International Journal of Innovative Science and Modern Engineering (IJISME) ISSN: 2319-6386, Volume-3 Issue-8, July 2015

On the Homogeneous Biquadratic Equation with 5 2 2 2 2 2 2 2 Unknowns xy4 kxy  1   4 kxy  2  2 4 kpqz  1      

M. A. Gopalan, K. Geetha, Manju Somanath

Abstract: The homogeneous biquadratic equation with five In this context one may refer [4-12] for various problems on unknowns given by the biquadratic Diophantine equations. However, often we come across non-homogeneous biquadratic equations and as such one may require its integral solution in its most general is considered and analyzed for finding its non zero distinct form. This paper concern with the homogeneous biquadratic integral solutions. Introducing the linear transformations equation with five unknowns x u v, y u v, p2, u v q2 u v and employing the method of factorization different patterns of non zero distinct integer solutions of the equation under the above for determining its infinitely many non-zero integral equation are obtained. A few interesting relations between the solutions. Also a few interesting properties among the integral solutions and the special numbers namely Polygonal numbers, Pyramidal numbers, Centered Polygonal numbers, solutions are presented. Centered Pyramidal numbers, Thabit-ibn-Kurrah number, Carol number, Mersenne number are exhibited. II. METHOD OF ANALYSIS Keywords: Homogeneous equation, Integral solutions, Polygonal The biquadratic equation with five unknowns to be solved numbers, Pyramidal numbers and Special numbers. 2010 Mathematics Subject Classification Code: 11D25 for its non-zero distinct integral solution is Notations: tmn, = Polygonal number of rank n with sides m n (1) pm = Pyramidal number of rank n with sides m Consider the transformations

ctmn, = Centered Polygonal number of rank n with sides m x u v n y u v cpm = Centered Pyramidal number of rank n with sides m (2) p2 u v gn = Gnomonic number q2 u v Tha = Thabit-ibn-Kurrah number n On substituting (2) in (1), we get

car ln = Carol number cu l = Cullen number 2 2 2 n ku v3 k  1  4 k  1 z (3) mern = Mersenne number Pattern 1: wo = Woodhall number 22 n Assume z ka 31 k  b   (4) p = Pronic number n Write (4k-1) as

I. INTRODUCTION 4k 1  k  i 3 k  1 k  i 3 k  1 The theory of Diophantine equations offers a rich variety of    (5) fascinating problems. In particular biquadratic Diophantine Substituting (4) and (5) in (3) and employing the method of equations, homogeneous and non-homogeneous have factorization, aroused the interest of numerous mathematicians since antiquity [1-3].

Revised Version Manuscript Received on July 07, 2015. M. A. Gopalan, Professor, Department of Mathematics, Shrimathi Indira Gandhi College, Trichy, Tamil Nadu, India. K. Geetha, Asst. Prof., Department of Mathematics, Cauvery College for Women, Trichy, Tamil Nadu, India. Manju Somanath, Asst. Prof., Department of Mathematics, National College, Trichy. Tamil Nadu, India.

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222  kuikvkuikv3 1  3 1  kik 3 1 kik 3 1 ka  3 k 1 b  Equating the real and imaginary parts, we get 22 22 u ka 3 k  1 b  2 3 k  1 ab v ka 2 ab  3 k  1 b       Substituting u and v in (2) the values of x, y, p and q are given by 22 xxabk(,,)2  ka  2 b 31231 k   k  abab  2 y y( a , b , k )   6 kab 22 ppabk(,,)3  ka  3 b 3112 k   kab  6 ab   (6) 22 qqabk( , , )  3 ka  3 b 3 k  1  12 kab Thus (4) and 6 represent the non-zero distinct integer solutions of 1 Properties: n 1) x1,1,2 1  10 mern  Nasty number 2) p a1,1, k  9  t6,aa  2 t 3, 3) q(1,2 b ,1)  t18,b 3 mod31 2 4 22 4) znn 1,,1  ynn 21,,23   cpn  2 cp n  t32, n  1mod31  qn2 1,2,1  pnn 2  1,2  1,1  ynn 2,2  1,1  2  tt   36 g 5) 58,n 42, n n 6) Pattern 2: 2 2 2 2 v a 32 k b  kb  kab The assumption u X 31 k   T and v X kT (7) Substituting u and v in (2) the values of x, y, p and q are 2 2 2 given by in (3) gives X k31 k  T  z 2 2 2 2 (8) xxabk( , , )  2 a  6 kb  2 kb  4 abk  2 ab Case 1: y y( a , b , k )  8 abk  2 ab kk31 square (9) ppabk( , , )  3 a2  9 kb 2 2  3 kb 2  10 abk  4 ab 22 Assume z a  k31 k   b (10) (11) 2 2 2 2 qqabk( , , )  a  3 kb  kb  14 abk  4 ab From (8) and (10), we have, on factorization 2 Thus (10) and 11 represent the non-zero distinct integer X i k3 k  1 T  a  i k 3 k  1 b       solutions of

