Vol.5.Issue.2.2017 (April-June) ©KY PUBLICATIONS BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH

A Peer Reviewed International Research Journal

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RESEARCH ARTICLE

ON NON-HOMOGENEOUS SEXTIC EQUATION WITH FIVE UNKNOWNS x  yx 3  y3  26z2  w 2 T4

1 2 3* K.MEENA , S.VIDHYALAKSHMI , S. AARTHY THANGAM 1Former VC, Bharathidasan University, Trichy-620 024, Tamil Nadu, India. email: [email protected] 2Professor, Department of Mathematics, Shrimati Indira Gandhi College,Trichy, Tamil Nadu, India. email: [email protected] 3*Assistant Professor, Department of Mathematics, Chidambaram Pillai College for Women, Mannachanallur, Tamil Nadu, India. email: [email protected]

ABSTRACT The non-homogeneous sextic equation with five unknowns given by x  yx3  y3  26z2  w2 T4 is considered and analysed for its non-zero distinct integer solutions. Employing the linear transformation x  u  v,y  u  v,z  2u  v,w  2u  v,(u  v  0) and applying the method of factorization, three different patterns of non-zero distinct integer solutions are obtained. A few interesting relations between the solutions and special numbers namely Four dimensional numbers, Polygonal numbers, Octahedral numbers, Pyramidal numbers, Centered Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers, Kynea numbers and Star numbers are presented. Keywords: Non-homogeneous sextic equation, sextic equation with five unknowns, Integer solutions. 2010 Mathematics Subject Classification: 11D241

1. INTRODUCTION The theory of Diophantine equations offers a rich variety of fascinating problems [1-4] particularly, in [5, 6] Sextic equations with three unknowns are studied for their integral solutions. [7-12] analyze Sextic equations with four unknowns for their non-zero integer solutions. [13, 14] analyze Sextic equations with five unknowns for their non-zero integer solutions. This

45 Vol.5.Issue.2.2017 (April-June) Bull.Math.&Stat.Res (ISSN:2348-0580) communication analyzes a Sextic equation with five unknowns given by 3 3 2 2 4 x  yx  y  26z  w T . Infinitely many Quintuples (x, y, z, w, T) satisfying the above equation are obtained. Various interesting properties among the values of x, y, z, w and T are presented. Notations:  of rank n with size m  n 1m  2 Tm,n  n1   2   of rank n with size m 1 Pm  nn 1m  2n  5  m n 6  of rank n

Sn  6nn 11  of rank n 1 OH  n2n2 1 n 3  Centered Pyramidal number of rank n with size m mn 1nn 1 6n CP  m,n 6  Four dimensional of rank n whose generating polygon is a square n4  5n3  8n2  4n F  4,n,4 12  Four dimensional Figurate number of rank n whose generating polygon is a pentagon 3n4 10n3  9n2  2n F  4,n,5 4!  Jacobsthal number of rank n 1 J  2n  1n  n 3  Jacobsthal- of rank n n n jn  2  1  Kynea number of rank n

n 2 Ky n  2 1  2 2. METHOD OF ANALYSIS The non-homogeneous sextic equation with five unknowns to be solved is given by x  yx3  y3  26z2  w2 T4 (1) The substitution of the linear transformations x  u  v, y  u  v, z  2u  v, w  2u  v, u  v  0 (2) in (1) leads to 3u2  v2  52T4 (3) Assume T  Ta,b a2 3b2; a,b  0 (4)

