International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
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ON THE BINARY QUADRATIC DIOPHANTINE EQUATION y2 =272x2 + 16
Dr. G. Sumathi1, S. Gokila2
1Assistant
Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
2PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
---------------------------------------------------------------------***----------------------------------------------------------------------
Abstract – The
binary
y  272 x  16 is considered
properties among the solutions
the
integral solutions of
considerations a few patterns of
observed.
2
2
quadratic
equation
and a few interesting
are presented .Employing
the equation under
Pythagorean triangle are
whose smallest positive integer solutions of
x0  8 , y0  132
~
x0  2 , ~
y0  33
1. INTRODUCTION
whose general solution ~
xn , ~
yn  of (3) is given by,
The binary quadratic equation of the form y 2  Dx 2  1
where D is non –square positive integer has been
studied by various mathematicians for its non-trivial
integral solutions when D takes different integral values
[1-5]. In this context one may also refer [4,10]. These
results have motivated us to search for the integral
solutions of yet another binary quadratic equation
y 2  272 x 2  16 representing a hyperbola .A few
interesting properties among the solution are presented.
Employing the integral solutions of the equation
consideration a few patterns of Pythagorean triangles are
obtained.
1
1
~
xn 
gn , ~
yn  f n
272
2
where,
f n  (33  2 272 ) n1  (33  272) n1
g n  (33  2 272 ) n1  (33  2 272 ) n1
Applying Brahmagupta lemma between x0 , y0  and ~
xn , ~
yn 
the other integer solution of (1) are given by,
1.1 Notations
66
xn1  4 f n 
272
t m,n : Polygonal number of rank n with size m
Pn m : Pyramidal number of rank n with size m
yn1  9 f n 
Prn : Pronic number of rank n
S n : Star number of rank n
Ctm,n : Centered Pyramidal number of rank n with
20
(5)
gn
272 xn1  4 272 f n  66 g n
(6)
Replace n by n  1 in (6),we get
2. METHOD OF ANALYSIS
272 xn 2  4 272 f n1  66 g n1
 4 272 (33 f n  2 272 g n )  66(33g n  2 272 f n )
Consider the binary quadratic Diophantine equation is
272 xn 2  264 272 f n  4354 g n
(1)
Replace
|
40
(4)
gn
Therefore (4) becomes
size m
GFn (k , s) : Generalized Fibonacci sequence of rank n
GLn (k , s) : Generalized Lucas sequence of rank n
© 2019, IRJET
(2)
To obtain the other solutions of (1),Consider pellian
equation is
(3)
y 2  272 x 2  1
The initial solution of pellian equation is
Key Words: Binary, Quadratic, Pyramidal numbers, integral
solutions
y 2  272 x 2  16
( x0 , y0 ) is,
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(7)
n by n  1 in (7),we get
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Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
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7.528 yn3  4 xn1  8708xn3  0
272 xn3  264 272 f n1  4354 g n1
 264 272 (33 f n  2 272 g n )  4354(33g n  2 272 g n )
272 xn3  17420 272 f n  287298g n
8.2 yn 2  1088xn1  66 yn1  0
(8)
For simplicity and clear understanding the other choices of
9.2 yn3  71808xn1  4354 yn1  0
integer solutions are presented below:
10. 66 yn3  1088xn1  2 yn 2  0
272 yn1  66 272 f n  1088g n
(9)
272 yn 2  4354 272 f n  71808g n
(10)
272 yn3  287298 272 f n  4738240 g n
(11)
11. 8 yn1  8708xn 2  132 xn3  0
12. 8 yn 2  132 xn 2  4 xn3  0
13. 8 yn3  4 xn 2  132 xn3  0
14. 132 yn 2  2176xn 2  4 yn1  0
These are representing the non-zero distinct integer
15.132 yn3  143616 xn 2  132 yn1  0
solutions of (1)
16.4 yn3  2176 xn 2  132 yn 2  0
Some few numerical examples are given in the following
table:
17.8708 yn 2  2176 xn3  132 yn1  0
Table 1: Numerical examples
n
xn 1
yn 1
-1
0
1
2
3
4
8
528
34840
2298912
151693352
10009462320
132
8708
574596
37914628
2501790852
165080281604
18.8708 yn3  143616 xn3  4 yn1  0
19.2176 xn3  4 yn 2  132 yn3  0
20. 2176 yn1  143616 yn 2  2176 yn3  0
21.Each of the following expression represents a cubical
integer:
1 66 x3n 4  4354 x3n3  198 xn 2 

