Integral Solutions of Ternary Quadratic Diophantine Equation

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume 2, Issue 4 Ver. IV (Jul. - Aug.206), PP Integral Solutions of Ternary Quadratic Diophantine Equation 7 x 2 + y 2 3xy = 27z 2 R.Anbuselvi, S.JamunaRani 2 Associate Professor, Department of Mathematics, ADM College for women, Nagapattinam, Tamilnadu,India 2 Asst Professor, Department of Computer Applications, Bharathiyar college ofengineering and Technology, Karaikal, Puducherry, India Abstract: The ternary quadratic Diophantine equation given by 7(x 2 + y 2 ) 3xy = 27z 2 is analyzed for its patterns of non-zero distinct integral solutions. A few interesting relations between the solutions and special polygonal numbers are exhibited. Keywords: Ternary quadratic, integral solutions, polygonal numbers. I. Introduction Ternary quadratic equations are rich in variety [-3].For an extensive review of sizable literature and various problems, one may refer [-20]. In this communication, we consider yet another interesting ternary quadratic equation 7(x 2 + y 2 ) 3xy = 27z 2 and obtain infinitely many non-trivial integral solutions. A few interesting relations between the solutions and special polygonal numbers are presented. Notations Used t m,n - Polygonal number of rank n with size m P m n - Pyramidal number of rank n with size m F m,n Figurative number of rank n with size m Methods of Analysis The Quadratic Diophantine equation with three unknowns to be solved for its non zero distinct integral solutions is 7(x 2 + y 2 ) 3xy = 27z 2 On substituting the linear transformations x = u + v ; y = u v (2) in, leads to u 2 = 27(z 2 v 2 ) 3 Pattern I Equation 3 can be written as u v 2 = 27 z 2 4 Assume z 2 = A (5) 27 can be written as 27 = (i 27) i 27 (6) Substitute 4, 5 in 3 and applying the method of factorization, we get u + i 27v = 54AB + i 27 A 2 27B 2 7 Equating real and imaginary parts u = 54 AB ; v = A 2 27 B 2 8 Using 8, 5 and 2 we obtain the integer solutions to as presented below x A, B = x = A 2 27 B 2 54AB y A, B = y = A B 2 54AB z A, B = z = A B 2 9 DOI: / Page

2 . z, B x, B 08t 3,B z, B(B + 2) x, B(B + 2) 324P B 0 3. z, B(B + 2)(B + 3) x, B(B + 2)(B + 3) 296 P B z, B 2 x, B 2 08P B z, B(B + )(B + 2) x, B(B + )(B + 2) 648 F 4,B y, + z, 0 7. x A, A + y A, A + z(a,a) can be expressed as a sum of two squares 8. x,2 + y,2 is a cube no 9. Each of the following expression represents a perfect number a. y, b. z(,) 0. Each of the following expression represents a Nasty number a. b. 2 2 [x, z, ] [x, + y, ] c. y 0, + z(0,) d. x 2,2 y 2,2 + z(2,2) e. x A, A + y A, A + z(a, A). Each of the following can be expressed as a perfect squares a. y,3 + z,3 b. x,3 + y,3 c. x 2, z(2,) Pattern II write 3 as Assume u v 2 = 27 z 2 0 z 2 = A B 2 () Also 27 can be written as 3 + i 27 (3 i 27) = 36 (2) Applying 7, 6 in 5 and employing the method of factorization, define u + i 27v = 6 i 27 3A2 8B 2 54AB + 6 ( 27A B 2 62AB) 3 Equating real and imaginary parts u = 6 27A B 2 62AB v = 6 (3A2 8B 2 54AB) (4) Using 4, and 2 we obtain the integer solutions to as presented below. 5z A, y A, 20t 3,A 0( mod 8) 2. 4z A, x A, 6t 3,A 0 (mod 36) x A, B = x = 4A B 2 36AB y A, B = y = 5A B 2 8AB z A, B = z = 27 B 2 + A 2 5 DOI: / Page

