International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
www.irjet.net
ON BINARY QUADRATIC EQUATION 2 x 2  3 y 2   4
S. Vidhyalakshmi1, A. sathya2, A. Nandhinidevi3
1Professor,
Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India.
Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil nadu, India.
3PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil nadu, India.
---------------------------------------------------------------------***---------------------------------------------------------------------2 Assistant
Abstract- The binary quadratic Diophantine equation
To obtain the other solutions of (3), consider the Pell
equation
represented by the positive Pellian 2 x 2  3 y 2   4 is
analyzed for its non-zero distinct integer solutions. A few
interesting relations among the solutions are given.
Employing the solutions of the above hyperbola, we have
obtained solutions of other choices of hyperbolas and
parabolas and Pythagorean triangle.
X 2  6T 2  1
~ ~
Whose smallest positive integer solution is ( X 0 , T0 )  (5, 2)
the general solution of (4) is given by
~
~
1
1
Tn 
gn , X n  fn
2
2 6
Key Words: Binary quadratic, Hyperbola, Parabola, Pell
equation, Integer solutions
Where
1. INTRODUCTION
f n  (5  2 6 ) n1  (5  2 6 ) n1
The binary quadratic Diophantine equations of the
form ax 2  by 2  N , a, b, N  0 are rich in variety and have
been analyzed by many mathematicians for their respective
integer solutions for particular values of a, b and N . In this
context, one may refer [1-14].
g n  (5  2 6 ) n 1  (5  2 6 ) n 1 ,
n  10,1....
Applying Brahmagupta lemma between ( ~
x0 , ~
y0 ) and
~
~
( x , y ) the other integer solutions of (3) are given by
n
This communication concerns with the problem of obtaining
non-zero distinct integer solutions to the binary quadratic
equation given by 2 x 2  3 y 2   4 representing hyperbola. A
few interesting relations among its solutions are presented.
Knowing an integral solution of the given hyperbola, integer
solutions for other choices of hyperbolas and parabolas are
presented. Also, employing the solutions of the given
equation, special Pythagorean triangle is constructed.
5
6
gn
(5)
gn
(6)
27
6
22
6
gn
gn
The recurrence relations satisfied by x and y are given
by
(2)
xn3  10xn2  xn1  0
yn3  10 yn2  yn1  0
(3)
Some numerical examples of xn and yn satisfying (1) are
whose smallest positive integer solution is
X 0  10 , T0  4
|
y n1  2 f n 
6
(1)
From (1) and (2), we have
X 2  6T 2  4
12
y n1  9 f n 
Consider the linear transformations
x  X  3T , y  X  2T
xn1  5 f n 
xn1  11 f n 
The Diophantine equation representing the binary quadratic
equation to be solved for its non-zero distinct integral
solution is
2x 2  3 y 2   4
n
From (2), (5) and (6) the values of x and y satisfying (1)
are given by
2. METHOD OF ANALYSIS:
© 2019, IRJET
(4)
Impact Factor value: 7.211
given in the Table: 1 below,
|
ISO 9001:2008 Certified Journal
| Page 973
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
www.irjet.net
Table: 1 Numerical Examples
n
xn
yn
0
1
2
3
4
5
22
218
2158
21362
211462
2093258
18
178
1762
17442
172658
1709138
 Each of the following expressions represents a
nasty number:

Both
1
60  162 y2n3  1308x2n2 
5
1

588  162 y2n4  12948x2n2 
49
 12  267x2n4  2643x2n3 

xn and yn values are even.
1
60  1602 y2n2  132x2n3 
5
 12  1602 y2n3  1308x2n3 

xn3 10xn2  xn1  0

yn2  5 yn1  4xn1  0
1
60  1602 y2n4  12948x2n3 
5
1

588  15858y2n2  132x2n4 
49
1

60  15858 y2n3  1308x2n4 
5
 12  15858y2n4  12948x2n4 

xn3  xn1 12 yn2  0

 5xn1  xn2  6 yn1  0

 xn1  5xn2  6 yn1  0
 5xn1  49xn2  6 yn3  0
 60 yn3  49xn3  xn1  0
 xn3  49xn1  60 yn1  0
 5 yn 3  49 yn  2  4xn1  0
 7 xn3 1580 yn3 125857xn1  0