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(3) is solved through different approaches and different patterns of solutions thus obtained for (1) are illustrated below: 2.1 Pattern: 1 Write 52 as 52  7  i 37  i 3 (5) Using (4) and (5) in (3) and employing the method of factorization and equating positive factors we get 4 v  i 3u 7  i 3a  i 3b Equating real and imaginary parts, u  ua,b  a4  28a3b 18a2b2 84ab 3  9b4 v  va,b  7a4 12a3b 126a2b2  36ab 3  63b4 Employing (2), the values of x, y, z, w and T are given by x  xa,b  u  v  8a4 16a3b 144a2b2  48ab 3  72b4 y  ya,b  u  v  6a4  40a3b 108a2b2 120ab 3 54b4 z  za,b  2u  v  9a4  44a3b 162a2b2 132ab 3 81b4 w  wa,b  2u  v  5a4  68a3b  90a2b2  204ab3  45b4 T  Ta,b  a2 3b2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties:

 104SO a 3x(a,1) 4y(a,1)  0mod5 n  T1,2  9Jn  3jn  4  3Ky n 72T 2  x1,b 48CP  24S  0mod2  4,b 6,b b  wa,a ya,a za,ais a nasty number  2T1,b1 is a nasty number 2.2 Pattern: 2 One may write (3) as v2  3u2  52T4 *1 (6) Also, write 1 as 1 i4 31 i4 3 1  (7) 49 Substituting (4), (5) and (7) in (6) and employing the method of factorization and equating positive factors we get 7  i 31 i4 3 4 v  i 3u a  i 3b 7 Equating real and imaginary parts, we have

1 4 3 2 2 3 4 u  ua,b  29a  20a b  522a b  60ab  261b  7 (8)

1 4 3 2 2 3 4 v  va,b   5a  348a b  90a b 1044ab  45b  7 (9)

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The choices a=7A and b=7B in (8), (9) lead to 4 3 2 2 3 4 u  uA,B  9947A 6860A B179046A B  20580AB 89523B 4 3 2 2 3 4 v  vA,B  1715A 119364A B 30870A B  358092AB 15435B In view of (2), the integer values of x, y, z, w and T are given by x  xA.B  8232A4 12622A3B148176A2B2 378672AB 3 74088B4 y  yA,B 11662A4 112504A3B209916A2B2 337512AB 3 104958B4 z  zA,B 18179A4 133084A3B327222A2B2 399252AB 3 163611B4 4 3 2 2 3 4 w  wA,B  21609A 105644A B388962A B 316932AB 194481B T  TA,B  49A2 147B2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties: 2  24F4,A,5  76829T4,A   xA,A3yA,A15OH A 3T8,A  0mod3

 218148F4,A,4  zA,1 223979CP6,A 135044T9,A  0mod7 5  w 1,B 194481T 2  633864P 12005S  0 mod2   4,B B B   n  T1,2 1473Jn  jn 196 147Ky n  2T1,B 49is a nasty number Remark: It is worth to note that 52 in (5) and 1in (7) are also represented in the following ways 1 i 3 1 i 3 1     52  5  i3 35  i3 3 4

 2  i4 3 2  i4 3 1 i15 3 1 i15 3        676 By introducing the above representations in (5) and (7), one may obtain different patterns of solutions to (1). 2.3 Pattern: 3 Write (3) as 2 4 4 2 3u  T  49T  v (10) Factorizing (10) we have 2 2 2 2 3u  T u  T  7T  v7T  v (11) This equation is written in the form of ratio as 3u  T2  7T2  v a   , b  0 7T2  v u  T2 b (12) which is equivalent to the system of double equations 2 3bu  av 3b  7aT  0 (13) 2  au  bv  7b  aT  0 (14) Applying the method of cross multiplication, we get 2 2 u  a  3b 14ab (15) 2 2 v  7a  21b  6ab (16)

