On Non-Homogeneous Sextic Equation with Five Unknowns

Asian Journal of Applied Science and Technology (AJAST) Page | 45 Volume 1, Issue 6, Pages 45-47, July 2017

3 3 2 2 4 2x  yx  y  39z  w T

1 2 3 S.Vidhyalakshmi , M.A.Gopalan and S.Aarthy Thangam

1Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India. 2Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India. 3Research Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, Tamil Nadu, India.

Article Received: 27 June 2017 Article Accepted: 10 July 2017 Article Published: 16 July 2017

ABSTRACT

The non-homogeneous sextic equation with five unknowns given by 2x  yx3  y3  39z2  w2 T4 is considered and analysed for its non-zero distinct integer solutions. Employing the linear transformations 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, Centered Pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas numbers, Kynea numbers and Star numbers are presented. Keywords: Integer solutions, Non-homogeneous sextic equation and Sextic equation with five unknowns.

1. INTRODUCTION  Four dimensional Figurate number of rank n whose The theory of Diophantine equations offers a rich variety of generating polygon is a pentagon fascinating problems [1-4] particularly, in [5, 6] Sextic 3n4 10n3  9n2  2n F  equations with three unknowns are studied for their integral 4,n,5 4! solutions. [7-12] analyze Sextic equations with four unknowns for their non-zero integer solutions. [13-16]  Jacobsthal number of rank n analyze Sextic equations with five unknowns for their 1 n n Jn  2  1  3 non-zero integer solutions. This communication analyzes a Sextic equation with five unknowns given by  Jacobsthal-Lucas number of rank n n n 2 x  y x3  y3  39 z2  w2 T4 . Infinitely many Quintuples (x, jn  2  1      y, z, w, T) satisfying the above equation is obtained. Various  Kynea number of rank n 2 interesting properties among the values of x, y, z, w and T are Ky  2n 1  2 n   presented.

3. METHOD OF ANALYSIS 2. NOTATIONS The non-homogeneous sextic equation with five unknowns to  Polygonal number of rank n with size m be solved is given by  n 1m  2 Tm,n  n1  3 3 2 2 4  2  2x  yx  y  39z  w T (1) The substitution of the linear transformations  Star number of rank n x  u  v, y  u  v, z  2u  v, w  2u  v, u  v  0 (2) Sn  6nn 11 in (1) leads to 3u2  v2  39T4 (3)  Octahedral number of rank n (3) is solved through different approaches and different 1 2 patterns of solutions thus obtained for (1) are illustrated OHn  n2n 1 3 below:  Centered Pyramidal number of rank n with size m 3.1 Pattern: 1 mn 1nn 1 6n CP  Assume T  Ta,b a2  3b2; a,b  0 (4) m,n 6 Write 39 as

Asian Journal of Applied Science and Technology (AJAST) Page | 46 Volume 1, Issue 6, Pages 45-47, July 2017

39  6  i 36 i 3 (5) Properties 2 Using (4) and (5) in (3) and employing the method of  w1,B x1,B 72T4,B  4320CP6,B  24SB  0mod2 factorization and equating positive factors, we get n  T2 ,112Jn  4jn 16  4Kyn 4 v  i 3u 6  i 3a  i 3b  3xA,A yA,A 2zA,Ais a nasty number Equating real and imaginary parts, we have u  ua,b a4  24a3b 18a2b2 72ab3  9b4 Remark It is worth to note that 39 in (5) and 1in (7) are also 4 3 2 2 3 4 v  va,b 6a 12a b 108a b 36ab 54b represented in the following ways Employing (2), the values of x, y, z, w and T are given by 3  i7 3 3  i7 3 1 i4 3 1 i4 3 39     1     x  xa,b u  v  7a4 12a3b 126a2b2 36ab3  63b4 4 49 9  i5 3 9  i5 3 11 i4 3 11 i4 3 4 3 2 2 3 4         y  ya,b u  v  5a 36a b 90a b 108ab  45b 4 169 z  za,b  2u  v  8a4  36a3b 144a2b2 108ab3  72b4 By introducing the above representations in (5) and (7), one may obtain different patterns of solutions to (1). w  wa,b 2u  v  4a4  60a3b  72a2b2 180ab3 36b4

