International Journal of Computational Engineering Research||Vol, 03||Issue, 4||
Integral Solutions of Non-Homogeneous Biquadratic Equation With Four Unknowns M.A.Gopalan 1 , G.Sumathi 2 , S.Vidhyalakshmi 3 1.
Professor of Mathematics, SIGC,Trichy Lecturer of Mathematics, SIGC,Trichy, 3 . Professor of Mathematics, SIGC,Trichy 2.
Abstract The non-homogeneous biquadratic equation with four unknowns represented by the diophantine equation 3 3 2 n 3
x y (k 3) z w
is analyzed for its patterns of non-zero distinct integral solutions and three different methods of integral solutions are illustrated. Various interesting relations between the solutions and special numbers, namely, polygonal numbers, pyramidal numbers, Jacobsthal numbers, Jacobsthal-Lucas number,Pronic numbers, Stella octangular numbers, Octahedral numbers, Gnomonic numbers, Centered traingular numbers,Generalized Fibonacci and Lucas sequences are exhibited.
Keywords: Integral solutions, Generalized Fibonacci and Lucas sequences, biquadratic nonhomogeneous equation with four unknowns
M.Sc 2000 mathematics subject classification: 11D25 NOTATIONS t m, n : Polygonal number of rank n with size m
Pnm
:
Sn
:
Star number of rank n
Prn
:
Pronic number of rank n
Son
:
Stella octangular number of rank n
jn
:
Jacobsthal lucas number of rank n
Jn
Jacobsthal number of rank n
:
OH n
Pyramidal number of rank n with size m
:
Octahedral number of rank n
Gnomicn : Gnomonic number of rank n
GFn
:
Generalized Fibonacci sequence number of rank n
GLn
:
Generalized Lucas sequence number of rank n
Ct m, n
: Centered Polygonal number of rank n with size m
I.
INTRODUCTION
The biquadratic diophantine (homogeneous or non-homogeneous) equations offer an unlimited field for research due to their variety Dickson. L. E. [1],Mordell.L.J. [2],Carmichael R.D[3]. In particular, one may refer Gopalan.M.A.et.al [4-13] for ternary non-homogeneous biquadratic equations. This communication concerns with an interesting non-homogeneous biquadratic equation with four unknowns represented by www.ijceronline.com
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Integral Solutions Of Non-Homogeneous Biquadratic…
x 3 y 3 (k 2 3) n z 3 w for determining its infinitely many non-zero integral points. Three different methods are illustrated. In method 1, the solutions are obtained through the method of factorization. In method 2, the binomial expansion is introduced to obtain the integral solutions. In method 3, the integral solutions are expressed in terms of Generalized Fibonacci and Lucas sequences along with a few properties in terms of the above integer sequences Also, a few interesting relations among the solutions are presented.
II.
METHOD OF ANALYSIS
The Diophantine equation representing a non-homogeneous biquadratic equation with four unknowns is (1) x 3 y 3 (k 2 3) n z 3 w Introducing the linear transformations (2) x u v, y u v, w 2u in (1), it leads to (3) u 2 3v 2 (k 2 3) n z 3 The above equation (3) is solved through three different methods and thus, one obtains three distinct sets of solutions to (1). 2.1. Method:1
z a 2 3b 2
Let
(4)
Substituting (4) in (3) and using the method of factorization,define n 3
(u i 3v) (k i 3) (a i 3b)
(5)
r n exp(in )(a i 3b) 3 r k 2 3,
where
tan 1
3 k
(6)
Equating real and imaginary parts in (5) we get n
u (k 2 3) 2 cos n (a 3 9ab 2 ) 3 sin (3a 2 b 3b 3 )
