Abstract Urban growth has resulted in increasing demand for energy primarily in form of electricity. With depleting energy resources, growing concern for promotion of alternative energy sources urban housing is going through a transition phase. There is a greater need for promotion of on-site microgeneration systems at community level to ensure a near zero sustainable domestic sector. The paper tries to define three major concerns that need to be addressed in ensuring near zero housing communities – the first being understanding the character of existing renewable energy strategies and promoting them in the context of changing housing typologies, second, identifying the need for development of hybrid energy generation systems in conjunction with renewable energy systems to balance energy demand at times of climatic inconsistencies, the third concern being the integration of on-site microgeneration infrastructure within the context of the built environment without altering the existing character and aesthetics. Keywords: Micro generation, Near Zero Housing Communities, Hybrid Energy Systems, Housing Typologies, Integrated Development

1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 449
INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
322
3519)(5 zyxxyyx 
S. Vidhyalakshmi1
, T.R. UshaRani2
, M.A.Gopalan3
1
Department of Mathematics, Shrimathi Indira Gandhi College Tiruchirappalli-2, Tamil Nadu
2
Department of Mathematics, Shrimathi Indira Gandhi College Tiruchirappalli-2, Tamil Nadu
3
Department of Mathematics, Shrimathi Indira Gandhi College Tiruchirappalli-2, Tamil Nadu
Abstract
The ternary cubic equation
322
3519)(5 zyxxyyx  is considered for determining its non-zero distinct integral
solutions Employing the linear transformations x=u+v,y=u-v (u≠v≠0),and employing the meyhod of factorization in complex
conjugates, different patterns of integral solutions to the ternary cubic equation under consideration are obtained.. In each
pattern, interesting relations among the solutions, some special polygonal , pyramidal numbers and central pyramidal numbers
are exhibited.
Keywords: Ternary cubic, Integral solutions, polygonal number, pyramidal number, Mathematics subject
classificationnumber: 11D09
-------------------------------------------------------------------***------------------------------------------------------------------
1. INTRODUCTION
Diophantine equations are numerously rich because of its
variety. The determination of integral solutions for cubic(
homogeneous or non-homogeneous) diophantine equations
with three variables has been an interest to mathematicians
since antiquity as can be seen from [1-3].In this context one
may refer [4-24]. In this Communication, the non-
homogeneous ternary cubic diophantine equation
represented by
322
3519)(5 zyxxyyx  is
considered for its non-zero distinct integral solutions. A few
interesting relations between special polygonal numbers and
pyramidal numbers are exhibited.
2. METHOD OF ANALYSIS:
The ternary cubic diophantine equation under consideration
is
322
3519)(5 zyxxyyx  (1)
Introduction of the transformations
vux  , vuy  (2)
in (1) leads to
322
3519)1( zvu  (3)
Equation (3) is solved through different methods and thus,
we obtain different patterns of solutions to (1)
2.1 Method 1
Take
22
19baz  (4)
Write )194)(194(35 ii  (5)
Using (4) and (5) in (3) and employing the method of
factorization, define
3
)19)(194(19)1( biaiviu  (6)
Equating real and imaginary parts of (6) on both sides we
get
3223
3223
765712
3612285741
babbaav
babbaau


Substituting the values of u,v in (2) the non-zero distinct
integral solutions to (1) are given by
22
3223
3223
19
1437171693
1285285455
baz
babbaay
babbaax



Properties:
1) )324(mod4030)1,( ,92
3
1   aa tPax
2) )92(mod603)1,()1,( ,50
3
 aa tCPaxay
3) 76]8219)1,(8)1,()1,([4 ,4  aa tSazayax
is a cubical integer

2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 450
4) )124(mod11382)1,()1,(23 ,3
9
 aa tCPayaz
5) ]1140280360),1([10 33
1   aaa PRCPPaay
is a perfect square
6) bbb tPRPBx ,4
5
52545570),1(  is a perfect
square
7) 0)(19]1),1([2  bb GnoCSbz
8) )19(mod02)1,( ,6,5  aa ttaz
Note:
Re-writing (5) as
)194)(194(35 ii 
and proceeding as in method1, the non-zero distinct integral
solutions to (1) are given by
22
3223
3223
19
1285285455
1437171693
baz
babbaay
babbaax