2 X i k3 k  1 T  a  i k 3 k  1 b      

On equating real and imaginary parts, in either of the above two equations we obtain 22 X a  k31 k   b

T 2 ab On substituting X and T in (4) we get the values of u and v to be 2 2 2 2

u a 3 k b  kb  6 abk  2 ab

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Properties: 1) y a, a ,1  2 t8,aa  g  1 2) q1,1, k  p 1,1, k  2 gk  1 xb1, ,1 3) 1 t6,b 2 4) p a,1,1  q a ,1,1  y a ,1,1  4  t18,aa  t 14, 5) ynn 1,  2,2  xnn  1,  2,2  ct28,n  ct 8, n  31 g n  87 Case 2: 2 Choosing k31 k w in (5), it is satisfied by 2 22 22 X 2 w RS , T w R S  and z w R S  Substituting X and T in (4) we get the values of u and v to be 2 2 2 2 2 u2 w RS  3 kwR  3 kwS  wR  wS 2 2 2 v2 w RS  kwR  kwS Substituting u and v in (2) we get the values of x, y, p and q are given by 2 2 2 2 2 x x R, S , w , k  4 w RS  2 kwR  2 kwS  wR  ws 2 2 2 2 y y R, S , w , k  4 kwR  4 kwS  wR  wS 2 2 2 2 2 p p R, S , w , k  6 w RS  5 kwR  5 kwS  2 wR  2 wS   (12) 2 2 2 2 q p R, S , w , k  2 w RS  7 kwR  6 wS  2 wR Thus (10) and (12) represent the non-zero distinct integer solutions of 1

Properties: nn 1) z2 ,2 ,1,1  2 mer2n 1 nn 2) p1,1,2,1  x 1,1,2,1  2 jal2n  1 n 3) q2 ,1,1,1  Tha22n  ky n  mer n  3 4) x1,1, wky ,  1,1, wk ,  2 q 1,1, wk ,  1  t10,w  g w  7 wg k 2 5) p2,1,1, n 1   q 2,1,1,2 n  2 t17,n  1 mod9 Pattern 3: Rewrite (5) as 2 2 2 z X  k31 k   T (13) z X z  X  k31 k  T2      (14) Case 1: (14) can be written as the system of double equations as z X 31 k  T   (11)

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z X kT (12) Which is satisfied by z41 k t X21 k t Tt 2   ,   , On substituting X and T in (7) we get the values of u and v which are given by u83 k t , v21 k t On substituting u and v in (2) we get the values of x, y, z, p and q. The non-zero distinct integrals values of x, y, z, p and q satisfying (1) are given by x x k, t  10 k  2 t y y k, t  6 k  4 t z z( k , t )  4 k  1 t p p k, t  18 k  5 t q q k, t  14 k  7 t

Properties: 26 1) x( n , n ) y ( n , n )  n  10 cpnn  t14, 2 18 2) pnn( , ) qnn ( , )  5  6 cpn  2 ct11, n  3 g n

3) x n,1 y n ,1  2 t62,nn  3 g   11 2 2 23 4) pnn ,  2 nnzn  1,  ,26  cpnn  2 ct25,  1mod12  2 20 16 5) q n,2 n 13  y n  1, n  6 cpnn  3 cp  7mod37  Case 2:

Also, (10) can be written as the system of double equations as z X 3 k2  k T   (13) z X T (14) Applying the same procedure as in case 1 we get the non-zero distinct integrals values of x, y, z, p and q satisfying (1) are given by x x( k , t )  6 k2  3 t  

y y( k , t )  4 k2  1 t  

z z( k , t )  3 k2  k  1 t  

p p( k , t )  9 k2  2 k  5 t  

q q( k , t )  3 k2  6 k  3 t   Properties:  x n,1 y n ,1  t2  t 12,n  3 mod 4 1) 50,n

