The document summarizes three methods for finding integral solutions to the non-homogeneous biquadratic equation wzkyx n 3233 )3(  with four unknowns. Method 1 uses factorization to obtain solutions in terms of polygonal numbers, pyramidal numbers, and other special numbers. Method 2 uses binomial expansion to express solutions in terms of binomial coefficients. Method 3 expresses solutions recursively in terms of Generalized Fibonacci and Lucas sequences. Several properties relating the solutions to special numbers are also presented.

1. International Journal of Computational Engineering Research||Vol, 03||Issue, 4||
www.ijceronline.com ||April||2013|| Page 51
Integral Solutions of Non-Homogeneous Biquadratic Equation
With Four Unknowns
M.A.Gopalan1
, G.Sumathi2
, S.Vidhyalakshmi3
1.
Professor of Mathematics, SIGC,Trichy
2.
Lecturer of Mathematics, SIGC,Trichy,
3
. Professor of Mathematics, SIGC,Trichy
M.Sc 2000 mathematics subject classification: 11D25
NOTATIONS
nmt , : Polygonal number of rank n with size m
m
nP : Pyramidal number of rank n with size m
nS : Star number of rank n
nPr : Pronic number of rank n
nSo : Stella octangular number of rank n
nj : Jacobsthal lucas number of rank n
nJ : Jacobsthal number of rank n
nOH : Octahedral number of rank n
nGnomic : Gnomonic number of rank n
nGF : Generalized Fibonacci sequence number of rank n
nGL : Generalized Lucas sequence number of rank n
nmCt , : 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
Abstract
The non-homogeneous biquadratic equation with four unknowns represented by the
diophantine equation
wzkyx n 3233
)3( 
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 non-
homogeneous equation with four unknowns

2. Integral Solutions Of Non-Homogeneous Biquadratic…
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wzkyx n 3233
)3( 
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
wzkyx n 3233
)3(  (1)
Introducing the linear transformations
uwvuyvux 2,,  (2)
in (1), it leads to
3222
)3(3 zkvu n
 (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
Let
22
3baz  (4)
Substituting (4) in (3) and using the method of factorization,define
3
)3()3()3( biaikviu n
 (5)
3
)3)(exp( biainrn
 
where ,32
 kr
k
3
tan 1
 (6)
Equating real and imaginary parts in (5) we get
 )33(sin3)9(cos)3( 322322
bbaabanku
n
 






 )9(sin
3
1
)33(cos)3( 233222
ababbankv
n

Substituting the values of u and v in (2), the corresponding values of , ,x y z and w are represented by






 )999(
3
sin
)339(cos)3(),,( 2332322322
ababbabbaabankkbay
n


22
3),( babaz 
),,( kbaw  )33(3)9(cos)3(2 322322
bbaabank
n
 
Properties:
1. 22
)3(),,(),,(
n
kkbaykbax 






 )(
3
sin
)2Pr42(cos ,38,4,4,12 aaaaaaaa ttSotGnomicSCtn








 )999(
3
sin
)339(cos)3(),,( 3223322322
bbaababbaabankkbax
n



3. Integral Solutions Of Non-Homogeneous Biquadratic…
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2. 0)1632((cos)3(),,(),,( ,4
52 2
 bb tPnkkbbykbbx
n

4.Each of the following is a nasty number
(a)












22
522
)3(
),,(),,(cos)3(32
cos6 n
b
n
k
kbbykbbxnPk
n


(b)  ),,(),,()4cos()3(32)4cos(6 522
kbbykbbxPk b   
3.  ),( 22
aaz is a biquadratic integer.
 aaaaaaa
n
tCtGnomicttPnkkaw ,3,6,8,4
522
62(sin3)8Pr92(cos)3(2),1,(.4  
5. 






3
tan
1
2
),,(
),,(
kaaw
kaax
2.2. Method 2:
Using the binomial expansion of
n
ik )3(  in (5) and equating real and imaginary parts, we have
)33)((3)9)(( 3223
bbagabafu  
)9)(()33)(( 2332
abagbbafv  
Where
.......................
)3()1()(
)3()1()(
2
1
1
1121
2
2
0
12
2



















 











n
r
rrn
C
r
rrn
C
n
r
r
kng
knf
r
r


(7)
In view of (2) and (7) the corresponding integer solution to (1) is obtained as
    )33()(3)()9()()( 3223
bbagfabagfx  
    )33()(3)()9()()( 3223
bbagfabagfy  
22
3baz 
)33)((6)9)((2 3223
bbagabafw  
2.3. Method 3:
Taking 0n  in (3), we have,
322
3 zvu  (8)
Substituting (4) in (8), we get
32222
)3(3 bavu  (9)
whose solution is given by
)9( 23
0 abau 
)33( 32
0 bbav 
Again taking 1n  in (3), we have
322222
)3)(3(3 bakvu  (10)

4. Integral Solutions Of Non-Homogeneous Biquadratic…
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whose solution is represented by
001
001 3
kvuv
vkuu


The general form of integral solutions to (1) is given by
.....3,2,1,
232
2
3
2
0
0

























s
v
u
A
i
B
i
BA
v
u
ss
ss
s
s
Where ss
s ikikA )3()3( 
ss
s ikikB )3()3( 
Thus, in view of (2), the following triples of integers sss wyx ,, interms of Generalized Lucas and
fiboanacci sequence satisfy (1) are as follows:
Triples:
)33)(3,2(6)9)(3,2(
)999)(3,2()339)(3,2(
2
1
)999)(3,2()339)(3,2(
2
1
322232
3223232232
3223232232
bbakkGFabakkGLw
bbaabakkGFbbaabakkGLy
bbaabakkGFbbaabakkGLx
nns
nns
nns