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2 2 2 T  a  3b (17) Here T 2 (a,b) is of the form z2  Dx 2  y2 (D>0 and square free). Then the solution for (17) is 2 2 2 2 a  3p  q , b  2pq, T  3p  q (18) Using (18) in (15) and (16), we get u  up,q  9p4 84p3q 18p2q2  28pq 3  q4 v  vp,q  63p4  36p3q 126p2q2 12pq 3  7q4 In view of (2), the integer values of x, y, z, w, T are given by x  xp,q u  v  54p4 120p3q 108p2q2  40pq 3  6q4 y  yp,q  u  v  72p4  48p3q 144p2q2 16pq 3 8q4 z  zp,q  2u  v  45p4  204p3q 90p2q2  68pq 3  5q4 w  wp,q  2u  v  81p4 132p3q 162p2q2  44pq 3 9q4 T  Tp,q  3p2  q2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties: w1,q y1,q T 2  28CP  3S  0mod2  4,q 6,q q

n  2, if n is odd T2 ,1 3ky n  6jn   10 , if n is even   yp,1 72T 2  72CP  48T  0mod2  4,p 8,p 8,p  2Tp,11is a nasty number 3. CONCLUSION First of all, it is worth to mention here that in (2), the values of z and w may also be represented by z  2uv 1,w  2uv 1 and z  uv  2,w  uv  2and thus will obtain other choices of solutions to (1). In conclusion, one may consider other forms of Sextic equation with five unknowns and search for their integer solutions.

4. REFERENCES [1] Dickson, L.E., History of theory of numbers, vol.11, Chelsea publishing company, Newyork (1952). [2] Mordell, L.J., Diophantine equations, Academic press, London (1969). [3] Carmichael, R.D., The theory of numbers and Diophantine analysis, Dover publications, Newyork (1959) [4] Telang, S.G., Number Theory, Tata MC Graw Hill Publishing Company, New Delhi (1996) [5] Gopalan, M.A., Sangeetha, G., On the Sextic Equations with three unknowns x2  xy  y2  (k2  3)n z6 Impact J.Sci.tech, Vol.4, No: 4, 89-93, (2010) [6] Gopalan, M.A., Manju Somanath and Vanitha, N., Parametric Solutions of x2  y6  z2 , ActaCiencia Indica XXXIII, vol.3, 1083-1085, (2007) [7] Gopalan, M.A., Vijayashankar, A., Integral solutions of the Sextic Equation x4  y4  z4  2w6 , Indian journal of Mathematics and Mathematical Sciences, Vol.6, No:2, 241-245, (2010)

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[8] Gopalan, M.A., Vidhyalakshmi, S., Vijayashankar, A., Integral Solutions of Non-Homogeneous Sextic equation xy  z2  w6 , Impact J.Sci..tech, Vol.6, No: 1, 47-52, (2012) [9] Gopalan, M.A., Vidhyalakshmi, S., Lakshmi, L., On the Non-Homogeneous Sextic Equation x4  2(x2  w)x2y2  y4  z4 , IJAMA, 4(2), 171-173, Dec (2012) [10] Gopalan, M.A., Vidhyalakshmi, S., Kavitha, A., Observations on the Homogeneous Sextic Equation with four unknowns x3  y3  2(k2  3)z5w , International Journal of Innovative Research in Science, Engineering and Technology, Vol.2, Issue: 5, 1301-1307, (2013) [11] Gopalan, M.A., Sumathi, G., Vidhyalakshmi, S., Integral Solutions of Non- homogeneous Sextic Equation with four unknowns x4  y4 16z4  32w6 , Antarctica J.Math, 10(6), 623-629, (2013) [12] Gaussian Integer Solutions of Sextic Equations with four unknowns x6  y6  4z(x4  y4  w4 ) , Archimedes, J.Math, 3(3), 263-266, (2013) [13] Gopalan, M.A., Vidyalakshmi, S., Lakshmi, K., Integral Solutions of Sextic Equation with Five unknowns x3  y3  z3  w3  3(x  y)t5 , IJERST, 1(10), 562-564, (2012) [14] Gopalan, M.A., Sumathi, G., Vidyalakshmi, S., Integral Solutions of Sextic Non-Homogeneous Equation with Five unknowns (x3  y3)  z3  w3  6(x  y)t5 , International Journal of Engineering Research-online, Vol.1, Issue.2, 146-150, (2013)

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