ON THE CUBIC EQUATION WITH FIVE UNKNOWNS

IJESM Volume 2, Issue 1 ISSN: 2320-0294
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International Journal of Engineering, Science and Mathematics
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227
March
2013
ON THE CUBIC EQUATION WITH FIVE UNKNOWNS
4t12wzyx3 23333
)(
M.A.Gopalan*
S. Vidhyalakshmi*
N.Thiruniraiselvi*
__________________________________________________________
ABSTRACT
The non-homogeneous Cubic Equation with five unknowns represented by
4t12wzyx3 23333
)( is analyzed for its patterns of non–zero integral solutions. Six
different patterns of non-zero distinct integer solutions are obtained. A few interesting properties
between the solutions and special number patterns namely Polygonal numbers, Centered
Polygonal numbers, Pyramidal numbers, Stella Octangular numbers, Star numbers and
Pentatope number are exhibited.
KEY WORDS
Cubic Equation with Five Unknowns, Integral solutions
MSC Subject Classification: 11D25
* Department of Mathematics, Shrimati Indira Gandhi College, Tiruchirappalli – 620 002.

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INTRODUCTION
Integral solutions for the homogeneous (or) non homogeneous Diophantine cubic equation
is an interesting concept as it can be seen from [1, 2, 3]. In [4-9] a few special cases of cubic
Diophantine equation with 4 unknowns are studied. In [10, 11], cubic equations with 5 unknowns
are studied for their integral solutions. In this communication we present the integral solutions of
an interesting cubic equation with 5 unknowns 4t12wzyx3 23333
)( . A few
remarkable relations between the solutions are presented.
NOTATIONS USED
nmt , - Polygonal number of rank n with sizem.
m
nP - Pyramidal number of rank n with sizem.
nmct , - Centered polygonal number of rank n with sizem.
agn - Gnomonic number of rank a
nso - Stella octangular number of rank n
ns - Star number of rank n
npr - Pronic number of rank n
npt - Pentatope number of rank n
METHOD OF ANALYSIS
The Cubic Diophantine equation with five unknowns to be solved for getting non-zero
integral solutions is
4t12wzyx3 23333
)( (1)
On substituting the linear transformations
1cx , 1cy , 1az , 1aw (2)
in (1), it leads to
222
t2ac3 (3)
In what follows, we present six ways of solving (3) and in view of (2), six different
solutions patterns to (1) are obtained.
PATTERN-1
Take 22
q2pc (4)
Write 3 as ))(( 2i12i13 (5)
Substituting (4) & (5) in (3) and using the method of factorization, define
2
q2ip2i1t2ia ))(()( (6)
Equating the real and imaginary parts of (6), we get
pq4q2pqpa 22
),( (7)

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2013
pq2q2pqpt 22
),( (8)
Substitute (4) & (7) in (2) ,the integral solutions of (1) are given by
1q2pqpx 22
),(
1q2pqpy 22
),(
1pq4q2pqpz 22
),(
1pq4q2pqpw 22
),(
pq2q2pqpt 22
),(
PROPERTIES:
q3t8q1zq1x ,),(),(
p12ct1ppw1ppt ,),(),(
ps1ppw1ppt ),(),(
ppr211pt1px ),(),(
2
p
gn81p2y1p2x1p2t2 ),(),(),(
Each of the following represents a nasty number
a) )},(),({ ppzppx3 b) )},(),({ ppwppy3 c) 1ppwppt ),(),(
d) 1ppzppt ),(),( e) )},(),(),({ pp3ypp3xpp3t26
PATTERN-2:
In equation (5), Writing 3 as
9
2i52i5
3
))((
The corresponding values of a and t satisfying (3) are
)102(
3
1
),(
)4105(
3
1
),(
22
22
pqqpqpt
pqqpqpa
(9)