3 3. 4y A A +, 5x A A +, 26t 3,A y A A + (A + 2), 5x A A + (A + 2), 648 P 3 A y A A + (A + 2)(A + 3), 5x A A + (A + 2)(A + 3), 2592 P 4 A y A 2 A +, 5x A 2 A +, 26 P 5 A y A A + 2 (A + 2), 5x A A + 2 (A + 2), 296F 4,n x, + 4z, is a nasty number. 9. Each of the following expression represents a perfect square a. y, 5z, [x, 4z(,)] b. y, x, z(,) c. y 2,2 x 2,2 z 2,2 d. y 2,2 x 2,2 + z(2,2) e. x 2,3 y 2,3 + z 2,3 f. y 2,3 x 2,3 z(2,3) 0. The following expression represents a perfect no a. y, 5z, b. y 2, x 2, z 2, c. x 2, + z 2, y(2,) Pattern III Equation 3 canbewritternas u z v = = A, 27(z + v) u B B 0 6 which is equivalent to the system of equations Au Bz + Bv = 0 Bu 27Az 27Av = 0 From which we get u = 54AB v = B 2 27A 2 z = 27A 2 + B 2 Substituting 7 and 6 in 2, the corresponding non-zero distinct integral solutions of are given by. x A, (A + ) + y A, (A + ) 26t 3,A 0 x A, B = x = B 2 27A AB y A, B = y = 27A 2 B AB z A, B = z = 27A 2 + B x A, A (A + ) + y A, A(A + ) 26P 5 a 0 3. x A, A + A + 2 A y A, A + A + 2 A P 4 A 0 4. x A, (A + )(A + 2) + y A, (A + ) (A + 2) 648P 3 A 0 5. y A, 54 t 3,A mod 3 6. z A, t 56,A mod x, B + z, B 4t 3,B 0 mod 3 8. x A A +, (A + )(A + 2) 296F 4,n 4, 0 9. y A, + z A, 08t 3,A 0 0. x, + y, is expressed as difference of two perfect squares. x, + y, + z(,) can be expressed as a sum of two perfect squares 2. Each of the following expression represents a perfect number a. y 3, x 3, b. x 3,2 y 3,2 + z 3, 2 c. y 2,3 z 2,3 3. Each of the following expression represents a Nasty number a.,2 y(3,3) DOI: / Page

4 b. y, x, z(,) c. y 2,2 x 2,2 z(2,2) d. 3 z 3,3 It is observed that6, by rewriting 3 suitably, one may arrive at the following pattern of solutions to () Pattern IV x A, B = x = 27 B AB A 2 y A, B = y = A AB 27 B 2 z A, B = z = A B 2 9. z A, B t 56,A mod y A, B 4t 3,A 27 (mod 53) 3. x, B + z, B 08 t 3,B 0 4. x A A +, + z A A +, 08 t 3,A 0 mod 3 5. x A A + A + 2, + z A A + A + 2, 324 P 3 A 0 (mod 3) 6. x A A + A + 2 A + 3, + z A A + A + 2 A + 3, 296 P 4 A 0 mod 3 7. x A 2 A +, + z A 2 A +, 08 P 5 A 0 mod 3 8. x A A + 2 A + 2, + z A A + 2 A + 2, 648 F 4,n 4 0 mod 3 9. y,2 + y 3,3 can be expressed as a sum of two squares 0. Each of the following expression represents a perfect number a. y, b. z(,). Each of the following expression represents a Nasty number a. x, y, z(,) b. x 2,2 y 2,2 z 2,2 c. x, + y(,2) 2. Each of the following can be expressed as a difference of two squares a. z 3,3 b. y(2,2) II. Conclusion In this paper, we have presented five different patterns of non- zero distinct integer solutions of ternary quadratic Diophantine equation 7 x 2 + y 2 3 xy = 27z 2 and relations between solutions and special numbers are also obtained. To conclude, one may search for other patterns of solutions and their corresponding properties. Journal Articles References [] GopalanMA, Sangeethe G. On the Ternary Cubic Diophantine Equation y Dx z Archimedes J.Math, 20,():7-4. [2] GopalanMA,VijayashankarA,VidhyalakshmiS.Integral solutions of Ternary cubic Equation, 2 x y xy 2( x y 2) ( k 3) z, Archimedes J.Math,20;(): [3] GopalanM.A,GeethaD,Lattice points on the Hyperboloid of two sheets J.Sci.Tech,200,4, [4] GopalanM.A,VidhyalakshmiS,KavithaA,Integral points on the Homogenous Cone J.Math,202,(2) [5] GopalanM.A,VidhyalakshmiS,SumathiG,Lattice points on the Hyperboloid of one sheet Diophantus J.Math,202,(2),09-5. [6] GopalanM.A, VidhyalakshmiS, LakshmiK, Integral points on the Hyperboloid of two sheets Diophantus J.Math,202,(2), x 6xy y 6x 2y 5 z 4 Impact z 2x 7y,The Diophantus 4z 2x 3y 4, The 3y 7x z 2, DOI: / Page

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