This document presents six infinite families of positive integral solutions to the biquadratic equation 1)( 2  xyzwwzyx with four unknowns. Linear transformations are used to put the equation in the form of a general Pell's equation. Six triples (a,b,c) that satisfy the necessary conditions are identified and used to generate the families of solutions. Each family is expressed parametrically in terms of two integers n and n+1. In total, six different parametric families of solutions to the original biquadratic equation are presented.

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 7, Issue 4 (Jul. - Aug. 2013), PP 11-13
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Integral Solutions to the Biquadratic Equation with Four
Unknowns 1)( 2
 xyzwwzyx
M. A. Gopalan1
, S. Vidhyalakshmi 2
, A. Kavitha3
1,2,3
(P.G & Research Department of Mathematics, Shrimati Indira Gandhi college, Trichy, Tamilnadu, India)
Abstract: The main thrust of this paper is to study the biquadratic equation with four
unknowns 1)( 2
 xyzwwzyx . We present six different infinite families of positive integral
solutions to this equation.
I. Introduction
Integral solutions to equations in three or more variables are given in various parametric forms [1,2,3].
In [4], the theory of general pell’s equation is used to generate four infinite families of positive integral solutions
to the equation .)( 2
xyzzyx  In[5] other choices of integral solutions to the above equation are
presented . In[6], the Diophantine equation xyzttzyx  2
)( is considered and nine different infinite
families of positive integral solutions are given.
In this paper, the biquadratic equation with four unknowns given by 1)( 2
 xyzwwzyx
is studied for its non-zero distinct integral solution. In particular, six different infinite families of positive
integral solutions are obtained.
II. Method of Analysis
The biquadratic equation with four unknowns to be solved is 1)( 2
 xyzwwzyx (1)
Introduction of the linear transformations
,avux  ,avuy  ,bz  .cw  (2)
where a,b,c,are positive integers
in (1) leads to the form
1)2()4( 2222
 bcacbabcvubc (3)
In which 4bc and abccba  2
)2(2 (4)
There are six triples (a,b,c) satisfying the above conditions (4) namely (5,2,3), (7,1,6), (12,1,5), (1,10,3),
(3,14,1), (4,1,9).
The following table contains the general pell’s equation (3) corresponding to the above triples (a,b,c), their
pell’s resolvent, both equations with their fundamental solutions.
),,( cba General pell’s equation(3) and its fundamental solution Pell’s resolvent and its fundamental solution
)3,2,5(
,373 22
 vu (7,2) 13 22
 sr , (2,1)
(7,1,6)
,733 22
 vu (10,3) 13 22
 sr , (2,1)
(12,1,5)
,1795 22
 vu (28,11) 15 22
 sr , (9,4)
(1,10,3)
,971513 22
 vu (7,6) 1195 22
 sr , (14,1)
(3,14,1)
,15775 22
 vu (10,7) 135 22
 sr , (6,1)
(4,1,9)
,17995 22
 vu (50,37) 145 22
 sr , (161,24)

2. Integral Solutions To The Biquadratic Equation With Four Unknowns
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In view of (2), the following are the six families of positive integer solutions to (1)
3,2,5
6
3
2
5
5
6
313
2
9
)(
1
1




wz
gf
y
g
f
xi
n
n
where ])32()32[( 11 
 nn
f
])32()32[( 11 
 nn
g
6,1,7
322
7
7
32
19
2
13
)(
1
1




wz
gf
y
g
f
xii
n
n
5,1,12
10
527
2
17
12
10
583
2
49
)(
11
1
1
1
1




wz
gf
y
g
f
xiii
n
n
where ])549()549[( 11
1

 nn
f
])549()549[( 11
1

 nn
g
3w,10z,1
1952
g
2
f5
y
1g
1952
181
2
f13
x)iv(
22
1n
2
2
1n






where ])19514()19514[( 11
2

 nn
f
])19514()19514[( 11
2

 nn
g
1,14,3
3522
3
3
352
99
2
17
)(
33
1
3
3
1




wz
gf
y
g
f
xv
n
n
where ])356()356[( 11
3

 nn
f
])356()356[( 11
3

 nn
g
9,1,4
452
83
2
13
4
452
583
2
87
)(
44
1
4
4
1






wz
gf
y
g
f
xvi
n
n
where ])4524161()4524161[( 11
4

 nn
f
])4524161()4524161[( 11
4

 nn
g

3. Integral Solutions To The Biquadratic Equation With Four Unknowns
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III Conclusion
To conclude, one may search for other patterns of solutions.
References
[1] L.E.Dickson, History of Theory of Numbers, Chelsea Publishing company, New York, Vol.11, (1952).
[2] L.J.Mordell, Diophantine equations, Academic Press, London(1969).
[3] Andreescu, T.Andrica, D., An Intrduction to Diophantine Equations, GIL Publishing house, 2002.
[4] Andreescu ,T., A note on the equation xyzzyx  2
)( , General Mathematics Vol.10, No.3-4 ,17-22,(2002).
[5] M.A.Gopalan, S.Vidhyalakshmi, A.Kavitha, “observations on xyzzyx  2
)( , accepted in IJMSA.