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n 2) z(2 ,1) Tha2nn  mer  1

3) p n1,2  t38,n  17 mod23 42 7p k , t  2 q k , t  2 z k , t  27 4) = Nasty Number 22,ynn n z 2,4 n  wo  cul  Tha  mer   1mod2  5)     2n 2 n 2 n n Case 3: 3. Gopalan.M.A., and Pandichelvi.V., On the solutions of the In addition to case 2, rewrite (14) as the pair of 2 222 2 4 Biquadratic equation x y  z 1  w , equations as     International conference on Mathematical methods and 2 Computation,Pg:24-25, july 2009. z X T (16) 4. Gopalan.M.A., and Sangeetha. G., Integral solutions of Ternary 2 2 4 2 Quartic equation x y 2 xy  z , Antartica J.Math., 7(1), z X 3 k  k 95-101, 2010. (17) 5. Gopalan.M.A., and Sangeetha. G., Integral solutions of non- homogeneous Quartic equation Repeating the procedure as in case 1, we get the 4 4 2 2 2 non-zero distinct integrals values of x, y, z, p and q x y 2  2  1 z  w  , impact satisfying (1) are given by J.Sci.Tech., 4(3), (July-Sep ), 15-21, 2010. 6. Gopalan.M.A., and Sangeetha. G., Integral solutions of Non- 4 2 2 2 homogeneous biquadratic x x  y  y  z  z , Acta x x k, t  4 t22  12 k  2 k  8 kt  2 t   Ciencia Indica, Vol.XXXVII M.No.4, 799-803, 2011. 7. Gopalan .M.A., and Sivkami.B., Integral solutions of quartic equation y y k, t  16 kt  2 t 3 3 3 3 with four unknowns x y  z 3 xyz  2( x  y ) w , z z( k , t )  2 t22  6 k  k Antartica J.Math, 10(2), 151-159, 2013. 8. Manju Somanath, Sangeetha. G., and Gopalan.M.A., Integral solutions of biquadratic equation with four unknowns given by 2 2 2 22 xy k 15 z  w , Pacific-Asian Journal of p p k, t  6 t  18 k  6 k  20 kt  4 t   Mathematics, 6(2),185-190, July- Dec 2012. q q k, t   6 k2  2 k  4 t  4 kt 9. Manju Somanath, Sangeetha. G., and Gopalan.M.A., Integral   solutions of non-homogeneous Quartic equation

Properties: 4 4 2 2 2 x y  k 1 z  w  , Archimedes J.Math., 1(1), 1) x k,1  y k ,1  ct  3 g  2 24,kk 51-57, 2011. n 10. Sangeetha. G., Manju Somanath., Gopalan.M.A., and Pushparani., 2) q2 ,1 3 car ln2  Tha n  8  0 Integral solutions of the homogeneous biquadratic equation with   3 3 2 2 2 five unknowns x y z  w  p R , International 3) p1, t t   12 mod21   14,t   conference on Mathematical methods and Computation,Pg:221-226, Feb 2014. 4) y n1, n  1  2 q n ,1  ct6,nn  p  1(mod8)11. Gopalan.M.A., Geetha.K., and Manju Somanath., On the non- homogeneous Biquadratic equation with 4unknowns

xy3 3 2 z 3  3 xyz  6( k 2  sxyw 2 )(  ) 3 6 , 5) z2,1 n y n ,18 q n ,1  384 cp  2 g  1 International journal of Physics and Mathematical Sciences, Vol.4(4),       nn  Oct- Dec 2014, Pp.1-5. III. CONCULSION 12. Gopalan.M.A., Geetha.K., and Manju Somanath., On the It is worth to note that in (2), the transformations for z and w homogeneous Biquadratic equation with 5 unknowns 4 4 2 2 2 may be considered as p21 uv and q21 uv . For x y 10( z  w ) R Jamal Academic Research this case, the values of x, y and z are the same as above Journal: An interdisciplinary”, International Conference on Mathematical Methods and computation, Pp.268-273, Jan 2015 . where as the values of p and q changes for every pattern. To conclude one may consider biquadratic equation with multivariables(5)and search for their non-zero distinct integer solutions along with their corresponding properties.

REFERENCE 1. Carmichael R.D., The Theory of numbers and Diophantine analysis, Dover publications, new York, 1959. 2. Dickson L.E., History of theory of Numbers, Chelsa Publishing company, New Yors, 1952.

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