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Statistics Question Paper [ Gen ] {Under Graduate}, 2019

This paper concerns with the problem of obtaining non-zero distinct integer solutions to the ternary quadratic Diophantine equation representing a cone given by z^2=15x^2-6y^2. Different sets of solutions are presented. A few interesting relations between the solutions and special polygonal numbers are obtained. Given a solution, a formula for generating sequence of solutions is illustrated

Vijayasankar, A

Krithika, V

Gopalan, M.A

Journal of Physical Sciences, Vol. 24, 2019, 107-113 
ISSN: 2350-0352 (print), www.vidyasagar.ac.in/publication/journal 
Published on 6 June 2019 
 
107 
 
Lattice Points on the Cone 222 615 yxz −=  
   A.Vijayasankar1, V.Krithika2 and M.A.Gopalan
3
 
1Department of Mathematics, National College, Trichy -620001 
Tamilnadu, India. e-mail: avsankar70@yahoo.com 
2Department of Mathematics, Urumu Dhanalakshmi College, Trichy – 620019  
Tamilnadu, India. e-mail: krithijay.24@gmail.com 
3Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620002  
Tamilnadu, India. e-mail: mayilgopalan@gmail.com  
Received 28 November 2018; accepted 23 November2018 
ABSTRACT 
This paper concerns with the problem of obtaining non-zero distinct integer solutions to 
the ternary quadratic Diophantine equation representing a cone given by  = 15 −
6	. Different sets of solutions are presented. A few interesting relations between the 
solutions and special polygonal numbers are obtained. Given a solution, a formula for 
generating sequence of solutions is illustrated. 
 
Keywords: Ternary quadratic, homogeneous quadratic, cone, integer solutions. 
 
Mathematical Subject Classification (2010): 11D09 
 
Notations used: 
• nmt ,  - Polygonal Number of rank n  with sides m  
• mnp  - Pyramidal Number of rank n  with sides m  
• nPr   - Pronic Number of rank n 
• nJ  - Jacobsthal Number of rank n 
• ngn - Gnomonic Number of rank n 
 
1. Introduction 
There is a great interest for mathematicians since ambiguity in homogeneous and 
non-homogeneous quadratic Diophantine Equations [1-3]. In this context, one may refer 
[4-13] for varieties of problems on the quadratic Dophantine equations with two or three 
variables. In this paper, ternary quadratic equation given by 222 615 yxz −=  is 
analyzed for its non-zero distinct integer solutions. A few interesting relations between 
the solutions and special  numbers are exhibited. 
 
2. Method of analysis 
The homogeneous Quadratic Diophantine equation with three unknowns to be solved for 
its non-zero distinct integral solutions is   
   A.Vijayasankar, V.Krithika and M.A.Gopalan 
108 
 
222 615 zyx =−                        (1) 
Different patterns of solutions of (1) are presented b low. 
Pattern –I   
Introducing the linear transformations 
 wztXytXx 3,15,6 =+=+=                (2)  
in (1), it leads to  222 90twX +=                 (3) 
which is satisfied by  
 





+=
−=
=
22
22
90
90
2
baX
baw
abt
              (4) 
From (2), the corresponding non zero distinct integral solutions to (1) are 
 
22
22
22
3270),(
3090),(
1290),(
babaz
abbabay
abbabax
−=
++=
++=
  
 
Properties: 
1. ( ) ( ) ataaxaay ,3361,1, =+−+  
2. ( ) ( ){ }aaxaay ,,3 −  is a nasty number. 
3. ( ) ( ) 522 721,1,3 aPaazaax −+++  is a nasty number.  
4. ( ) ( ) )6(mod0,,3 bbazbax ≡+  
5. ( ) ( ) { } 0Pr156,13,1 ,4 =+++−+ aataayaaz  
 
Pattern-II:  
Write (3) in the form of ratio as  
0,
2
45 ≠=+=
−
q
q
p
t
wX
wX
t
                      (5)  
which is equivalent to system of double equations  
 
02
045
=−−
=+−
qwqxpt
pwpxqt
  
Applying the method of cross multiplication, we have 
 
22
22
452
452
2
qpw
qpX
pqt
−=
+=
=
   
Substituting the values of X ,t and w in (2), the values of x , y and z are  
Lattice Points on the Cone
222 615 yxz −=  
109 
 
 
( ) 



−=
++=
++=
22
22
22
1356,
30452),(
12452),(
qpqpz
pqqpqpy
pqqpqpx
                    (6) 
Thus (6) represents the non-zero distinct integral solutions to (1). 
 
