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International Research Journal of Engineering and Technology (IRJET) E-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 P-ISSN: 2395-0072 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072 OBSERVATIONS ON X2 + Y2 = 2Z2 - 62W2 G.Janaki* and R.Radha* *Department of Mathematics, Cauvery college for women, Trichy ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - The Quadratic equation with four unknowns 1), (-y, y, ±1, -y) satisfy the equation under of the form X2 + Y2 = 2Z2 - 62W2 has been studied for its consideration. In the above quadruples, any two values non-trivial distinct integral solutions. A few interesting of the unknowns are the same. In [13], the Quadratic relations among the solutions and special polygonal equation with four unknowns of the form XY +X+ Y+1 = numbers are presented. Z2 -W2 has been studied for its non-trivial distinct integral solutions. Thus, towards this end we search for Key Words: Quadratic equation with four unknowns, non-zero distinct integral solutions of the equation Integral solutions. under consideration. A few interesting relations among NOTATIONS: the solutions are presented. nn( 1) T Triangular number of rank n. METHOD OF ANALYSIS: 3,n 2 nn(5 3) T Heptagonal number of rank n. The Quadratic Diophantine equation with four 7,n 2 unknowns under consideration is T10,n n(4 n 3) Decagonal number of rank n. T12,n n(5 n 4) Dodecagonal number of rank n. X2 + Y2 = 2Z2 - 62W2 (1) nn(11 9) T Tridecagonal number of rank n. 13,n 2 The substitution of the linear transformations nn(15 13) T Heptadecagonal number of rank n. 17,n 2 X= u + v, Y = v - u and W = u (2) T18,n n(8 n 7) Octadecagonal number of rank n. in (1) leads to Gnon (2 n 1) Gnomonic number of rank n. Z 2= 32 u2+v2 (3) INTRODUCTION: Four different choices of solutions to (3) are presented In [ 1 to 12] some special types of quadratic below. Once the values of u and v are known , using (2), equations with four unknowns have been analyzed for the corresponding values of X and Y are obtained. their non-trivial integral solutions. This communication concerns with another interesting quadratic equation PATTERN 1: with four variables represented by X2 + Y2 = 2Z2 - 62W2. 2 2 2 To start with, we observe that the following non-zero In (3), v32 u Z quadruples ( 2rs-1, 2rs-1, r2+s2, r2 -s2 ), (r2-s2 -1, r2-s2- The solutions of the above equation is of the form 1,2rs, r2+s2), (y+2, y, y+2, ±1), (-1, y, -1, ±1), (-1, y, ±1, - © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1026 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072 22 2 ZAB32 32a ib 32 i 2 u 2 AB (4) ( 32u iv ) 6 v32 A22 B Equating the real and imaginary parts in (7), we have Substituting the values of u and v in (2) , we get 1 u32 a22 4 ab b X X( A , B ) 32 A22 2 AB B 6 22 1 Y Y( A , B ) 2 AB 32 A B v64 a22 64 ab 2 b (5) 6 ZZABAB( , ) 32 22 W W( A , B ) 2 AB Since our interest is on finding integer solutions we have choose Thus (5) gives the distinct integral solution of (1) a and b suitably so that u and v are integers. OBSERVATIONS: 22 2 u32 A 4 AB B 1.W n1,8 Y n ,1 34 2 T17,n 6 T 13, n 19( Gno n ) (8) v64 A22 64 AB 2 B 2. X(,2 A A 1) Y (,2 A A 1)2 T10,AA Gno 1 Thus, using the values of u and v and performing PATTERN 2: a few calculations the values of XY, and Z are obtained (3) can be written as as follows: Z 2 .1 = 32 u2+v2 X X( A , B ) 96 A22 60 AB 3 B Y Y( A , B ) 32 A22 68 AB B Assuming Z = 32p2+q2 and write (9) W W( A , B ) 32 A22 4 AB B 22 ( 32ii 2)( 32 2) ZZABAB( , ) 192 6 1 = (6) 36 Thus (9) represents the non-trivial solution integral Substituting (6) in (3), using the method of solution of (1) factorization we get OBSERVATIONS: 22 1. 