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Research Article Volume 7 Issue No. 2 On the Quintic Non-Homogeneous Diophantine Equation x4 y 4 40(z 2 w2 ) p3 P.Saranya1, G.Janaki2 Assistant Professor1, 2 Cauvery College for Women, Tiruchirapalli, Tamilnadu, India
Abstract: The non-homogeneous quintic diophantine equation having five unknowns given by x 4 y 4 40(z 2 w2 )p3 is analyzed for its non-zero distinct integer solutions. A few interesting relations among the solutions and the special numbers are also presented.
Keywords: integral solutions, non-homogeneous, quintic diophantine equation.
2010 MATHEMATICAL S UBJECT
CLASSIFICATION: 11D41
I. INTRODUCTION III. METHOD OF ANALYSIS
While individual equations present a kind of puzzle and have The considered quintic diophantine equation with five been considered throughout history, the formulation of general unknowns is theories of diophantine equations was an achievement of the x4 y4 40(z 2 w2 ) p3 (1) twentieth century. Detailed and clear study on diophantine Equations can be seen from [1-3]. For various problems and Introducing the linear transformations, ideas on the quintic diophantine equations one may refer [4-9]. x u v This paper concerns with the problem of determining non- y u v (2) trivial integral solutions of the non-homogeneous quintic z 2u v diophantine equation with five unknowns given by w 2u v x4 y4 40(z 2 w2 )p3 . in (1) and simplifying we get 2 2 3 u v 40 p (3) Also, a few interesting relations among the solutions and some Solving the above equation in different methods, we obtain special numbers like Jacobsthal, Mersenne, Kynea, Riesel, different patterns of infinite number of integer solutions of Thabit-ibn-kurrah, Woodall, Carol, Sierpinski, Polygonal and equation (1) Pyramidal numbers are presented. PATTERN-1: II. NOTATIONS Let p a2 b2 (4)
Write 40 (6 2i)(6 2i) (5) Tm,n Triangular Number of rank n. Using (4) and (5) in (3) and using the method of factorization, n P Pyramidal Number of rank n. Define m P(n) Pronic Number of rank n. ( u iv)(u -iv) (6 2i)(6 - 2i)[(a ib)(a -ib)]3
Starn Star Number of rank n. Equating real and imaginary parts we get CHn Centered Hexagonal Number of rank n. u 6a3 2b3 18ab2 6a2b CCn Centered Cube Number of rank n. v 2a3 6b3 6ab2 18a2b RD Rhombic Dodecagonal Number of rank n. n Gnon Gnomonic Number of rank n. Employing the values of u and v in (2) we get the non-zero
J n Jacobsthal Number of rank n. distinct integer solutions of (1) to be x 8a3 4b3 24ab2 12a2b H n Hex Number of rank n. 3 3 2 2 Wn Woodall Number of rank n. y 4a 8b 12ab 24a b 3 3 2 2 HRDn Hauy Rhombic Dodecagonal Number of rank n. z 14a 2b 42ab 6a b SO Stella octangula Number of rank n. 3 3 2 2 n w 10a 10b 30ab 30a b 2 2 TTn Truncated tetrahedral Number of rank n. p a b
International Journal of Engineering Science and Computing, February 2017 4685 http://ijesc.org/ OBSERVATIONS: 3 3 2 2 x 1100A 200B 3300AB 600A B y(a,a) z(a,a) 3 3 2 2 1. 1 represents Kynea number y 200A 1100B 600AB 3300A B 2a6 z 1550A3 850B3 4650AB 2 2550A2B 6 x(a,a) w(a,a) 97a w 250A3 1750B3 750AB 2 5250A2B 2. 6 represents Carol number a 2 2 p 25A 25B
3.6(y(a,a) z(a,a)) represents Nasty number OBSERVATIONS: x(A, A) 1.[ 3] Kynea number y(a,a) 3 4. [ 1] represents Thabit - ibn - kurrah number 100A a3
z(A, A) x(a,a) w(a,a) 2. [ 1] Carol no. 100A3 5. 6 63 represents Woodall number a 5 3.50[2PA 43T10,A 66GnoA ] [y(A,1) w(A,1)] 67 Wn 6.p(2n,1) 2 3J 2n