Properties: 
1. ( ) ( ) 0Pr181,1, =−+−+ pppxppy  
2. ( ) ( ){ }ppxppy ,,12 22 −  is a Cubical integer. 
3. ( ) ( ) ( ){ }ppzppyppx ,,,7 −+  is a Perfect square.  
4. ( ) ( ) ( ) 05270,13,1 28,4 =+++− JJqtqyqz q  
5. ( ) ( ) 01812,112,1 =++−+ pgnpypx  
 
It is worth to mention here that (5) may also be expr ssed in the form of ratio as follows: 
 
Ratio 1: 
 0,
90 ≠=−=
+
q
q
p
t
wX
wX
t
 
 
Ratio 2: 
 0,
3
30 ≠=+=
−
q
q
p
t
wX
wX
t
 
 
Ratio 3: 
 0,
5
18 ≠=+=
−
q
q
p
t
wX
wX
t
 
 
Ratio 4: 
 0,
6
15 ≠=+=
−
q
q
p
t
wX
wX
t
 
 
Ratio 5: 
 0,
9
10 ≠=+=
−
q
q
p
t
wX
wX
t
 
 
Following the procedure presented above , the corresponding Gaussian integer solutions of 
(1) are given below: 
 
 
   A.Vijayasankar, V.Krithika and M.A.Gopalan 
110 
 
Solution for Ratio 1: 
 
( ) 22
22
22
2703,
3090),(
1290),(
qpqpz
pqqpqpy
pqqpqpx
−=
−−−=
−−−=
 
 
Solution for Ratio 2: 
 
( ) 22
22
22
909,
30303),(
12303),(
qpqpz
pqqpqpy
pqqpqpx
−=
++=
++=
 
 
Solution for Ratio 3: 
 
( ) 22
22
22
5415,
30185),(
12185),(
qpqpz
pqqpqpy
pqqpqpx
−=
++=
++=
 
 
Solution for Ratio 4: 
 
( ) 22
22
22
4518,
30156),(
12156),(
qpqpz
pqqpqpy
pqqpqpx
−=
++=
++=
 
 
Solution for Ratio 5: 
 
( ) 22
22
22
3027,
30109),(
12109),(
qpqpz
pqqpqpy
pqqpqpx
−=
++=
++=
  
 
Remarkable observations 
 If the non-zero integer triplet ),,( 000 zyx  is any solution of (1) then each of the 
following three triplets of integer based on 00, yx  and 0z  also satisfies (1).  
Triplet 1: ),,( 0zyx nn  
Let 0,9,9,9 010101 ≠=+=−= hzzhyyxhx                  (7) 
be the solutions of (1).  
Substitute (7) in (1) and performing a few calculations, we have 
 00 1230 yxh +=  
and then 001001 2130,1221 yxyyxx +=+=  which is written in the form of matrix as 
Lattice Points on the Cone
222 615 yxz −=  
111 
 






=





0
0
1
1
y
x
M
y
x
             
where 





=
2130
1221
M  
Replacing the above process, the general solution ),( nn yx to (1) is given by  






=





0
0
y
x
M
y
x n
n
n  
The Eigen values of M are 10621,10621 −=+= βα , it is well known that  
)()( IMIMM
nn
n α
αβ
ββ
βα
α −
−
+−
−
=  
Using the above formula, we have 






−++−−+
−−+−++=
])10621()10621[(106])10621()10621[(30
])10621()10621[(12])10621()10621[(106
1012
1
nnnn
nnnn
nM
 
Thus, the general solution ),,( nnn zyx to (9.10) is given by 
[ ]
[ ]
0
00
00
9
])10621()10621[(106])10621()10621[(30
1012
1
])10621()10621[(12])10621()10621[(106
1012
1
zz
yxy
yxx
n
n
nnnn
n
nnnn
n
=
−+++−−+=
−−++−++=
  