1 v (k 2 3) 2 cos n (3a 2 b 3b 3 ) sin (a 3 9ab 2 ) 3 Substituting the values of u and v in (2), the corresponding values of x, y, z and w are represented by n
sin 3 x(a, b, k ) (k 2 3) 2 cos n (a 3 9ab 2 3a 2 b 3b 3 ) (a 9ab 2 9a 2 b 9b 3 ) 3 n sin y(a, b, k ) (k 2 3) 2 cos n (a 3 9ab 2 3a 2 b 3b 3 ) (9a 2 b 9b 3 a 3 9ab 2 ) 3 2 2 z (a, b) a 3b n
w(a, b, k ) 2(k 2 3) 2 cos n (a 3 9ab 2 ) 3 (3a 2 b 3b 3 ) n
Properties: 1. x(a, b, k ) y (a, b, k ) (k
2
n 3) 2
sin ( Soa t4, a t38, a ) cos n (Ct12, a S a 2Gnomica 4 Pra 2t4, a ) 3 www.ijceronline.com
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Integral Solutions Of Non-Homogeneous Biquadratic… n
2. x(b, b, k ) y (b, b, k ) (k 2 3) 2 (cos n (32 P 5 16t 4, b ) 0 b 4.Each of the following is a nasty number n 2 5 2 (a) 6 cos n 32(k 3) Pb cos n x(b, b, k ) y (b, b, k ) n (k 2 3) 2
(b) 6 cos(4 ) 32(k 3)
2
2
Pb5 cos(4 ) x(b, b, k ) y(b, b, k )
3. z (a , a ) is a biquadratic integer. n 2 4.w(a,1, k ) 2(k 3) 2 cos n (2 Pa5 9 Pra 8t 4,a ) 2
2
5. x(a, a, k )
3 sin (t8,a Gnomica 2Ct 6,a 6t 3,a
w(a, a, k ) tan 1 2 3
2.2. Method 2:
(k i 3 ) n in (5) and equating real and imaginary parts, we have u f ( )(a 3 9ab 2 ) 3g ( )(3a 2 b 3b 3 ) v f ( )(3a 2 b 3b 3 ) g ( )(a 3 9ab 2 )
Using the binomial expansion of
Where
r n 2r r f ( ) (1) nC 2 r k (3) (7) r 0 ....................... n 1 2 g ( ) ( 1) r 1 nC 2 r 1 k n 2r 1 (3) r 1 r 1 In view of (2) and (7) the corresponding integer solution to (1) is obtained as 3 2 2 3 n 2
x f ( ) g ( )(a 9ab ) f ( ) 3g ( )(3a b 3b )
y f ( ) g ( )(a 3 9ab 2 ) f ( ) 3g ( )(3a 2b 3b 3 ) z a 2 3b 2
w 2 f ( )(a 3 9ab 2 ) 6 g ( )(3a 2b 3b 3 ) 2.3. Method 3: Taking n 0 in (3), we have, 2 2 3
u 3v z
(8)
Substituting (4) in (8), we get
u 2 3v 2 ( a 2 3b 2 )3 whose solution is given by u0 (a 3 9ab 2 ) v0 (3a 2b 3b 3 ) Again taking n 1 in (3), we have 2 2
(9)
u 3v (k 2 3)(a 2 3b 2 ) 3
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(10)
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Integral Solutions Of Non-Homogeneous Biquadratic… whose solution is represented by
u1 ku0 3v0 v1 u 0 kv0 The general form of integral solutions to (1) is given by As us 2 v s Bs 2i 3
3Bs 2i As 2
u 0 , s 1,2,3..... v0
Where As (k i 3 ) s (k i 3 ) s
Bs (k i 3 ) s (k i 3 ) s Thus, in view of (2), the following triples of integers x s , y s , ws interms of Generalized Lucas and fiboanacci sequence satisfy (1) are as follows: Triples:
1 GLn (2k ,k 2 3)(a 3 9ab 2 3a 2 b 3b 3 ) GFn (2k ,k 2 3)(a 3 9ab 2 9a 2 b 9b 3 ) 2 1 y s GLn (2k ,k 2 3)(a 3 9ab 2 3a 2 b 3b 3 ) GFn (2k ,k 2 3)(a 3 9ab 2 9a 2 b 9b 3 ) 2
xs
ws GLn (2k ,k 2 3)(a 3 9ab 2 ) 6GFn (2k ,k 2 3)(3a 2 b 3b 3 ) The above values of x s , y s , ws satisfy the following recurrence relations respectively x s 2 2kxs 1 (k 2 3) x s 0 y s 2 2kys 1 (k 2 3) y s 0 ws 2 2kws 1 (k 2 3) ws 0 Properties 1. xs (a, a, k ) y s (a, a, k ) 16 As t3, a 8 As t 4, a 2. x s (a, a 1, k )
Bs
As 6 Pa5 9 Pa51 18t 4, a 1 3Ct 6, a 24 Pa3 4t 4, a 2
6Ct 6, a 18Pa5 42 Pa3 14 Pra 7t 4, a 3
2i i 3 x s ( a , a , k ) y s ( a, a , k ) 3. is a biquadratic integer. Bs Bs 24(OH a ) t 20, a 9t 4, a 0 4. x s (a, a, k ) y s (a, a, k ) 2i 3 4 Bs (2 Pa5 t 4,a ) 5. x s (a, a, k ) ws (a, a, k ) i 3 As (18(OHa ) Sa Ct4,a 2CT 6,a6t4,a ) 0 6. x s (2a, a, k ) y s (2a, a, k )
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Bs 2i 3
(10 Pa5 5t 4, a ) 5 As (t 4, a Soa Pra )
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Integral Solutions Of Non-Homogeneous Biquadratic… 7. x s (a,1, k ) y s (a,1, k )
Bs