 49 yn1  yn 3  40xn 1  0

 6 yn1  5xn3  49xn2  0
 6 yn2  21365 xn3  211489 xn2  0
 44xn2  5xn1  539 yn1  0
 5 yn2  yn1  4xn2  0

yn3  yn1  8xn2  0
 xn3  5xn2  6 yn2  0

yn3  5 yn2  4xn2  0
 49 yn2  5 yn1  4xn3  0
 49 yn3  yn1  40xn3  0
5y
y
 4x
0
n3
n 2
n3

 5 yn1  yn2  4xn1  0

yn3 10 yn2  yn1  0
 4xn1  49 yn2  5 yn3  0
|
Impact Factor value: 7.211
1
48  1308 y2n2  132 y2n3 
4
1
480  12948y2n2  132 y2n4 
40
1
48  12948y2n3  1308y2n4 
4
 Each of the following expressions represents
a cubical integer:
 6 yn3  5xn3  xn2  0
© 2019, IRJET
 267x2n2 
1
120  27x2n4  2643x2n2 
10
 12  162 y2n2  132x2n2 
 Relations among the solutions are given below:

2 n3

From the above table, we observe some interesting relations
among the solutions which are presented below:

12  27x
|

1  27 x3n4  267 x3n3  81xn2 


6   801xn1


1  27 x3n5  2643x3n3  81xn3 


60   7929xn1


 27 y3n3  22 x3n3  81yn1 


  66 xn1


1  27 y3n4  218x3n3  81yn2 


5   654xn1


1  27 y3n5  2158 x3n 3  81y n3 


49   6474xn1


1  267 x3n 5  2643x3n  4  801xn 3 


6   7929xn  2

ISO 9001:2008 Certified Journal
| Page 974
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
www.irjet.net
1  267 y3n 3  22 x3n  4  801y n1 


5   66 xn  2


1  267 y 4 n6  2158x4 n5  1068 y 2 n 4 


5   8632 x2 n3  30

 267 y3n 4  218 x3n 4  801y n 2 

  654 xn 2


1  2643 y 4 n 4  22 x4 n6  10572 y 2 n 2 


49   88 x2 n 4  294


1  267 y3n 5  2158x3n 4  801y n 3 


5   6474xn  2


1  2643 y 4 n 5  218 x4 n 6  10572 y 2 n 3 


5   872 x2 n  4  30


1  2643 y3n 3  22 x3n5  7929 y n 1 


49   66 xn3

 

1  2643y3n  4  218 x3n5  7929 y n  2 


5   654xn3


1  218 y 4 n  4  22 y 4 n 5  872 y 2 n  2 


4   88 y 2 n 3  24

 2643 y3n 5  2158 x3n 5  7929 y n 3 

  6474xn 3


1  2158 y 4 n 4  22 y 4 n6  8632 y 2 n 2 


40   88 y 2 n 4  240


1  218 y3n 3  22 y3n  4  654 y n 1 


4   66 y n  2


1  2158 y 4 n5  218 y 4 n 6  8632 y 2 n3 


4   872 y 2 n  4  24


1  2158 y3n3  22 y3n 5  6474 y n 1 


40   66 y n3


1  2158 y3n  4  218 y3n 5  6474 y n  2 


4   654 y n 3


 
 2643 y 4 n 6  2158 x4 n6  10572 y2 n  4 

  8632 x2 n 4  6

 
 Each of the following expression is a quintic
integer:
 Each of the following expressions is a
biquadratic integer:

1  27 x5n 6  267 x5 n 5  135 x3n  4


6   1335 x3n 3  270 xn  2  2670 xn 1 

 27 x5 n7  2643 x5 n5  135 x3 n5 

1 
  13215 x3 n3  270 xn3

60 

  26520 xn1

 27 y5 n 5  22 x5 n 5  135 y3n 3 

  110 x3n 3  270 y n 1  220 xn 1 

1  27 x4 n 5  267 x4 n  4  108 x2 n 3 


6   1068 x2 n  2  36

 

