Vol.5.Issue.2.2017 (April-June) ©KY PUBLICATIONS BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH
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RESEARCH ARTICLE
ON NON-HOMOGENEOUS SEXTIC EQUATION WITH FIVE UNKNOWNS x yx 3 y3 26z2 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 yx3 y3 26z2 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 yx y 26z 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: Polygonal number of rank n with size m n 1m 2 Tm,n n1 2 Pyramidal number of rank n with size m 1 Pm nn 1m 2n 5 m n 6 Star number of rank n
Sn 6nn 11 Octahedral number of rank n 1 OH n2n2 1 n 3 Centered Pyramidal number of rank n with size m mn 1nn 1 6n CP m,n 6 Four dimensional Figurate number 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 1n n 3 Jacobsthal-Lucas number 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 yx3 y3 26z2 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 Ta,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 37 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 3a i 3b Equating real and imaginary parts, u ua,b a4 28a3b 18a2b2 84ab 3 9b4 v va,b 7a4 12a3b 126a2b2 36ab 3 63b4 Employing (2), the values of x, y, z, w and T are given by x xa,b u v 8a4 16a3b 144a2b2 48ab 3 72b4 y ya,b u v 6a4 40a3b 108a2b2 120ab 3 54b4 z za,b 2u v 9a4 44a3b 162a2b2 132ab 3 81b4 w wa,b 2u v 5a4 68a3b 90a2b2 204ab3 45b4 T Ta,b a2 3b2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties:
104SO a 3x(a,1) 4y(a,1) 0mod5 n T1,2 9Jn 3jn 4 3Ky n 72T 2 x1,b 48CP 24S 0mod2 4,b 6,b b wa,a ya,a za,ais a nasty number 2T1,b1 is a nasty number 2.2 Pattern: 2 One may write (3) as v2 3u2 52T4 *1 (6) Also, write 1 as 1 i4 31 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 31 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 ua,b 29a 20a b 522a b 60ab 261b 7 (8)
1 4 3 2 2 3 4 v va,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 uA,B 9947A 6860A B179046A B 20580AB 89523B 4 3 2 2 3 4 v vA,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 xA.B 8232A4 12622A3B148176A2B2 378672AB 3 74088B4 y yA,B 11662A4 112504A3B209916A2B2 337512AB 3 104958B4 z zA,B 18179A4 133084A3B327222A2B2 399252AB 3 163611B4 4 3 2 2 3 4 w wA,B 21609A 105644A B388962A B 316932AB 194481B T TA,B 49A2 147B2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties: 2 24F4,A,5 76829T4,A xA,A3yA,A15OH A 3T8,A 0mod3
218148F4,A,4 zA,1 223979CP6,A 135044T9,A 0mod7 5 w 1,B 194481T 2 633864P 12005S 0 mod2 4,B B B n T1,2 1473Jn jn 196 147Ky n 2T1,B 49is 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 35 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 3u T 49T v (10) Factorizing (10) we have 2 2 2 2 3u T u T 7T v7T v (11) This equation is written in the form of ratio as 3u 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 7aT 0 (13) 2 au bv 7b aT 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 up,q 9p4 84p3q 18p2q2 28pq 3 q4 v vp,q 63p4 36p3q 126p2q2 12pq 3 7q4 In view of (2), the integer values of x, y, z, w, T are given by x xp,q u v 54p4 120p3q 108p2q2 40pq 3 6q4 y yp,q u v 72p4 48p3q 144p2q2 16pq 3 8q4 z zp,q 2u v 45p4 204p3q 90p2q2 68pq 3 5q4 w wp,q 2u v 81p4 132p3q 162p2q2 44pq 3 9q4 T Tp,q 3p2 q2 which represent non-zero distinct integer solutions of (1) in two parameters. Properties: w1,q y1,q T 2 28CP 3S 0mod2 4,q 6,q q
n 2, if n is odd T2 ,1 3ky n 6jn 10 , if n is even yp,1 72T 2 72CP 48T 0mod2 4,p 8,p 8,p 2Tp,11is 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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