(4 Pa5 2Ct 36, a 36t 4, a )
2i 3 A s (4 Pa5t 4, a Gnomica Ct 4, a 4t3, a ) 2 8 , if a is odd a a 8. ws (2 ,2 , k ) 24 As J 3a - 8 , if a is even 8 , if a is odd a a a a 9. x s (2 ,2 , k ) y s (2 ,2 , k ) 8 As j3a - 8 , if a is even 10. xs (a, b, k ) y s (a, b, k ) ws (a, b, k ) 3 3 3 11. xs (a, b, k ) y s (a, b, k ) 3x s (a, b, k ) y s (a, b, k ) ws (a, b, k ) ws (a, b, k )
5 Bs 2 12. (9 As ) x s (a,2a, k ) y s (a, a, k ) (2 Pa5 t 4, a ) is a cubic integer
2i 3
13.Each of the following is a nasty number (a) 3 As xs (a, a, k ) y s (a, a, k ) 16 As t3, a
(b) 6z (a, a)
(c) 4 As x s (a, a, k ) ws (a, a, k )
4 Bs
(2 Pa 5 t 4, a )
i 3 As (18(OH a ) S a 3Ct 4, a 2Ct 6, a ) (d) 6x s (a, a, k ) y s (a, a, k )ws (a, a, k
Bs
2i 3
(e) 30 As x s (a, a, k ) ws (a, a, k )
14. xs (a, b, k ) y s (a, b, k ) GLn (2k ,k
(10 Pa5 5t 4, a ) 5 As ( Soa Pra )
2
3)u0 6GFn (2k ,k 2 3)v0
15.xs 1 (a, b, k ) y s 1 (a, b, k ) 2k GLn (2k ,k 2 3) (k 2 3) GLn 1 (2k ,k 2 3) u0
(12k )GF (2k,k n
2
3) 6(k 2 3)GFn 1 (2k ,k 2 3) vo
16.xs 2 (a, b, k ) y s 2 (a, b, k ) (3k 2 1) GLn (2k ,k 2 3) (2k 3 6k ) GLn 1 (2k ,k 2 3) u0
(18k
2
1)GFn (2k ,k 2 3) (12k 3 36k )GFn 1 (2k ,k 2 3) vo III.
CONCLUSION:
To conclude, one may search for other pattern of solutions and their corresponding properties.
REFERENCES: [1] [2] [3]
Dickson. L. E. “History of the Theory of Numbers’’, Vol 2. Diophantine analysis, New York, Dover,2005. Mordell .L J., “ Diophantine Equations Academic Press’’, New York,1969. Carmichael. R.D. “ The Theory of numbers and Diophantine Analysis’’ ,New York, Dover, 1959.
[4]
Gopalan.M.A., Manjusomnath and Vanith N., “Parametric integral solutions of Indica,Vol.XXXIIIM(No.4)Pg.1261-1265(2007). Gopalan.M.A.,and Pandichelvi .V., “On Ternary biquadratic Diophantine equation
[5] [6] [7]
x 2 y 3 z 4 ’’,
Acta Ciencia
x 2 ky3 z 4 ’’, Pacific- Asian Journal of Mathematics, Vol-2.No.1-2, 57-62, 2008. Gopalan.M.A., and Janaki.G., “Observation Indica,Vol.XXXVM(No.2)Pg.445-448,(2009).
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on
2( x 2 y 2 ) 4 xy z 4 ’’,
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Acta
Ciencia
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Integral Solutions Of Non-Homogeneous Biquadratic…
x 2 xy y 2 (k 2 3) n z 4 ’’,
[8]
Gopalan.M.A., Manjusomnath and Vanith N., “Integral solutions of Applied Mathematika Sciences,Vol.LXIX,NO.(1-2):149-152,(2009).
[9]
Gopalan.M.A., and Sangeetha.G., “Integral solutions of ternary biquadratic equation ’’, Antartica J.Math.,7(1)Pg.95-101,(2010). 2 2 4 Gopalan.M.A.,and Janaki.G., “Observations on 3( x y ) 9 xy z ’’, Antartica J.Math.,7(2)Pg.239-245,(2010).
[10] [11] [12]
( x 2 y 2 ) 2 xy z 4
Gopalan.M.A., and Vijayasankar.A., “Integral Solutions of Ternary biquadratic Impact.J.Sci.Tech.Vol.4(No.3)Pg.47-51,2010. Gopalan.M.A,Vidhyalakshmi.S., and Devibala.S., “Ternary
2
[13] [14] [15]
Pure and
4n 3
3
3
(x y ) z
equation
x 2 3 y 2 z 4 ’’,
biquadratic
diophantine
4
equation ’’,Impact. J.Sci.Tech.Vol(No.3) Pg.57-60,2010. 12.Gopalan.M.A,Vidhyalakshmi.S., and SumathiG., “ Integral Solutions of Ternary biquadratic Non-homogeneous Equation
( 1)( x 2 y 2 ) (2 1) xy z 4 ’’,JARCEvol(6),No.2, Pg.97-98,July-Dec 2012.
[16] [17] [18]
13. Gopalan.M.A, SumathiG.,and Vidhyalakshmi.S., “ Integral Solutions of Ternary biquadratic Non-homogeneous Equation
[19]
(k 1)( x 2 y 2 ) (2k 1) xy z 4 ’’,Archimedes J.Math,3(1),67-71,2013.
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