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1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 3 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 665
INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
322
972)1(24)(3 zyxxyyx 
G.Janaki1, C.Saranya2,*
Assistant Professors, Department of Mathematics, Cauvery College for Women, Tiruchirappalli, Tamil Nadu, India.
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Abstract:The non-homogeneous cubic equation with three unknowns represented by the Diophantine equation
322
972)1(24)(3 zyxxyyx  is analyzed for its patterns of non – zero integral solutions. A few
interesting properties among the solutions and some special polygonal numbers are presented.
Keywords: Cubic Equation with Three Unknowns, Integral solutions
Introduction:
Mathematics is the language of patterns and relationships and is used to describe anything that can be quantified.
Diophantine equations has been matter of interest to various mathematicians. The problem of finding all integer
solutions of a diophantine equation with three or more variables and equations of degree at least three, in general
presents a good deal of difficulties. In [1-3], theory of numbers were discussed. In [4,5], a special Pythagorean
triangle problem have been discussed for its integral solutions. In [6-10], higher order equations are considered for
integral solutions.
In this communication, the non-homogeneous cubic equation with three unknowns represented by the equation
322
972)1(24)(3 zyxxyyx  is considered and in particular a few interesting relations among the
solutions are presented.
Notations:
nObl = Oblong number of rank ‘n’.
nmT , = Polygonal number of rank ‘n’ with sides ‘m’.
nCS = Centered Square number of rank ‘n’.
nSO = Stella octangula number of rank ‘n’.
nO = Octahedral number of rank ‘n’.
nGno = Gnomonic number of rank ‘n’.
nStar = Star number of rank ‘n’.
nTO = Truncated octahedral number of rank ‘n’.
m
nP = Pyramidal number of rank ‘n’ with sides ‘m’.

2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 3 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 666
Method of Analysis:
The Cubic equation to be solved for its non-zero integral solution is
322
972)1(24)(3 zyxxyyx  (1)
On substitution of the transformations,
vux  , vuy  (2)
in (1) leads to,   322
48651 zvu  (3)
We illustrate below four different patterns of non-zero distinct integer solutions to (1).
Pattern: 1
Assume 22
5),( babazz  (4)
where a and b are non-zero integers.
and write )5519)(5519(486 ii  (5)
Substituting (4) & (5) in (3), and using factorization method,
    33
)5()5()5519)(5519(5)1(5)1( biabiaiiviuviu  (6)
Equating the like terms and comparing real and imaginary parts, we get
3223
3223
9557755),(
11257528519),(
bbaababavv
bbaababauu


Substituting the above values of vu & in equation (2), the corresponding integer solutions of (1) are
given by
22
3223
3223
5),(
122013221014),(
1301836024),(
babazz
bbaababayy
bbaababaxx



Properties:
1. ),(),( aaxaay  is a cubical integer.
2. )1,1()1,1( xy  is a nasty number.
3. )36(mod02650),1(38),1(),1( ,8  aa GnoTazaayax
4. )215(mod025195)1,()1,( ,14  bbb GnoTSObybx
5. )260(mod0505515)1,(4)1,()1,( ,6  aaa GnoTOazaxay
6. )189(mod021420),(),(),( ,22
5
 aa TPaazaaayaax
7. )1,1(18)1,1()1,1( zxy  is a perfect square.
Pattern: 2
Instead of (5), Write )599)(599(486 ii  (7)
Substituting (7) and (4) in (3) and employing the method of factorization, following the procedure presented in
pattern 1, the corresponding integer solutions of (1) are represented by

3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 3 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 667
22
32
3223
5),(
1270162),(
118010827018),(
babazz
bbabayy
bbaababaxx



Properties:
1. ),(12),(),( aazaaaxaay  is a cubical integer.
2. )1,1(12)1,1()1,1( zxy  is a nasty number.
3. )220(mod01202236)1,(2)1,()1,( ,22,16
5
 aaaa GnoTTPazaxay
4. )8(mod01716232)1,(2)1,( ,6
5
 aaa GnoTPazaax
5. )124(mod01392090),1(2),1( ,30  bbb GnoTSObzbx
6. )115(mod010443155),1(),1(),1( ,22  bbbb GnoTOTObzbybx
7. )1,1(6)1,1()1,1( zxy  is a perfect square.
Pattern: 3
Instead of (5), Write )5321)(5321(486 ii  (8)
Substituting (8) and (4) in (3) and following the same procedure presented in pattern 1, the corresponding integer
solutions of (1) are represented by
22
3223
3223
5),(
118010827018),(
1301836024),(
babazz
bbaababayy
bbaababaxx