Triplet 2: ),,( 0 nn zyx  
Let 0,4,, 010101 ≠+==−= hhzzyyxhx       
Following the procedure presented in triplet: 1, the corresponding general solution to (1) 
is given by  
[ ]
[ ]00
0
00
])15831()15831[(158])15831()15831[(120
1516
1
])15831()15831[(8])15831()15831[(158
1516
1
zxz
yy
zxx
nnnn
n
n
nnnn
n
−−++−+−−−+−−=
=
−−−+−−−−++−=
 
Triplet 3: ),,( 0 nn zyx  
Let 0,53,5,5 010101 ≠−=+== hzhzhyyxx      
Repeating the above process, the corresponding general solution to (1) is given by  
   A.Vijayasankar, V.Krithika and M.A.Gopalan 
112 
 
[ ]
[ ]00
00
0
])621()621[(62])621()621[(12
64
1
])621()621[(2])621()621[(62
64
1
5
ziiiyii
i
z
ziiyiii
i
y
xx
nnnn
n
nnnn
n
n
n
−++−−−+−=
−−++−++−=
=
, 
 
3. Conclusion 
In this paper, we have made an attempt to find different patterns of  non-zero distinct 
integer solutions to the ternary cubic equation, given by 222 615 yxz −= . As ternary 
quadratic equations are rich in variety, one may search for integer solutions to other 
choices of quadratic equations with multivariates along with suitable properties. 
 
REFERENCES 
1. A.H.Beiler, Recreations in the theory of numbers, Dover Publications Inc., New York, 
1963. 
2. A.Weil, Basic number theory, Third Edition, Springer Verlag, Berlin, 1995. 
3. D.M.Burton, Elementary number theory, Tata McGraw-Hill Publishing Company 
Limited, New Delhi. 
4. M.A.Gopalan, S.Vidhyalakahmi and E.Premalatha, On the non-homogeneous binary 
quadratic equation  − 3	 + 	 + 2 = 0, International journal of Engineering 
& Scientific Research, 2(2) (2014) 83-88. 
5. M.A.Gopalan, S.Vidhyalakahmi and E.Premalatha, Observations on the binary 
quadratic diophantine equation 1688 22 ++= xxy , International Journal Of 
Latest Research in Science and Technology, 1(4) (2012) 379-382. 
6. M.A.Gopalan, S.Vidhyalakahmi and E.Premalatha, On ter ary quadratic equation 
222 12zyxyx =++ ,  Diophantus J.Math., 1(2) (2012) 69-76. 
7. M.A.Gopalan, S.Vidhyalakahmi and E.Premalatha, On ter ary quadratic diophantine 
equation  
222 73 zyx =+  , Diophantus J. Math., 1(1) (2012) 51-57.   
8. M.A.Gopalan, S.Vidhyalakahmi, E.Premalatha and R.Malathi, On the ternary 
quadratic diophantine equation
222 1637 zyx =− , Proceedings of ReDeEM, 
Thiagarajar College, Madurai, 2015. 
9. M.A.Gopalan, S.Vidhyalakahmi, M.Manjula, E.Premalatha and N.Thiruniraiselvi, 
On the ternary quadratic equation 
222 5 yxz += , Retell, 12(2) (2012) 151-154.  
10. M.A.Gopalan, S.Vidhyalakahmi, E.Premalatha and V.Krithika, Integral solutions of 
ternary quadratic Diophantine equation 
12222 ++=+ αdzdyx , Proceedings 
of ReDeEM, Thiagarajar College, Madurai, 2015. 
Lattice Points on the Cone
222 615 yxz −=  
113 
 
11. M.A.Gopalan, S.Vidhyalakahmi and E.Premalatha, On the homogeneous quadratic 
equation with three unknowns 
2222 )3( zkyxyx +=+− , Bulletin of 
Mathematics and Statistics Research, 1(1) (2013) 38-41. 
12. M.A.Gopalan, S.Vidhyalakahmi, M.Manjula, E.Premalatha and N.Thiruniraiselvi, 
On the diophantine equation with three unknowns 
222 95 zyx =+  Retell, 12(2) 
(2012) 147-150. 
 


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