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Observations On X2 + Y2 = 2Z2 - 62W2
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 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1026 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 of the form X2 + Y2 = 2Z2 - 62W2 has been studied for its non-trivial distinct integral solutions. A few interesting relations among the solutions and special polygonal numbers are presented. Key Words: Quadratic equation with four unknowns, Integral solutions. NOTATIONS: 3, 7, 10, 12, 13, ( 1) Triangular number of rank n. 2 (5 3) Heptagonal number of rank n. 2 (4 3) Decagonal number of rank n. (5 4) Dodecagonal number of rank n. (11 9) Tridecagonal numb 2 n n n n n n n T n n T T n n T n n n n T 17, 18, er of rank n. (15 13) Heptadecagonal number of rank n. 2 (8 7) Octadecagonal number of rank n. (2 1) Gnomonic number of rank n. n n n n n T T n n Gno n INTRODUCTION: In [ 1 to 12] some special types of quadratic equations with four unknowns have been analyzed for theirnon-trivialintegralsolutions.Thiscommunication concerns with another interesting quadratic equation with four variables represented by X2 + Y2 = 2Z2 -62W2. To start with, we observe that the following non-zero quadruples ( 2rs-1, 2rs-1, r2+s2, r2 -s2 ), (r2-s2 -1, r2-s2- 1,2rs, r2+s2), (y+2, y, y+2, ±1), (-1, y, -1, ±1), (-1, y, ±1, - 1), (-y, y, ±1, -y) satisfy the equation under consideration. In the above quadruples,anytwovalues of the unknowns are the same. In [13], the Quadratic equation with four unknowns of the form XY+X+Y+1= Z2 -W2 has been studied for its non-trivial distinct integral solutions.Thus,towardsthisendwesearchfor non-zero distinct integral solutions of the equation under consideration.Afewinterestingrelationsamong the solutions are presented. METHOD OF ANALYSIS: The Quadratic Diophantine equation with four unknowns under consideration is X2 + Y2 = 2Z2 - 62W2 (1) The substitution of the linear transformations X= u + v, Y = v - u and W = u (2) in (1) leads to Z 2= 32 u2+v2 (3) Four differentchoices of solutions to (3) are presented below. Once the values of u and v are known ,using(2), the corresponding values of X and Y are obtained. PATTERN 1: In (3), 2 2 2 32v u Z The solutions of the above equation is of the form
2.
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 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1027 2 2 2 2 32 2 32 Z A B u AB v A B (4) Substituting the values of u and v in (2) , we get 2 2 2 2 2 2 ( , ) 32 2 ( , ) 2 32 ( , ) 32 ( , ) 2 X X A B A AB B Y Y A B AB A B Z Z A B A B W W A B AB (5) Thus (5) gives the distinct integral solution of (1) OBSERVATIONS: 1. 2 17, 13,1,8 ,1 34 2 6 19( )n n nW n Y n T T Gno 2. 10,( ,2 1) ( ,2 1) 2 1A AX A A Y A A T Gno PATTERN 2: (3) can be written as Z 2 .1 = 32 u2+v2 Assuming Z = 32p2+q2 and write 1 = ( 32 2)( 32 2) 36 i i (6) Substituting (6) in (3), using the method of factorization we get 2 2 32 32 32 2 32 2 32 32 36 a ib ib i i u iv u iv Now define, 2 32 32 2 ( 32 ) 6 a ib i u iv (7) 2 32 32 2 ( 32 ) 6 a ib i u iv Equating the real and imaginary parts in (7), we have 2 2 2 2 1 32 4 6 1 64 64 2 6 u a ab b v a ab b Since our interest is on finding integer solutions we have choose a and b suitably so that u and v are integers. 2 2 2 2 32 4 64 64 2 u A AB B v A AB B (8) Thus, using the values of u and v and performing a few calculations the values of ,X Y and Z are obtained as follows: 2 2 2 2 2 2 2 2 ( , ) 96 60 3 ( , ) 32 68 ( , ) 32 4 ( , ) 192 6 X X A B A AB B Y Y A B A AB B W W A B A AB B Z Z A B A B (9) Thus (9) represents the non-trivial solution integral solution of (1) OBSERVATIONS: 1. 18,,1 ,1 8 24( ) 22A AX A Y A T Gno 2. 12, 7,(1, ) (1,2 1) 2 11 157n nZ n W n T T n PATTERN 3: (3) can be written as Z 2 - 32 u2 = v2 (10) Assuming Z = a2- 32b2 and
3.