 8  13062 xn1

The recurrence satisfied by the values of xn 1 and yn 1 are
respectively
xn3  66 xn 2  xn1  0 n  1,0,1,......
,
yn3  66 yn 2  yn1  0 n  1,0,1,.....
,

1 66 x3n5  287298 x3n3  198 xn1 


528  861894 xn1


1 66 y3n3  1088x3n 3  198 y n1 


4   3264 xn1


1  y3n 4  1088 x3n3  3 y n 2 


2  3264 xn1


1 66 y3n5  4738240 x3n3  198 y n3 


8708  14214720 xn1

3. 4 yn 2  2 xn1  66 xn 2  0

1 4354 x3n5  287298 x3n 4  13062 xn1 


8  861894 xn 2

4. 4 yn3  66 xn1  4354 xn 2  0

1 4354 y3n3  1088 x3n 4  13062 y n1 


132  3264 xn 2


1 4354 y3n 4  71808 x3n 4  13062 y n 2 


4  215424 xn 2

A few interesting relations among the solutions are
presented below:
1. 2 xn3  132 xn 2  2 xn1  0
2. 4 yn1  66 xn1  2 xn 2  0
5. 528 yn1  8708xn1  4 xn3  0
6.528 yn 2  132xn1  132xn3  0
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1 4354 y3n5  4738240 x3n 4  13062 y n1 


132  14214720 xn 2


1 287298 y3n 3  1088x3n 5 



8708  861894 y n1  3264 xn 3 
1 287298 y3n 4  71808 x3n5  861894 y n 2 



132  215424 xn3

1 287298 y3n5  4738240 x3n5  861894 y n3 
 

4  14214720 xn3

1 718088 y3n3  1088 y3n 4  215424 y n1 


2176  3264 y n 2



1 287298 y 4n5  71808 x4n 6


132  1149192 y 2n3  287232 x2n 4  798


1 287298 y 4n 6  4738240 x4n 6


4  1149192 y 2n 4  18952960 x2n 4  24


1 71808 y 4n 4  1088 y 4n5


2176  287232 y 2n 2  4352 y 2n3  13056

 4738240 y 4n 4  1088 y 4n 6 
1 

 18952960 y 2 n 2  4352 y 2 n 4 
143616 
  861696


 4738240 y 4 n 5  71808 y 4n 6

1 

 18952960 y 2 n 3  287232 y 2 n 4 

2176
  13056

23. Each of the following expression represents quintic
integer:
1 4738240 y3n 3  1088 y3n5 



143616  14214720 y n1  3264 y n3 
1 4738240 y3n 4  71808 y3n5 


2176  14214720 y n 2  215424 y n3 


22. Each of the following expression represents biquadratic integer:


1 66 x5n 6  4354 x5n5  330 x3n 4


8  21770 x3n3  660 xn 2  43540 xn1 

66 y5n 7  287298 x5n 5  330 x3n 5 
1 

 1436490 x3n 3  2872980 xn1


528
 660 xn 3


1 66 x4n5  4354 x4n 4  264 x2n3 


8  17416 x2n 2  48


1 66 y5n5  1088 x5n5  330 y3n3 


4  5440 x3n3  10880 xn1  660 y n1 

1 66 y 4n 6  287298 x4n 4  264 x2n 4 


528  1149192 x2n 2  1056


1  y5n 6  1088 x5n5  5440 x3n 3 


2  5 y3n 4  10880 xn1  10 y n 2 

1 66 y 4n 4  1088 x4n 4  264 y 2n 2 


4  4352 x2n 2  24


1  y 4n5  1088 x4n 4  4 y 2n3 


2  4352 x2n 2  12





66 y 4n 6  4738240 x4n 4  264 y 2n 4  
1 


8708 18952960 x2n 2  52248


1 4354 x4n5  287298 x4n5  17416 x2n 4 


8  1149192 x2n3  48



1 4354 y 4n 4  1088 x4n5  17416 y 2n 2 


132  4352 x2n3  792


1 4354 y 4n5  71808 x4n5  17416 y 2n3 


4  287232 x2n3  54


1 4354 y 4n 6  4738240 x4n5  17416 y 2n 4 


132  18952960 x2n3  792


1 287298 y 4n 4  1088 x4n 6  1149192 y 2n 2 


8708  4352 x2n 4  52248

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
|

66 x5n 7  4738240 x5n 5  330 y3n 5 
1 

 23691200 x3n 3  47382400 xn1 
8708 
 660 y n 3

4354 y5n 7  287298 x5n 6  1436490 x3n 4 
1

 21770 x3n 5  2872980 xn 2

8
 43540 xn 3

 4354 y5n 5  1088 x5n 6  5440 x3n 4 
1 

 21770 y3n 3  1088 xn 2

132 
  43540 y n1

 4354 y5n 6  71808 x5n 6  359040 x3n 4 
1

 21770 y3n 4  718080 xn 2

4
  43540 y n 2

4738240 y5n 5  1088 y5n 7 
1 

 5440 y3n 5  23691200 y3n 3 
143616 
 10880 y n 3  47382400 y n1 
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