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Since our aim is to find the integral solutions, it is seen that a and t are integers for suitable
choices of p and q.
CHOICE -1:
Let Q3qP3p ,
The corresponding integral solutions of (1) are obtained as below
1Q18P9QPx 22
),(
1Q18P9qPy 22
),(
1PQ12Q30P15QPz 22
),(
1PQ12Q30P15QPw 22
),(
PQ30Q6P3QPt 22
),(
CHOICE -2:
Let 1Q3q1P3p ,
For this choice, the non-zero integral solutions of (1) are found to be
4Q12P6Q18P9QPx 22
),(
2Q12P6Q18P9qPy 22
),(
2PQ12Q24P6Q30P15QPz 22
),(
4PQ12Q24P6Q30P15QPw 22
),(
3PQ30Q6P12Q6P3QPt 22
),(
PROPERTIES:
2
p3
gn3ppxppt3 ),(),(
4
pp32431p21ppw31p21ppt15 )),(()),((

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3
pp32412p1ppz2p1ppt53 })))((,()))((,({
p
222
OH1081p2pw1p2pz1p2pt103 )},(),(),({
2
p
gn81p2y1p2x1p2t2 ),(),(),(
a) 2ppz3ppx5 ),(),( b) 8ppw3ppx5 ),(),(
c) 3ppw3ppt15 ),(),( d) 1ppxppt3 ),(),(
e) ),(),( ppwppz
PATTERN-3:
In equation (5), Writing 3 as
81
211i1211i1
3
))((
Repeating the above process, we’ve
)22211(
9
1
),(
)442(
9
1
),(
22
22
pqqpqpt
pqqpqpa
CHOICE-1:
Let Q3qP3p ,
The corresponding integral solutions of (1) are obtained as below
1Q18P9QPx 22
),(
1Q18P9qPy 22
),(
1PQ44Q2PQPz 22
),(
1PQ44Q2PQPw 22
),(
PQ2Q22P11QPt 22
),(
PROPERTIES:
p
pr4481ppz91ppx ),(),(

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5
pp88101pppz91pppy ))(,())(,(
2SO40211p2pz111p2pt9 p
22
}),(),({
)(mod),(),( 448qpw9qpy
)(mod),(),(),(),( 880qpw9qpz9qpyqpx
a) )},(),({ ppyppx3
b) )},(),({ pp2wpp2z30
c) }),(),(),({ 1ppzppyppx6
PATTERN – 4
Rewriting (3) as 222
t2ac3 (10)
Assume 22
qp3t (11)
Write 2 as ))(( 13132 (12)
Substitute (11) & (12) in (10) and using the method of factorization, define
2
qp313ac3 ))(()(
Equating the rational and irrational parts , we get
pq6qp3qpa 22
),(
pqqpqpc 23),( 22
The corresponding integral solutions of (1) are given by
1pq2qp3qpx 22
),(
1pq2qp3qpy 22
),(
1pq6qp3qpz 22
),(
1pq6qp3qpw 22
),(
22
qp3qpt ),(
PROPERTIES:

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2013
p
pt963p2p1ppx3p2p1ppz )))((),(()))((),((
p3t161ppy1ppx1ppw1ppz ,))(,())(,())(,())(,(
3
pp242p1ppy2p1ppw )),(()),((
pq3gnrNastynumbeqpwqpt ),(),(
ppr611pw1pt ),(),(
a) )},(),({ ppyppw6 b) }1)p,p(w)p,p(t{2
c) )},(),(),({ ppyppxppt21 d) )},(),({ ppzppy6
PATTERN – 5
Write 2 as ))(( 5335332
Repeating the above process the non-zero distinct integral solutions of (1) are
1pq10q3p9qpx 22
),(
1pq10q3p9qpy 22
),(
1pq18q5p15qpz 22
),(
1pq18q5p15qpw 22
),(
22
qp3qpt ),(
PROPERTIES:
p
pr10rNastynumbe1p1pt3p1px ),(),(
pqgn2qpx5qpz3 ),(),(
)(mod),(),( 40qpzqpy
p
22
SO421p2py51p2pw3 ),(),(