2.2 Method 2:
Equation (3) can be written as
1.3519)1( 322
zvu 
Write „1‟ as
2
10
)199)(199(
1
ii 

Define
10
)199()19)(194(
)191(
3
ibiai
viu


(7)
Equating real and imaginary parts, we have
]469396974117[
10
1
1 3223
babbaau 
(8)
]3237415113[
10
1 3223
babbaav  (9)
Since our aim is to find integer solutions, assuming a=10A
and b=10B in (4),(8) and (9) and substituting the values of
u,v in (2), we obtain the distinct non-zero integral solutions
to (1) as
]19[10
1]50162287924[10
1]4370171069030[10
222
2232
2232
BAz
ABBAAy
ABBAAx



Properties:
1) )667(mod45797236)9)1,()1,(10[
10
1 3
13
  AA PRPAxAy
2) )133.10)1)1,((30)1)1,((4(
10
14 5
5
 AyAx is a nasty number
3) )466(mod545317223]11)1,(10)1,([
10
1
,3
14
3
 AA tCPAyAx
4) )1095(mod0]5015994[
10
1
,15862
 AtyAz
2.3 Method 3:
Instead of (5),35can be written as
2
2
)1911)(1911(
35
ii 

Proceeding as in method 1 and performing some algebra, we
obtain the distinct non-zero integral solutions to (1) as
]19[2
1]5705709010[2
1]1526842412[2
222
32232
32232
BAz
BABBAAy
BABBAAx




3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 451
Properties:
1) )2807(mod60724)1,( ,194  AA tSOAx
2) ]5089066[4)1,()1,(  AA GnoPRAyAx i
s a cubical integer.
3) 3131122)1,( ,12,18  AAA GnottAz is a
perfect square
4) )1490(mod39101140),1( ,458  BB tSOBy
Note:
Rewriting (7) as
)
10
199
()19)(194()19)1(( 3 i
biaiviu


and proceeding as in method 2, the non-zero distinct integral
solutions to (1) are given by
]19[2
1]1526842412[2
1]5705709010[2
222
32232
32232
BAz
BABBAAy
BABBAAx



2.4 Method 4:
Equation (7) can be written as
)
10
199
()19(
2
)1911(
)19)1(( 3 i
bia
i
viu




Equating real and imaginary parts on both sides and
assuming a=2A, b=2B, we get
]152114242[2
]7224561148[21
32232
32232
BABBAAv
BABBAAu


Also (4) implies
)19(4 22
BAz  (10)
Substituting the values of u,v in (2) we obtain the non-zero
integral solutions to (1) as
1]437171693[8
1]57579[40
3223
3223


BABBAAy
BABBAAx
along with (10).
3. CONCLUSION
In this paper, we have made an attempt to obtain a complete
set of non-trivial distinct integral solutions for the non-
homogeneous ternary cubic equation. To conclude, one
may search for other choices of solutions to the considered
cubic equation and further cubic equations with
multivariables.
NOTATIONS:
 



 

2
)2)(1
1,
mn
nt nm
    mnm
nn
Pm
n 


 
 5()2
6
)1
1)1(6  nnSn
)12( 2
 nnSOn
22
)1(  nnCSn
12  nGnOn
)1(  nnPRn
2
)1( 2
3 

nn
CPa
2
)13( 2
9 

nn
CPa
3
)47( 2
14 

nn
CPa
REFERENCES
[1] Dickson. L.E. “History of Theory of Numbers and
Diophantine Analysis”, Vol.2, Dover Publications,
New York 2005
[2] Mordell L.J., “Diophantine Equations” Academic
Press, New York, 1970
[3] R.D. Carmicheal, “The Theory of Numbers and
Diophantine Analysis”, Dover Publications, New
York 1959
[4] M.A. Gopalan, ManjuSomnath and N. Vanitha, “On
ternary cubic Diophantine equation x2 + y2 = 2z3
Advances in Theoretical and Applied Mathematics”,
Vol. 1, No.3,227-231, 2006.
[5] M.A. Gopalan, ManjuSomnath and N. Vanitha,
“Ternary cubic Diophantine equation x2 - y2 = z3”,
ActaCienciaIndica”, Vol.XXXIIIM, No. 3, 705 –
707, 2007.

