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), v32 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, -

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 ZAB32  32a ib  32 i 2 u 2 AB (4) ( 32u iv ) 6 v32 A22 B Equating the real and imaginary parts in (7), we have Substituting the values of u and v in (2) , we get 1 u32 a22  4 ab  b  22 6 X X( A , B )  32 A  2 AB  B 22 1 Y Y( A , B )  2 AB  32 A  B v64 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 u32 A  4 AB  B 1.W n1,8  Y n ,1  34  2 T17,n  6 T 13, n  19( Gno n ) (8) v64 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 ( 32ii 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

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.WAYAT2 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 a32 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 ab32  6 32  (Zu 32 ) 32u2  Z  v Z  v 2 (12)    2 ab32  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 (15) (Z v ) A uB (16) 1 Z6 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 u2 AB choose a and b suitably so that u and Z are integers.

Z6 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: v2 A22 64 B X 32 A22  2 AB  B Thus, using the values of u and v and performing a few Y 32 A22  2 AB  B (17) calculations the values of XY, and Z are obtained as follows: W2 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).

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. She completed her Ph.D. degree National College, Bharathidasan University. She has published many papers in international and national level journals. Her research area is “Number Theory”.

R.Radha received M.Sc. degree in mathematics from Bharathidasan University, Trichy and M.Phil degree in mathematics from Alagappa University, Karaikudi, South India. Her ongoing research focusing on the subject of “Number Theory”.