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2013
a) }),(),({ 1ppt3ppx6
b) 1ppz3ppt3ppyppx5 ),(),(),(),(
PATTERN – 6
Consider (3) as 222
t2c3a (13)
Substitute the linear transformations,
q3ptq2pc , (14)
in (13), we have 222
q6pa
which is satisfied by
22
22
sr6a
rs2q
sr6p
Hence , the corresponding non-zero integral solutions of (1) are seen to be
1rs4sr6srx 22
),(
1rs4sr6sry 22
),(
1sr6srz 22
),(
1sr6srw 22
),(
rs6sr6srt 22
),(
PROPERTIES:
rS1rr3y1rr3t ),(),(
rsgnsrxsrt ),(),(
5
rP81rrry1rrrx1rrrt2 ))(,())(,())(,(
)(mod),(),( 40srwsrx
a) )},(),(),(),({ rrwrrzrryrrx2

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2013
b) )},(),(),({ rryrrxrrt26
REMARKABLE OBSERVATIONS:
Employing the solutions ),,,,( twzyx of (1), a few observations among the special
polygonal and pyramidal numbers are exhibited below
1. )(mod
,,
304
t
P
t
P3
3
w3
5
w
3
z3
3
z
2. )(mod
,,
32
t
P6
t
P6
3
2w23
4
1w
3
1z6
4
z
3. )(mod
,,,
32
t
P6
t
P
t
P
2w23
4
1w
3
z3
3
1z
1z3
4
1z
4. 8
1ct
P4
2
t
P3
2
p
Pt4
t
P6
6
w4
5
w
3
z3
3
z
3
3
3y
3y
3
1x6
4
x
,,,
is a nasty number
5. 4
1s
p12
12
gn
t
1s
p36
3
t
pp3
3
2
1t
5
2t
3
w
1w23
3
1y
3
2y
3
2x3
3
1x
4
1x ,
,
is a cubical integer
CONCLUSION:
To conclude, one may search for other patterns of solutions and their corresponding
properties.
REFERENCE:
[1]. L.E. Dickson, History of Theory of numbers, vol.2, Chelsea Publishing company,
New York, 1952.
[2]. L.J. Mordell, Diophantine Equations, Academic press, London, 1969.
[3]. Carmichael.R.D, The Theory of numbers and Diophantine Analysis,
New York, Dover, 1959.

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2013
[4] M.A.Gopalan and S.Premalatha , Integral solutions of
322
z1k2wxyyx )())(( , Bulletin of pure and applied
sciences,Vol.28E(No.2),197-202, 2009.
[5] M.A.Gopalan and V.pandiChelvi, Remarkable solutions on the cubic equation with
four unknows 2333
wzyx28zyx )( , Antarctica J.of maths, Vol.4 , No.4,
393-401, 2010.
[6] M.A. Gopalan and B.Sivagami, Integral solutions of homogeneous cubic equation
with four unknows 3333
wyx2xyz3zyx )( ,Impact.J.Sci.Tech, Vol.4,
No.3 ,53-60, 2010.
[7] M.A.Gopalan and S.Premalatha , On the cubic Diophantine equation with four
unknows 322
zn2n2wxyyx )())(( , International Journal of mathematical
sciences,Vol.9,No.1-2, Jan-June,171-175, 2010.
[8] M.A.Gopalan and J.KaligaRani, Integral solutions of
zwwzwzxyyxyx 3333
)()( , Bulletin of pure and applied
sciences,Vol.29E(No.1),169-173, 2010.
[9] M.A.Gopalan and S.Premalatha , Integral solutions of
32
z1k2wxyyx )())(( ,The Global Journal of applied mathematics and
Mathematical sciences, Vol.3, No.1-2,51-55, 2010.
[10] M.A.Gopalan, S.Vidhyalakshmi and T.R.Usha Rani , On the cubic equation with
five unknows )( yxtwzyx 23333
, Indian Journal of science,
Vol.1,No.1,17-20, Nov 2012.
[11] M.A.Gopalan, S.Vidhyalakshmi and T.R.Usha Rani , Integral solutions of the
cubic equation with five unknows 33333
t3vuyx ,IJAMA, Vol.4(2), 147-
151, Dec 2012.

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