1  27 x4 n 6  2643 x4 n  4  108 x2 n  4 


60   10572 x2 n  2  360



1  27 y5n6  218 x5 n5  135 y3n 4


5   1090 x3n3  270 y n 2  2180 xn1 
 27 y 4 n  4  22 x4 n  4  108 y 2 n  2 

  88x2 n  2  6


1  27 y 4 n 5  218x4 n  4  108 y 2 n 3 


5   872 x2 n  2  30

 27 y5 n7  2158x5 n5  135 y3 n5 

1 
  10790x3 n3  270 yn3

49 

  21580xn1



1  267 x5 n7  2643 x5 n6  1335 x3n5


6   13215 x3n 4  2670 xn3  26430 xn 2 
1  267 y5 n5  22 x5n6  1335 y3n3 


5   110 x3n 4  2670 y n1  220 xn 2 
 


1  27 y 4 n 6  2158 x4 n  4  108 y 2 n  4 


49   8632 x2 n 2  294


1  267 x4 n 6  2643 x4 n 5  1068 x2 n  4 


6   10572 x2 n 3  36



1  267 y 4 n 4  22 x4 n5  1068 y 2 n 2 


5   88x2 n3  30

 
 267 y5 n 6  218 x5 n 6  1335 y3n  4 

  1090 x3n  4  2670 y n  2  2180 xn  2 
 267 y4 n5  218x4 n5  1068 y2 n3 

 
  872 x2 n3  6

© 2019, IRJET

|
Impact Factor value: 7.211

|

1  267 y5 n7  2158 x5 n6  1335 y3n5


5   10790 x3n 4  2670 y n3  21580 xn 2 
ISO 9001:2008 Certified Journal
| Page 975
International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
www.irjet.net

1  2643 y5 n 5  22 x5 n 7  13215 y3n3 


49   110 x3n 5  26430 y n 1  220 xn 3 

1  2643 y5 n 6  218 x5 n7  13215 y3n  4 


5   1090 x3n5  26430 y n 2  2180 xn 3 
 2643 y5 n7  2158 x5 n 7  13215 y3n5 

 
  10790 x3n5  26430 y n 3  21580 xn3 

1  218 y5 n5  22 y5 n6  1090 y3n3 


4   110 y3n 4  2180 y n1  220 y n 2 

1  2158 y5 n5  22 y5n7  10790 y3n3 


40   110 y3n5  21580 y n1  220 y n3 

1  2158 y5 n 6  218 y5 n 7  10790 y3n  4 


4   1090 y3n 5  21580 y n  2  2180 y n 3 
6
X 2  6Y 2  144
 267 xn 3  2643xn  2 , 


 1079xn  2  109xn3 
7
X 2  6Y 2  100
 267 yn1  22 xn 2 ,

 9 xn 2  109 yn1
8
X 2  6Y 2  4
 267 y n 2  218xn  2 , 


 89 xn  2  109 y n 2 
9
X 2  6Y 2  100
 267 y n 3  2158xn  2 , 


 881xn  2  109 y n 3 
10
X 2  6Y 2  9604
 2643 y n 1  22 xn 3 , 


 9 xn 3  1079 y n 1 
11
X 2  6Y 2  100
 2643y n 2  218xn3 , 


 89 xn3  1079 y n 2 
12
X 2  6Y 2  4
 2643yn3  2158xn3 , 


 881xn 3  1079 yn 3 
13
X 2  6Y 2  64
 218 yn1  22 yn2 , 


 9 yn2  89 yn1

14
X 2  6Y 2  6400
 2158 yn1  22 yn3 , 


 9 yn3  88 yn1

15
X 2  6Y 2  64
 2158 yn2  218 yn3 , 


 89 yn3  881yn2

3. REMARKABLE OBSERVATIONS
3.1 Employing linear combinations among the solutions of
(1), one may generate integer solutions for other choices of
hyperbolas which are presented in Table 2 below:
Table 2: Hyperbolas
 X ,Y 
S.No Hyperbola
1
X  6Y  144
 27 xn 2  267 xn1 , 