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 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1028 1 = (6 32)(6 32) 4 (11) Substituting (11) in (10) , using the method of factorization 2 2 32 32 6 32 6 32 32 32 4 a b a b Z u Z u Now define, 2 2 32 6 32 ( 32 ) 2 32 6 32 ( 32 ) 2 a b Z u a b Z u (12) Equating the like terms in (12), we get 2 2 2 2 1 6 64 192 2 1 12 32 2 Z a ab b u a ab b Since our interest is on finding integer solutions we have choose a and b suitably so that u and Z are integers. 2 2 2 2 2 2 6 64 192 12 32 2 64 Z A AB B u A AB B v A B (13) Thus, using the values of u and v and performing a few calculations the values of ,X Y and Z are obtained as follows: 2 2 2 2 2 2 2 2 ( , ) 3 12 32 ( , ) 12 96 ( , ) 12 32 ( , ) 6 64 192 X X A B A AB B Y Y A B A AB B W W A B A AB B Z Z A B A AB B (14) Thus (14) represents the non-trivial solution integral solution of (1) OBSERVATIONS: 1. 3,2 1,1 2 1,1 95 16 AW A Y A T 2. When B=1, X+Y+Z+192 is a Nasty number. PATTERN 4: (3) can be written as 2 2 2 2 2 2 2 32 32 32 (32 )( ) 32 ( ) 32 (15) ( ) (16) u v Z u Z v u Z v Z v u u Z v Z v Z v u A u Z v B Z v B uA Z v A uB Solving the equations (15) and (16) we get the solutions 2 2 2 2 32 32 2 Z B A v A B u AB Thus, using the values of u and v and performing a few calculations the values of ,X Y and Z are obtained as follows: 2 2 2 2 2 2 32 2 32 2 (17) 2 32 X A AB B Y A AB B W AB Z A B Thus (17) represents the non-trivial solution integral solution of (1). OBSERVATIONS: 1. ( , ) ( , ) ( , )Y A A X A A W A A is a nasty number.
4.
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 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1029 2. ( , ) ( , )Y A A X A A is a square number. 3. ( ,1) ( ,1) 2 ( ,1)AX A W A Gno Z A 4. 2 ( ,1) (2 , ) ( ,1) 6 2AY A W A A Z A O REFERENCE : [1] Dickson.L.E., “History of the Theory of Numbers”, Chelsea Publishing Company, New York, Vol II, 1952. [2] M.A.Gopalan, Manju Somanath and N. Vanitha, “one Quadratic Diophantine equation with four variables”, Bulletin of pure applied Sciences, Vol.24E, No.2, pp.389-391 (2005). [3] M.A. Gopalan, Manju Somanath and N. Vanitha, “Integral Solutions of ax2 + by2 = w2 – z2”, Advances in Theoretical and Applied Mathematics, Vol.l, No.3.pp.223-226(2006). [4] M.A. Gopalan and S. Vidyalakshmi, “Integral solutions of x2+y2 =w2+Dz2”, Advances in Theoretical and Applied Mathematics, Vol.l, No.2,pp.115-118(2006). [5] M.A. Gopalan and S. Vidyalakshmi, “Integral solutions of kxy – w(x+y)=z2”,Advances in Theoretical and Applied Mathematics, Vol.1No.2,pp.167-172(2006) [6] Mordell.L.J., Diophantine equation, Academic Press, New York (1969) [7] M.A. Gopalan and S. Vidyalakshmi, “On the Diophantine equation kxy +yz+zx=w2”Pureand Applied Mathematika Sciences, Vol. LXVIII, No. 1-2, September 2008. [8] M.A. Gopalan, Manju Somanath and N. Vanitha, “On a Quadratic Diophantine equationwithfour variables”,AntarticaJ.math.,4(1),41-45,(2007). [9] M.A. Gopalan, Manju Somanath and N. Vanitha, “Homogeneous Quadratic equation with four unknowns”, Acta ciencia Indica, Vol.XXXIIIM, No.3, 915(2007). [10] M.A. Gopalan, Manju Somanath and N. Vanitha, “On Space Pythagorean equationx2+y2+z2 =w2”, International journalofMathematics,Computer Sciences and Information Technology, Vol.1,No.1, January-June, pp.129-1332008. [11] M.A. Gopalan and S.Vidyalakshmi, “Quadratic Diophantine equation with four variables x2+y2+xy+y=u2+v2uv+u-v”. Impact J.Sci. Tech: Vol 2(3), 125-127, 2008. [12] M.A. Gopalan, Manju Somanath andN.Vanitha,“ On Quadratic Diophantine equation with four unknowns x2+y2=z2+2w2-2w+1 Impact J.Sci. Tech: Vol 2(2), 97-103, 2008. [13] G.Janaki andS.Vidyalakshmi,“Integralsolutions of xy+x+y+1 = z2 - w2 ”, Antartica J.math., 7(1), 31-37, (2010).
5.
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 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1030 BIOGRAPHIES Dr.G.JanakireceivedthePh.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”.
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