Properties:
1. ),(8),(),( aazaaaxaay  is a cubical integer.
2. )1,1(8)1,1()1,1( zxy  is a nasty number.
3. )222(mod012633)1,()1,( ,6  aaa GnoTSOaaxay
4. )647(mod0)(162)1,(3)1,(4 ,8  aa GnoTaxay
5. )215(mod018446)1,()1,( ,8,6
5
 aaaa GnoTTPazaax
6. )13(mod01921224)1,(2)1,( ,20  aaa GnoTOazaay
7. )1,1(4)1,1()1,1( zxy  is a perfect square.
Pattern: 4
Instead of (5), Write
9
)52913)(52913(
486
ii 
 (9)
and using the same procedure as in pattern 1, the corresponding solutions of (3) are represented by
22
3223
3223
5),(
)653943529(
3
1
),(
1)72543519513(
3
1
),(
babazz
bbaababavv
bbaababauu




4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 3 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 668
Since our interest is on finding integer solutions, we have choose a and b suitably so that vu, and z are integers.
Let us take a = 3A and b = 3B, we have
22
3223
3223
459),(
5853153915261),(
1652539151755117),(
BABAzz
BBAABABAvv
BBAABABAuu



In view of (2), the integer solutions of (1) are given by
22
3223
3223
459),(
1711042662160144),(
1594035645670378),(
BABAzz
BBAABABAvv
BBAABABAxx



Properties:
1. ),(140),(),( AAzAAAxAAy  is a cubical integer.
2. )1,1()1,1( xy  is a nasty number.
3. )8605(mod044432688468)1,()1,( ,8
5
 AAA GnoTPAyAx
4. )539(mod05400324)1,(42)1,( ,24  AA GnoTAzAAx
5. )6653(mod0456474)1,(16)1,( ,20  AA GnoTAzAAy
6. )807(mod05331894468)1,(2)1,()1,( ,20
5
7 AAA GnoTPAzAyAx
7. )1,1(17)1,1()1,1( zyx  is a perfect square.
Note:
In addition, one may write 486 as


























49
)5693()5693(
49
)56737()56737(
49
)55787()55787(
49
)547113()547113(
49
)545117()545117(
49
)59153()59153(
486
ii
ii
ii
ii
ii
ii
For these choices, one may obtain different patterns of solutions of (1).

5. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 3 | Mar -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 669
Conclusion:
In this paper, we have presented four different patterns of non-zero distinct integer solutions of the non-
homogeneous cone given by
322
972)1(24)(3 zyxxyyx  .
To conclude, one may search for other patterns of non-zero integer distinct solutions and their
corresponding properties for other choices of cubic diophantine equations.
References:
[1] Carmichael, R.D., The theory of numbers and Diophantine Analysis, Dover Publications,New York, 1959.
[2] Dickson L.E, History of Theory of Numbers, Vol.11, Chelsea Publishing company, New York,1952.
[3] Mordell. L.J, Diophantine equations, Academic Press,London,1969 Telang, S.G., Number theory, Tata Mc
Graw Hill publishing company, New Delhi, 1996
[4] Gopalan. M.A, Manju Somnath and Vanitha. N, Parametric Solutions of 262
zyx  , Acta ciencia indica,
XXXIII, 3, 1083-1085, 2007.
[5] Gopalan.M.A and Janaki.G, Integral solutions of 3222222
)(2233)(( pwzxyyxyx  , Impact
J.Sci.,Tech., 4(1), 97-102, 2010.
[6] Gopalan M.A., Pandichelvi.V, observations on the ternary cubic equation )(4 232233
zzyxyx 
Archimedes J.Math 1(1), 31-37, 2011.
[7] Gopalan M.A, Srividhya.G., Integral solutions of ternary cubic diophantine equation
233
zyx  Acta Ciencia Indica,Vol XXXVII, No.4, 805-808, 2011.
[8] Gopalan.M.A., and Janaki.G., Integral solutions of 3222222
)(2)233)(( pwzxyyxyx  , Impact
journal of Science & Technology, Vol-4, No.97-102, 2010.
[9] Janaki.G and Saranya.C., Observations on the Ternary Quadratic Diophantine Equation
222
7293311)(6 zyxxyyx  , International Journal of Innovative Research in Science,
Engineering and Technology, Vol-5, Issue-2, Pg.no: 2060-2065, Feb 2016.
[10] Janaki.G and Saranya.P., On the ternary Cubic diophantine equation
322
404)(46)(5 zyxxyyx  , International Journal of Science and Research- online, Vol 5,
Issue3, Pg.No:227-229, March 2016.