On ternary quadratic equation 6(x 2 + y 2 ) 11xy = 23z 2

Transcription

1 On ternary quadratic equation 6(x 2 + y 2 ) 11xy = 23z 2 R.Nandhini Assistant Professor and Head Dept of Mathematics Bharathidasan University Model College, Thiruthuraipoondi ABSTRACT The ternary homogeneous quadratic equation given by 6(x 2 + y 2 ) 11xy = 23z 2 representing a cone is analyzed for finding its nonzero distinct integral solutions. A few interesting relations and special polygonal and pyramidal numbers are presented. Keywords: Ternary quadratic, integer solutions, homogeneous polygonal number, pyramidal number. Notations used Introduction 1.t m,n - Polygonal number of rank n with size m. 2.P n m - Pyramidal number of rank n with size m. 3.ct m,n Centered polygonal number of n with size m. 4.Pr n Pronic number of rank n. 5.Pt n Pentatope number of rank n. 6.obl n oblong number of rank n. 7.Sqp n Square pyramidal number of rank n. 8.S n Star number of rank n. 9.Gno n gnomic number of rank n. The Diophantine equation offers an unlimited field for research due to their variety [1-5]. In Particular, one may refer [6-20] for quadratic equations with three unknowns. This communication concerns with yet another interesting equation 6(x 2 + y 2 ) 11xy = 23z 2 representing homogeneous cubic equation with three unknowns for determining its infinitely many non-zero integral solutions. Also a few interesting relations among the solutions are presented. 142

2 Method of Analysis The Ternary quadratic diophantine equation representing a cone under consideration is 6(x 2 + y 2 ) 11xy = 23z 2 (1) The substitution of the linear transformations x = u + v, y = u v (2) in (1) gives u v 2 = 23z 2 (3) Now (3) is solved through different choices and thus different patterns of solutions to (1) are obtained. Pattern I Write (3) as u 2 = 23 (z 2 v 2 ) = 23 [(z + v)(z - v)] (4) Choice (i) (4) is written in the form of ratio as 0 (5) This is equivalent to the following two equations Bu Av Az =0 -Au -23Bv + 23Bz =0 (6) Applying the method of cross multiplication, the above system of equation is satisfied by v = A 2 23B 2 z = A B 2 (7) x = A 2-46AB 23B 2 y = 23B 2 46AB A 2 (8) Thus (7) and (8) represent non zero distinct integral solutions of (1) in two parameters. 1. y(1,b) t 62,B + t 16,B -1 (mod 23) 2. y(2,b) + 2z(2,B) t 152,B + t 14,B 4(mod 23) 3. 2x(A,1) + z(a,1) t 12,A + t 6,A -23 (mod 89) 4. y(a,a +1) + z(a,a + 1) t 102,A + t 10,A + 46 obl A 46 (mod 138) 5. 3 x[a (A + 1), A + 2] + y [A (A + 1), A + 2] = -552P A Choice (ii) 143

3 (4) is written in the form of ratio as (9) Bu + Av Az =0 Au 23Bv 23Bz=0 (10) By the method of cross multiplication, the above system of equation is satisfied by v = 23B 2 A 2 z = -A 2-23B 2 (11) Substituting the values of u and v in (2) we get x= 23B 2 46 AB A 2 y = -23B 2-46 AB + A 2 (12) Thus (11) and (12) represent non zero distinct integral solutions of (1) in two parameters. 1. y(a,2) t 16,A + t 14,A -1 (mod 91) 2. x(1,b) z(1,b) + t 82,B - s B -1 (mod 1) 3. x[a,(a + 1)(2A + 1)] + z[a,(a + 1)(2A + 1)] + t 20,A - t 16,A Sqp A 0 (mod 2) 4. y[b + 1, B] 2x [B +1, B] 46 Pr B + s B + t 122, B 2 (mod 59) 5. 2x(2,B) + y(2,b) t 56,B + t 10,B -4 (mod 253) Choice (iii) Also (4) is written in the form of ratio as Bu + 23Av 23 Az =0 (13) Au Bv Bz=0 (14) Applying the method of cross multiplication, the above system of equation is satisfied by v = B 2 23 A 2 z = -23A 2 - B 2 (15) x = B 2 23 A 2 46AB y = -23A 2 - B 2-46AB (16) Thus (15) and (16) represent non zero distinct integral solutions of (1) in two parameters. 144