13 x(A,1) PATTERN 2: 4.6PA 2T11,A 16GnoA 1 Mersenne no. Write (3) as 100 u 2 v 2 40 p3.1 (6) x(A,1) 5.6P13 2T 16Gno 3 Thabit ibn kurrah no. and take A 11,A A 100 (6 +8i) (6 -8i) 1 = (7) 10 2 PATTERN 4: (11 + 60i) (11- 60i) Using (4), (5) and (7) in (6) and proceeding as in pattern1 we Write 1 = (9) 2 get the non-zero distinct integer solutions of (1) to be 61 3 3 2 2 x 8a 4b 12a b 24ab Using (4), (5) and (9) in (6) and proceeding as in pattern 3 we y 4a3 8b3 24a2b 12ab2 get 1 3 3 2 2 z 10a3 10b3 30a2b 30ab 2 u [54a 382b 162ab 1146a b] 61 3 3 2 2 w 2a 14b 42a b 6ab 1 v [382a3 54b3 1146ab2 162a2b] p a 2 b2 61
OBSERVATIONS: Since our interest is to find integer solutions, taking a = 61A;b = 61B , in the above equations and 1..x(a,1) y(a,1) 10Ha [RDa 2Pr(a) 6Gnoa ] 0 (mod27) proceeding as in earlier patterns we get the distinct non-zero integer solutions of (1) to be 2.HRDa Stara [26Gnoa 3SOa z(a,1)] 0(mod13)
3 3 2 2 16 x 1220488A 1622356B 4867068A B 3661464AB 3.6Pb Hb 17Gnob w(1,b) 3 Wn y 1622356A3 1220488B3 3661464A2B 4867068AB 2
4.w(a,1) 2p(a,1) CC CH 6TT 2T 18Gno 1 Kynea no. 3 3 2 2 a a a 19,a a z 1019554A 3043778B 9131334A B 3058662AB w 1823290A3 2641910B3 7925730A2B 5469870AB 2 5.x(a,1) w(a,1) 45Pr(a) [10GnoaRDa CCa 7] Wn p 3721A2 3721B2
The solutions presented in pattern1 and pattern 2 are non- OBSERVATIONS: trivial only when a b x(A, A) 1. [ 1] Mersenne no. 710711A3 PATTERN 3: (4 + 3i) (4 - 3i) Write 1 = (8) 2 y(A, A) 5 2. 1 Thabit ibn kurrah no. Using (4), (5) and (8) in (6) and proceeding as in pattern2 we 33489A3 get 1 3.The following expressions represent the Nasty number u [18a3 26b3 54ab2 78a2b] 5 i)y(1,1) 1 3 3 2 2 ii) [z(1,1) 11250] v [26a 18b 78ab 54a b] 5 iii) [x(1,1) w(1,1) 10032] Since our interest is to find integer solutions, z(1,1) 4.[ 1] Sierpinski no. taking a = 5A;b = 5B , in the above equations and proceeding 2 as in earlier patterns we get the distinct non-zero integer 61 solutions of (1) to be
International Journal of Engineering Science and Computing, February 2017 4686 http://ijesc.org/ IV. CONCLUSION
In this paper, we have obtained infinitely many non-zero distinct integer solutions to the non-homogeneous quintic diophantine equation having five unknowns given by x4 y4 40(z 2 w2 )p3 One can also search for other patterns of solutions for the above equation.
V. REFERENCES
[1]. L.E.Dickson, History of the Theory of Numbers, Chelsea Publishing Company, New York, Vol II, 1952.
[2]. Niven, Ivan, Zuckermann, S.Herbert and Montgomery, L.Hugh, An Introduction to the Theory of Numbers , John Wiley and Sons, Inc, New York, 2004.
[3]. L.J. Mordell, Diophantine equations, Academic Press, New York, 1969.
[4]. Gopalan MA, Janaki.G ,Integral solutions of 2 2 2 2 2 2 3 Impact J.Sci Tech,vol4 No.97-102,2010 (x y )(3x 3y 2xy) (z w )p
[5]. M.A.Gopalan, S.Vidhyalakshmi, D.Maheswari, Observations on the Quintic Equation with five unknowns 4 4 2 2 3 International Journal of Applied Research x y 37(z w )p , vol-1(3): 78-81,2015
[6]. Gopalan MA, Vidhyalakshmi S, Kavitha A. on the Quintic equation with five unknowns, 2(x y)(x3 y3) 19(z2 w2)p3 International Journal of Engineering Research 2013; 1(2):279-282.
[7]. Vidhyalakshmi S, Mallika S, Gopalan MA. Observations on the non-homogeneous quintic equation with five unknowns x4 y4 2(k 2 s2 )(z 2 w2 )p3 International Journal of Innovative Research in Science, Engineering and Technology 2013; 2(4):1216-1221.
[8]. Gopalan MA, Vidhyalakshmi S , Premalatha E and Manjula S ,The Non- Homogeneous Quintic Equation With Five Unknowns x4 y4 2(x2 y2 )(x y)2 14(z 2 w2 )p3 International Journal of Physics and Mathematical Sciences ISSN: 2277- 2111 (Online) Vol.5(1)January- March, pp. 64-69,2015
[9]. Gopalan MA., Vidhyalakshmi S , Thiruniraiselvi N And Malathi R,The Non-homogeneous Quintic Equation With Five Unknowns x4 y4 65(z 2 w2 ) p3 International Journal Of Physics And Mathematical Sciences Issn: 2277-2111 (Online) Vol. 5 (1) January- March, Pg.No. 70-77,2015
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