4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 03 Issue: 11 | Nov-2014, Available @ http://www.ijret.org 452
[6] M.A. Gopalan and R.Anbuselvi, “Integral Solutions
of ternary cubic Diophantine equation x2 + y2 + 4N
= zxy”, Pure and Applied Mathematics Sciences
Vol.LXVII, No.1-2, 107-111, March 2008.
[7] M.A. Gopalan, ManjuSomnath and N. Vanitha,
“Ternary cubic Diophantine equation 22-1 (x2 +
y2) = z3, ActaCienciaIndica”, Vol.XXXIVM, No. 3,
1135 – 1137, 2008.
[8] M.A.Gopalan,V.Pandichelvi On the ternary cubic
equation
3222
)( xgkgzy  Impact J.
Sci.Tech.,Vol 4 No 4, 117-123, 2010
[9] M.A.Gopalan,J.Kaligarani Integral solutions
333
)12()(8 zkyxkyx  Bulletin of
Pure and Applied Sciences, Vol 29E, No.1, 95-
99,2010
[10] M.A.Gopalan, S.Premalatha On the ternary cubic
equation
)(2 )232323
zzyyxx  Cauvery
Research Journal Vol 4,iss 1&2.,87-89,July 2010-
Jan 2011
[11] M.A.Gopalan, V.Pandichelvi,observations on the
ternary cubic equation
)(4 232233
zzyxyx  Archimedes
J.Math 1(1),31-37,2011
[12] M.A.Gopalan, G.Srividhya Integral solutions of
ternary cubic diophantine equation
233
zyx  ActaCienciaIndica,Vol XXXVII
No.4 ,805-808,2011
[13] M.A.Gopalan,A.Vijayashankar, S.Vidhyalakshmi
Integral solutions of ternary cubic
equation
3222
)3()(2 zkzyxxyyx 
Archimedes J.Math 1(1),59-65,2011
[14] M.A.Gopalan,G.Sangeetha on the ternary cubic
diophantine equation
322
zDxy  Archimedes
J.Math 1(1),7-14,2011
[15] M.A. Gopalan and B.Sivakami, “Integral Solutions
of the ternary cubic Diophantine equation 4x2 – 4xy
+ 6y2 =[(k+1)2+5] w3” Impact J. Sci. Tech., Vol. 6,
No.1, 15 – 22, 2012.
[16] M.A.Gopalan, S.Vidhyalakshmi and G.Sumathi, on
the non-homogeneous equation with three unknowns
“
 "314 333
yxzyx  , Discovery
Science, Vol.2, No:4,37-39. oct:2012
[17] M.A.Gopalan,B.Sivakami Integral solutions of the
ternary cubic equation
3222
)5)1((644 wkyxyx  Impact J.
Sci.Tech.,Vol 6 No 1, 15-22,2012
[18] M.A.Gopalan, B.Sivakami on the ternary cubic
diophantine equation
)(2 2
zxyxz 
Bessel.J.Math 2(3),171-177,2012.
[19] S.Vidhyalakshmi,T.R.Usha Rani and M.A.Gopalan
Integral solutions of non-homogenous ternary cubic
equation
322
)( zbabyax  Diophantus J.
Math.,2(1), 31-38, 2013
[20] M.A.Gopalan, K.Geetha On the ternary cubic
diophantine equation
322
zxyyx  Bessel
J.Math., 3(2),119-123,2013
[21] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha
Observations on the ternary cubic
equation
322
12zxyyx  Antartica J.
Math.10(5),453-460, 2013
[22] M.A.Gopalan, S.Vidhyalakshmi and K.Lakshmi
Lattice points on the non-homogeneous ternary
cubic equation
0)(333
 zyxzyx
Impact J. Sci.Tech.,Vol 7 No 1, 21-25,2013
[23] M.A.Gopalan, S.Vidhyalakshmi and K.Lakshmi
Lattice points on the non-homogeneous ternary
cubic equation
0)(333
 zyxzyx
Impact J. Sci.Tech.,Vol 7 No 1, 51-55,2013
[24] M.A.Gopalan, S.Vidhyalakshmi and S.Mallika on
the ternary non-homoogenous cubic equation
3233
)23(2)(3 zkyxyx  Impact J.
Sci.Tech.,Vol 7 No 1, 41-45,2013