4 1. 3 y[a(a + 1), A + 2] + z [A(A + 1), A + 2] t 24,A + t 28,A + 276P A -8(mod 10) 2. z(2,b) x(2,b) t 36,B + t 40,B 0(mod 90) 3. x(a 2,A + 1) + y (A 2 5, A + 1) -184P A 4. x[a,b(b + 1)] + z[a, B(B + 1)] t 26,A + t 118,A + 92 ct A,B -2 (mod 46) 5. y(a,a + 1) x (A,A +1) t 130,A + t 42,A -2 (mod 40) Choice (iv) Again (4) is written in the form of ratio as Bu - 23Av 23 Az =0 (17) Au + Bv Bz=0 (18) Applying the method of cross multiplication, the above system of equations is satisfied by u = 46AB v = -23A 2 + B 2 19) z = 23A 2 + B 2 x = -23A 2 + B AB y = 23A 2 + B AB (20) Thus (19) and (20) represent non-zero distinct integral solutions of (1) in two parameters. Pattern -II :- 1. x[a,(a+1) (A+2)] 2[A, (A+1) (A+2)] t 86,A + t 10,A 276P 3 A 0 (mod 41) 2. x(a,a+1) y(a,a+1) t 70,A + t 22,A 2 (mod 38) 3. z(a,a+1) t 44,A t 8,A 1 (mod 24) 4. z(b,b+1) y(b,b+1) t 16,B + t 12,B Gno B + 46Obl B 3 (mod 4) 5. x[a(a+1), (A+2)(A+3)] + y[a(a+1), (A+2)(A+3)] = 2208Pt A Assume z = A B 2 (21) Write 23 as 23 = (i 23) (-i 23) (22) Substituting (21) and (22) in (3) (u+i 23v) (u-i 23v) = (i 23) (-i 23)(A 2 +23B 2 ) 2 = (i 23) (-i 23)[(A+i 23B) (A-i 23B)] 2 (23) Equating the positive or negative roots u+i 23v = i 23 (A+i 23B) 2 (24) Equating real and imaginary parts in (24), we get v = A 2 23B 2 (25) z = A B 2 x= A 2 23B 2 46AB (26) y= 23B 2 - A 2 46AB Thus (25) and (26) represent non-zero distinct integral solutions of (1) in two parameters. 145

5 :- 1. x(a,2) y(a,2) t 76,A + t 72,A 0 (mod2) 2. x[a,(a+1)(a+2)(a+3)] + z[a,(a+2)(a+3)] t 88,A + t 84,A Pt A 0 (mod2) 3. x(a,a+1) + 2z(A,A+1) t 42,A S A + 46 Pr A 22 (mod71) 4. y[a,b(b+1)] Z[A,B(B+1)] - t 16,A + t 20,A + 92ct A,B 0(mod 2) 5. x(4,b) + t 60,B S B 15 (mod 206) Conclusion Reference To conclude one may search for other patterns of solutions and their corresponding properties. 1. Dickson LE. History of 1l1eory of numbers, Chelsea Publishing Company, New York, 1952, Mordell LJ. Diophantine Equations, Academic press, London, Andre Weil, Number Theory: An approach_thmllgh history: from- hammurapi to legendre I Andre "eil: Boston (Birkhauser Boston, Nigel Smart P. The algorithmic Resolutions of Diophantine equations, Cambridge unh ersiry press, Smith DE. History of mathematics Dover publications, New York, (II). 6..Gopalan MA. Note on the Diophantine equation x2 +axy + by2 = z2 Acta Ciencia lndica 2000; XXVJM(2): Gopalan MA. Note on the Diophantine equationx2 +xy +y2 = 3z2 Acta Ciencia Indica 2000; XXVJM(3): Gopalan MA. Ganapathy R. Srikanth R. on the Diophantine equation z2 = Ax 2 - By 2 Pure and Applied Mathematika Sciences 2000; LU( 1-2): Gopalanand MA, Anbuselvi R. On Ternary Quadratic Homogeneous Diophantine equation x2 + Pxy + y2 = z2, Bulletin of Pure and Applied Sciences 2005; 24E(2): Gopalan MA, Vidhyalakshmi S, Krishnamoorthy A. Integral solutions Ternary Quadraticax 2 + by2 = c(a + b)z 2,Bulletin of Pure and Applied Sciences 2005; 24E(2): Gopalan MA, Vidhyalakshmiands S. Devibala, Integral solutions of ka(x 2 +y2) + bxy = 4ka2z 2, Bulletin of Pure and Applied Sciences 2006; 25E(2): Gopalan MA, Vidhyalakshmiands S. Devibala, Integral solutions of 7x2 + 8y2 = 9 z2, Pure and Applied Mathematitn.I.Sciences, 2007; LXVI( l-2): Gopalan MA, Vidhyalnkshmi S. An observation on kax2 + by2 = cz2, Acta Cienica lndica 2007; XXXIITM(l): Gopalan MA, Manjusomanath, Vanitha N. Integral solutions of kxy + m(x + y) = z2, Acta Cienica lndica 2007;XXXIIIM(4): Gopalan MA, Kaliga Rani J. Observation on the Diophantine Equation y2 = Dx 2 + y 2, Impact J Sci Tech. 2008; 2(2): Gopalan MA, Pondichelvi V. On Ternary Quadratic Equation x2 + y2 = z2 + 1, Impact J.Sci. Tech,Vol (2), No:2,2008, Gopalan MA, Gnanam A. Pythagorean triangles and special polygonal numbers, International Journal of Mathema1ical Science.2010; 9(!-2): Gopalan MA, Vijayasankar A. Observations on a Pythagorean Problem, Acta Cienica lndica 2010; XXXVIM(4):5 l Gopalan MA, Pandichelvi V. Integral Solutions of Ternary Quadratic Equation Z(X - Y) = 4XY, Impact J Sci Tech. 2011; 5(1): Gopalan MA, Kaligarani J. On Ternary Quadratic Equation X 2 +Y2 = Z2 + 8, Impact J Sci Tech.201 1; 5(1):