2 2 T  Ta,b a  3b 3.3 Pattern: 3 Which represent non-zero distinct integer solutions of (1) in Write (3) as two parameters. 2 4 4 2 3u  T  36T  v (10) Factorizing (10) we have Properties 2 2 2 2 3u  T u  T  6T  v6T  v (11)  234OH  z(a,1)  2w(a,1)  0mod2 a This equation is written in the form of ratio as  504F4,b,5  x1,b 246CP6,b 105T8,b  1mod2 3 u  T2 6T2  v a       , b  0  2T1,b1is a nasty number 6T2  v u  T2 b (12) Which is equivalent to the system of double equations 2 3.2 Pattern: 2 3bu  av 6a  3bT  0 (13) One may write (3) as 2  au bv   a  6bT  0 (14) v2 3u2  39T4 *1 (6) Applying the method of cross multiplication, we get Also, write 1 as 2 2 u  a 3b 12ab (15) 1 i 31 i 3 1 (7) v  6a2 18b2  6ab 4 (16) 2 2 2 Substituting (4), (5) and (7) in (6) and employing the method T  a  3b (17) of factorization and equating positive factors we get Now, the solution for (17) is 2 2 2 2 6  i 31 i 3 4 a  3p q , b  2pq, T  3p  q (18) v  i 3u a  i 3b 2 Using (18) in (15) and (16), we get Equating real and imaginary parts, we have u  up,q 9p4  72p3q 18p2q2  24pq3 q4 1 4 3 2 2 3 4 u  ua,b 7a 12a b 126a b 36ab  63b  v  vp,q 54p4 36p3q 108p2q2 12pq3  6q4 2 (8) 1 In view of (2), the integer values of x, y, z, w, T are given by v  va,b 3a4 84a3b 54a2b2  252ab3  27b4  2 (9) x  xp,q u  v  45p4 108p3q 90p2q2 36pq3 5q4 The choices a=2A and b=2B in (8), (9) lead to y  yp,q u  v  63p4 36p3q 126p2q2 12pq3 7q4 u  uA,B 56A4 96A3B1008A2B2  288AB3 504B4 z  zp,q 2u  v  36p4 180p3q 72p2q2 60pq3  4q4 v  vA,B 24A4 672A3B 432A2B2  2016AB3  216B4 w  wp,q 2u  v  72p4 108p3q 144p2q2 36pq3 8q4

In view of (2), the integer values of x, y, z, w and T are given 2 2 by T  Tp,q 3p  q x  xA,B 80A4 576A3B1440A2B2 1728AB3  720B4 Which represent non-zero distinct integer solutions of (1) in two parameters. y  yA,B 32A4  768A3B576A2B2  2304AB3  288B4

z  zA,B136A4  480A3B 2448A2B2 1440AB3 1224B4 Properties w  wA,B 88A4 864A3B1584A2B2  2592AB3  792B4  2z1,q w1,q156CP6,q  0mod3 T  TA,B  4A2 12B2 n 2, if n is odd Which represent non-zero distinct integer solutions of (1) in  T2 ,1 3ky n  6jn   10 , if n is even two parameters. © 2017 AJAST All rights reserved. www.ajast.net

Asian Journal of Applied Science and Technology (AJAST) Page | 47 Volume 1, Issue 6, Pages 45-47, July 2017

 468CP6,p  2xp,1 2yp,1 zp,1  0mod2 [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, 4. CONCLUSION First of all, it is worth to mention here that in (2), the values of 10(6), 623-629, 2013. z and w may also be represented by z  2uv 1,w  2uv 1 and z  uv  2,w  uv  2 and thus, will obtain other choices of [12] Gopalan, M.A., Sumathi, G., Vidhyalakshmi, S., solutions to (1). In conclusion, one may consider other forms “Gaussian Integer Solutions of Sextic Equations with four of Sextic equation with five unknowns and search for their unknowns x6  y6  4z(x4  y4  w4) ”, Archimedes, J.Math, integer solutions. 3(3), 263-266, 2013.

REFERENCES [13] Gopalan, M.A., Vidyalakshmi, S., Lakshmi, K., [1] Dickson, L.E., History of theory of numbers, vol.2, “Integral Solutions of Sextic Equation with Five unknowns Chelsea publishing company, Newyork (1952). x3  y3  z3  w3  3(x  y)t5 ” , IJERST, 1(10), 562-564, 2012.

[2] Carmichael, R.D., The theory of numbers and Diophantine analysis, Dover publications, Newyork (1959). [14] Gopalan, M.A., Sumathi, G., Vidyalakshmi, S., “Integral Solutions of Sextic Non-Homogeneous Equation [3] Mordell, L.J., Diophantine equations, Academic press, with Five unknowns (x3  y3)  z3  w3  6(x  y)t5 ”, London (1969). International Journal of Engineering Research, Vol.1, Issue.2, 146-150, 2013. [4] Telang, S.G., Number Theory, Tata MC Graw Hill Publishing Company, New Delhi (1996). [15] Gopalan, M.A., Aarthy Thangam, S., Kavitha, A., “On Non-homogeneous Sextic equation with five unknowns [5] Gopalan, M.A., Manju Somanath and Vanitha, N., 2x  yx3  y3 28z2  w2 T4 ”, Jamal Academic Research “Parametric Solutions of x2  y6  z2 ”, Acta Ciencia Indica Journal (JARJ) , special Issue, 291-295, ICOMAC- 2015. XXXIII, vol.3, 1083-1085, 2007. [16] Meena, K., Vidhyalakshmi, S., Aarthy Thangam, S., [6] Gopalan, M.A., Sangeetha, G., “On the Sextic Equations “On Non-homogeneous Sextic equation with five with three unknowns x2  xy  y2  (k2  3)n z6 ”, Impact unknowns x  yx3  y3 26z2  w2 T4 ”, Bulletin of J.Sci.tech, Vol.4, No: 4, 89-93, 2010. Mathematics and Statistics Research, Vol.5, Issue.2, 45-50, 2017. [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.

[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.