The above values of sss wyx ,, satisfy the following recurrence relations respectively
0)3(2
0)3(2
0)3(2
2
12
2
12
2
12






sss
sss
sss
wkkww
ykkyy
xkkxx
Properties
1. asasss tAtAkaaykaax ,4,3 816),,(),,( 
2.  aaaaaa
s
s tPCttPP
A
kaax ,4
3
,61,4
5
1
5
42431896
2
),1,(  
 aaaaa
s
tPPCt
i
B
,4
35
,6 7Pr1442186
32

3.
 
s
ss
B
kaaykaaxi ),,(),,(3 
is a biquadratic integer.
4.   09)(24
32
),,(),,( ,4,20  aaa
s
ss ttOH
i
B
kaaykaax
5.  )2(
3
4
),,(),,( ,4
5
aa
s
ss tP
i
B
kaawkaax
0)62)(18( ,4,6,4  aaaaas tCTCtSOHA
6. )Pr(5)510(
32
),,2(),,2( ,4,4
5
aaasaa
s
ss SotAtP
i
B
kaaykaax 

5. Integral Solutions Of Non-Homogeneous Biquadratic…
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7. )3624(
32
),1,(),1,( ,4,36
5
aaa
s
ss tCtP
i
B
kaykax 
)44(
2
,3,4,4
5
aaaaa
s
tCtGnomictP
A

8.  as
aa
s JAkw 324),2,2(



evenisaif,8-
oddisaif,8
9.  as
aa
s
aa
s jAkykx 38),2,2(),2,2(



evenisaif,8-
oddisaif,8
10. ),,(),,(),,( kbawkbaykbax sss 
11. ),,(),,(),,(),,(3),,(),,( 333
kbawkbawkbaykbaxkbaykbax ssssss 
12. 





 )2(
32
5
),,(),2,()9( ,4
52
aa
s
sss tP
i
B
kaaykaaxA is a cubic integer
13.Each of the following is a nasty number
(a)  assss tAkaaykaaxA ,316),,(),,(3 
(b)  ),(6 aaz
(c)   )2(
3
4
),,(),,(4 ,4
5
aa
s
sss tP
i
B
kaawkaaxA
)23)(18( ,6,4 aaaas CtCtSOHA 
(d)    kaawkaaykaax sss ,,(),,(),,(6 
(e) 



 )Pr(5)510(
32
),,(),,(30 ,4
5
aasaa
s
sss SoAtP
i
B
kaawkaaxA
14. 0
2
0
2
)3,2(6)3,2(),,(),,( vkkGFukkGLkbaykbax nnss 
       0
2
1
22
11 )3,2()3()3,2(2),,(),,(.15 ukkGLkkkGLkkbaykbax nnss
 onn vkkGFkkkGFk )3,2()3(6)3,2()12( 2
1
22
 
       0
2
1
322
22 )3,2()62()3,2()13(),,(),,(.16 ukkGLkkkkGLkkbaykbax nnss
 onn vkkGFkkkkGFk )3,2()3612()3,2()118( 2
1
322
 
III. CONCLUSION:
To conclude, one may search for other pattern of solutions and their corresponding properties.
REFERENCES:
[1] Dickson. L. E. “History of the Theory of Numbers’’, Vol 2. Diophantine analysis, New York, Dover,2005.
[2] Mordell .L J., “ Diophantine Equations Academic Press’’, New York,1969.
[3] 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
432
zyx  ’’, Acta Ciencia
Indica,Vol.XXXIIIM(No.4)Pg.1261-1265(2007).
[5] Gopalan.M.A.,and Pandichelvi .V., “On Ternary biquadratic Diophantine equation
[6]
432
zkyx  ’’, Pacific- Asian Journal of Mathematics, Vol-2.No.1-2, 57-62, 2008.
[7] Gopalan.M.A., and Janaki.G., “Observation on
422
4)(2 zxyyx  ’’, Acta Ciencia
Indica,Vol.XXXVM(No.2)Pg.445-448,(2009).

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[8] Gopalan.M.A., Manjusomnath and Vanith N., “Integral solutions of
4222
)3( zkyxyx n
 ’’, Pure and
Applied Mathematika Sciences,Vol.LXIX,NO.(1-2):149-152,(2009).
[9] Gopalan.M.A., and Sangeetha.G., “Integral solutions of ternary biquadratic equation
422
2)( zxyyx  ’’,
Antartica J.Math.,7(1)Pg.95-101,(2010).
[10] Gopalan.M.A.,and Janaki.G., “Observations on
422
9)(3 zxyyx  ’’, Antartica J.Math.,7(2)Pg.239-245,(2010).
[11] Gopalan.M.A., and Vijayasankar.A., “Integral Solutions of Ternary biquadratic equation
422
3 zyx  ’’,
Impact.J.Sci.Tech.Vol.4(No.3)Pg.47-51,2010.
[12] Gopalan.M.A,Vidhyalakshmi.S., and Devibala.S., “Ternary biquadratic diophantine
equation
43334
)(2 zyxn

’’,Impact. J.Sci.Tech.Vol(No.3)
[13] Pg.57-60,2010.
[14] 12.Gopalan.M.A,Vidhyalakshmi.S., and SumathiG., “ Integral Solutions of
[15] Ternary biquadratic Non-homogeneous Equation
[16]
422
)12())(1( zxyyx   ’’,JARCEvol(6),No.2, Pg.97-98,July-Dec 2012.
[17] 13. Gopalan.M.A, SumathiG.,and Vidhyalakshmi.S., “ Integral Solutions of
[18] Ternary biquadratic Non-homogeneous Equation
[19]
422
)12())(1( zxykyxk  ’’,Archimedes J.Math,3(1),67-71,2013.