32a ib 32 ib 32 i 2 32 i 2 X A,1 Y A ,1 8 T18,AA 24( Gno )22 32u iv 32 u iv 36 2. Z(1, n ) W (1,2 n 1) T12,nn 2 T 7, 11 n 157 Now define, PATTERN 3: 2 (3) can be written as 32a ib 32 i 2 ( 32u iv ) (7) 6 Z 2 - 32 u2 = v2 (10) Assuming Z = a2- 32b2 and © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1027 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072 (6 32)(6 32) OBSERVATIONS: 1 = (11) 4 1.WAYAT2 1,1 2 1,1 95 16 3,A Substituting (11) in (10) , using the method of 2. When B=1, X+Y+Z+192 is a Nasty number. factorization PATTERN 4: 22 a32 b a 32 b 6 32 6 32 Z 32 u Z 32 u (3) can be written as 4 Now define, 32u2 v 2 Z 2 2 2 2 2 32u Z v ab32 6 32 (Zu 32 ) a32u2b Z v Z v 2 (12) 2 ab32 6 32 (32u )( u ) Z v Z v (Zu 32 ) 2 Z v u A 32u Z v B Equating the like terms in (12), we get (Z v ) B 32 uA u v (15) (Z v ) A uB (16) 1 XY, Z Z6 a22 64 ab 192 b Solving the equations (15) and (16) we get the 2 solutions 1 22 u a 12 ab 32 b 22 2 ZBA 32 v 32 A22 B Since our interest is on finding integer solutions we have u2 AB choose and suitably so that u and Z are integers. Z6 A22 64 AB 192 B Thus, using the values of and and performing a few calculations the values of and are obtained as u A22 12 AB 32 B (13) follows: v2 A22 64 B X 32 A22 2 AB B Thus, using the values of and and performing a few Y 32 A22 2 AB B (17) calculations the values of and are obtained as follows: W2 AB ZAB 32 22 X X( A , B ) 3 A22 12 AB 32 B Y Y( A , B ) A22 12 AB 96 B Thus (17) represents the non-trivial solution integral (14) W W( A , B ) A22 12 AB 32 B solution of (1). Z Z( A , B ) 6 A22 64 AB 192 B OBSERVATIONS: Thus (14) represents the non-trivial solution integral 1.YAAXAAWAA(,)(,)(,) is a nasty number. solution of (1) © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1028 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072 2.YAAXAA(,)(,) is a square number. [7] M.A. Gopalan and S. Vidyalakshmi, “On the Diophantine equation kxy +yz+zx=w2” Pure and 3. X(,1) A W (,1)2 A Gno Z (,1) A A Applied Mathematika Sciences, Vol. LXVIII, No. 2 1-2, September 2008. 4.YAWAAZAO(,1) (2 ,) (,1)6 A 2 REFERENCE : [8] M.A. Gopalan, Manju Somanath and N. Vanitha, “On a Quadratic Diophantine equation with four [1] Dickson.L.E., “History of the Theory of variables”, Antartica J.math., 4(1), 41-45, (2007). Numbers”, Chelsea Publishing Company, New York, Vol II, 1952. [9] M.A. Gopalan, Manju Somanath and N. Vanitha, “Homogeneous Quadratic equation with four [2] M.A.Gopalan, Manju Somanath and N. Vanitha, unknowns”, Acta ciencia Indica, Vol.XXXIIIM, “one Quadratic Diophantine equation with four No.3, 915(2007). variables”, Bulletin of pure applied Sciences, Vol.24E, No.2, pp.389-391 (2005). [10] M.A. Gopalan, Manju Somanath and N. Vanitha, “On Space Pythagorean equation x2+y2+z2 = w2”, [3] M.A. Gopalan, Manju Somanath and N. Vanitha, International journal of Mathematics, Computer “Integral Solutions of ax2 + by2 = w2 – z2”, Sciences and Information Technology, Advances in Theoretical and Applied Vol.1,No.1, January-June, pp.129-1332008. Mathematics, Vol.l, No.3.pp.223-226(2006). [11] M.A. Gopalan and S.Vidyalakshmi, “Quadratic [4] M.A. Gopalan and S. Vidyalakshmi, “Integral Diophantine equation with four variables solutions of x2+y2 =w2+Dz2”, Advances in x2+y2+xy+y=u2+v2uv+u-v”. Impact J.Sci. Tech: Theoretical and Applied Mathematics, Vol.l, Vol 2(3), 125-127, 2008. No.2,pp.115-118(2006). [12] M.A. Gopalan, Manju Somanath and N. Vanitha, “ [5] M.A. Gopalan and S. Vidyalakshmi, “Integral On Quadratic Diophantine equation with four solutions of kxy – w(x+y)=z2”,Advances in unknowns x2+y2=z2+2w2-2w+1 Impact J.Sci. Theoretical and Applied Mathematics, Tech: Vol 2(2), 97-103, 2008. Vol.1No.2,pp.167-172(2006) [13] G.Janaki and S. Vidyalakshmi, “Integral solutions [6] Mordell.L.J., Diophantine equation, Academic of xy+x+y+1 = z2 - w2 ”, Antartica J.math., 7(1), Press, New York (1969) 31-37, (2010). © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1029 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 03 | Mar -2017 www.irjet.net p-ISSN: 2395-0072 BIOGRAPHIES Dr.G.Janaki received the Ph.D., M.Sc., and M.Phil., degree in mathematics from Bharathidasan University, Trichy, South India.
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