), we get two equtions Applying cross rtio method, we get the integer solutions re Observtions: 1. y,3) z,3) 41T 0mod 13). 10,. x,1) z,1) T 1 0mod 79). 8, 3. For ll the vlues of nd, x + y + z is divisible by. 4. Ech of the following expressions iii) y,) - x,) - z,) iv) T T 3x,1). represents Nsty number. Cse : Eqution 11) cn be written s 4, 1, -8x + y + z = 0 x + y z = 0 x = x,) = - y = y,) = 8 z = z,) = , IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 14

5 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: z y 8 x x z y From eqution 13), we hve two equtions 13) -x + y + z = 0 8x + y z = 0 Applying cross rtio method, we get the integer solutions re x = x,) = - y = y,) = 8 z = z,) = - 8 Observtions 1. x 1, ) y1, ) z1, ) T 0mod ). 6,. y, ). z, ) DF 0mod 4). 6, 3. T 10 x,1) represents Nsty Number. 4 14, ttern 4: Eqution 1) cn be written s z 8x = y * 1 14) nd write Assuming y = y,b) = 8b 15) we hve Using fctoriztion method, we get From eqution 16), we get z 8x. 8b b 8 9). z 8x 16) 8b 8 9). z 8x 17) z = b + 164b x = + 8b + 18b 017, IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 143

6 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: Thus, the corresponding non-zero distinct integer solutions re x = x,b) = + 8b + 18b y = y,b) = 8b z = b + 164b Observtions 1. x, ) 3y, ) 4T 0mod ). 3 7,. x,1) y,1) z,1) 5T 0mod 5). 11, 3. x, b) z, b) T 0mod 5). 9 1, b 4. CONCLUSION one my serch for other ptterns of non-zero integer solutions nd reltions mong the solutions. REFERENCES [1] Btt.B nd Singh.A.N, History of Hindu Mthemtics, Asi ublishing House [] Crmichel, R.D., The Theory of Numbers nd Diophntine Anlysis, Dover ublictions, New York,1959. [3] Dickson,.L.E., History of the theory of numbers, Chelsi ublishing Co., Vol.II, New York, 195. [4] Mollin.R.A., All solutions of the Diophntine eqution x Dy n,, For Est J.Mth. Sci., Socil Volume, 1998, rt III, pges [5] Mordell.L.J., Diophntine Equtions, Acdemic ress, London [6] Telng.S.G., Number Theory, Tt McGrw-Hill ublishing Compny, New Delhi [7] Nigel..Smrt, The Algorithmic Resolutions of Diophntine Equtions,Cmbridge University ress, London [8] Gopln.M.A., Mnju Somnth, nd Vnith.N., Integrl Solutions of kxy m x y) z, Act Cienci Indic, Volume 33;number 4, pges , 007. [9] Gopln.M.A., nd Anbuselvi.R., A specil ythgoren tringle, Act Cienci 017, IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 144

7 Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: Volume: 04 Issue: 03 Mr p-issn: Indic, Volume 31, number 1, pges 53-54, 005. [10] Gopln.M.A., nd Devibl.S., A specil ythgoren tringle, Act Cienci Indic, Volume 31, number 1, pges 39-40, 005. [11] Gopln.M.A. nd Anbuselvi.R, On Ternry udrtic Homogeneous Diophntine Eqution X XY Y Z, Bulletin of ure nd Applied Science, Vol.4E, No., p , 005. [1] Gopln.M.A, Vidylkshmi.S nd Devibl.S, Integrl Solutions of k x y ) bxy 4k z, Bulletin of ure nd Applied Science, Vol.5E, No., p , 006 [13] Gopln.M.A., nd Devibl.S, On Ternry udtrtic Homogeneous Eqution x y) x z) y 0, Bulletin of ure nd Applied Science, Vol.4E, No.1, p.5-8, , IRJET Impct Fctor vlue: ISO 9001:008 Certified Journl ge 145

Homogeneous Bi-Quadratic Equation With Five Unknowns

Homogeneous Bi-Quadratic Equation With Five Unknowns Interntionl Journl of Mthemtics Reserch. ISSN 0976-580 Volume 6, Number (01), pp. 5-51 Interntionl Reserch Publiction House http://www.irphouse.com Homogeneous Bi-Qudrtic Eqution With Five y p q Unknowns