 109 xn1  11xn 2 
2
X 2  6Y 2  14400
 27 xn3  2643xn1 , 


 1079xn1  11xn3 
2
2
3
X 2  6Y 2  4
4
X 2  6Y 2  100
 27 yn1  22 xn1 ,

 9 x  11y
n 1
 n1



3.2 Employing linear combination among the solutions of
(1), one may generate integer solutions for other choices of
parabolas which are presented in Table: 3 below,




Table 3: Parabolas
5
X  6Y  9604
2
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|
 27 yn 2  218 xn 1 ,

 89 xn1  11yn  2



 27 y n 3  2158xn 1 , 


 881xn 1  11y n 3

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 X ,Y 
S.No Parabola
|
1
Y  X  12
2
Y 2  10 X  1200
2
 27 x2 n3  267 x2 n 2 ,

109 xn1  11xn 2



 27 x2 n  4  2643x2 n  2 , 


1079xn1  11xn 3

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6Y 2  X  2
6Y 2  5 X  50
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 27 y2 n2  22 x2 n2 ,

 9 x  11y
n 1
 n1




 27 y2 n3  218x2 n 2 ,

 89 xn1  11yn 2



5
6Y 2  49 X  4802
6
Y 2  X  12
 267 x2 n  4  2643x2 n 3 , 


1079 xn  2  109 xn 3

7
6Y 2  5 X  50
 267 y2 n 2  22 x2 n3 ,

 9 xn 2  109 yn1
Pythagorean
triangle
T  ,  ,  
where
  2 pq ,   p 2  q 2 ,   p 2  q 2 , p  q  0 .
Taking p  xn1  yn1 , q  xn1 , it is observed that T  ,  ,   is
satisfied by the following relations:

3    2   4
A
   
P
A
 2 P  xn1 yn1
where A, P represent the area and perimeter of the
Pythagorean triangle.
 27 y 2 n 4  2158x2 n 2 , 


 881xn1  11y n 3


4
4. CONCLUSION



8
6Y 2  X  2
 267 y2 n3  218x2 n3 , 


 89 xn2  109 yn2

In this paper, we have presented infinitely many integer
solutions for the positive Pell equation 2x 2  3 y 2  4 . As the
binary quadratic Diophantine equations are rich in variety, one
may search for the other choices of Pell equations and determine
their integer solutions along with suitable properties.
9
6Y 2  5 X  50
 267 y 2 n  4  2158x2 n 3 , 


 881xn 2  109 y n 3

REFERENCES
10
6Y  49 X  4802
[1]
11
2
6Y 2  5 X  50
 2643 y 2 n 2  22 x2 n 4 , 


 9 xn3  1079 y n1

[2]
[3]
 2643y2 n3  218x2 n4 , 


 89 xn3  1079 yn 2

[4]
12
6Y 2  X  2
 2643 y 2 n  4  2158x2 n  4 , 


 881xn 3  1079 y n 3

13
6Y 2  4 X  32
 218 y2 n 2  22 y2 n3 , 


 9 yn 2  89 yn1

14
6Y 2  40 X  3200
 2158 y2 n 2  22 y2 n 4 ,

 9 yn3  88 yn1



 2158 y2 n3  218 y2 n 4 ,

 89 yn3  881yn 2



15
6Y 2  4 X  32
[5]
[6]
[7]
[8]
[9]
[10]
3.3 Let
p , q be two non-zero distinct integers such
that p  q  0 .
© 2019, IRJET
Treat
|
[11]
p , q as the generators of the
Impact Factor value: 7.211
|
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ISO 9001:2008 Certified Journal
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Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
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M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha,
Observationson
the
Hyperbola
2
2
Archimedes
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(2013),
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10y  3x  13,
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[12]
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