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 4354 y5n 7  4738240 x5n 6

1 

 23691200 x3n 4  21770 y3n 5 
132 
  47382400 xn 2  43540 y n 3 
287298 y5n 5  1088 x5n 7  5440 x3n 5 
1 

 1436490 y3n 3  10880 xn 3

8708 
 2872980 y n1

1
26112  430848 y2n 2  6528 y2n3 
2176
1 1723392  28429440 y 2n 2 



143616   6528 y 2n 4



25. REMARKABLE OBSERVATIONS
 287298 y5n 6  71808 x5n 7

1 

 1436490 y3n 4  3590940 x3n 5 
132 
  718080 xn 3  2872980 y n 2 
287298 y5n 7  4738240 x5n 7

1

 1436490 y3n 5  23691200 x3n 5 
4
 47382400 xn 3  2872980 y n 3 
I. 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: Hyperbola
S.N
O
Hyperbola
71808 y5n 5  1088 y 5n 6 
1 

 359040 y3n 3  5440 y3n 4 
2176 
  1088 y n 2  718080 y n1 
 4738240 y5n 6  71808 y5n 7

1 

 23691200 y3n 4  359040 y3n 5 
2176 
  47382400 y n 2  718080 y n 3 
1











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X n2  272Yn2
 1115136
3
1
96  396 x2n3  26124x2n 2 
8
1
6336  396 x2n 4  1723788x2n 2 
528
1
48  396 y2n 2  6528x2n 2 
4
1
24  6 y2n3  6528x2n 2 
2
1
104496  396 y2n 4  28429440 x2n 2 
8708
1
96  26124 x2n 4  1723788x2n3 
8
1
1584  26124 y2n 2  6528x2n3 
132
1
48  26124 y2n3  430848x2n3 
4
1
1584  26124 y2n 4  28429440 x2n3 
132
1
104496  1723788 y2n 2  6528x2n 4 
8708
1
1584  1723788 y2n3  430848x2n 4 
132
1
48  1723788 y2n 4  28429440 x2n 4 
4
|
X n2  4352Yn2
 256
24. Each of the following expression represents Nasty
number:

1 26112  28429440 y 2n3 


2176  430848 y 2n 2

X n2  272Yn2
 64
 66 xn 2  4354 xn1 , 


 66 xn1  xn 2

 17420 xn1  4 xn3 , 


 66 xn3  287298 xn1 
 66 y n1  1088 xn1 , 


 66 xn1  4 y n1

4
4356 X n2  272Yn2  y n 2  1088 x n1 , 


 69696
 4354 x n1  4 y n 2 
5
X n2  272Yn2
 303317056
6
X n2  272Yn2
 256
7
X n2  272Yn2
 69696
8
X n2  272Yn2
 64
9
X n2  272Yn2
 69696
|
( X n ,Yn )
 66 y n 3  4738240 x n1 , 


 287298 x n1  4 y n 3 
 4354 xn 3  287298 xn 2 , 


 17420 n 2  264 xn 3

 4354 y n1  1088 x n 2 , 


 66 x n 2  264 y n1

 4354 y n 2  71808 x n 2 , 


 4354 x n 2  264 y n 2 
 4354 yn  3  4738240 xn  2 , 


 287298 xn  2  264 yn  3 
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Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
X n2  272Yn2
 303317056
11
X n2  272Yn2
 69696
12
X n2  272Yn2
13
X  272Y
2
n
 18939904
14
X n2  272Yn2
 82502221824
15
 287298 y n1  1088 x n 3 , 


 66 x n 3  17420 y n1

8
 287298 y n 2  71808 x n 3 , 


 4354 x n 3  17420 y n 2 
9
X n2  272Yn2
132 X n  272Yn2
 34848
10
8708 X n  272Yn2
 151658528
 71808 yn 1  1088 y n  2 , 


 66 y n  2  4354 yn 1

11
 4738240 yn 1  1088 yn  3 , 


 66 yn  3  287298 yn 1

12
132 X n  272Yn2
 34848
4 X n  272Yn2
 32
 4738240 y n 2  71808 y n 3 , 


 4354 y n 3  287298 y n 2 
 18939904
4 X n  272Yn2
 32
 287298 yn  3  4738240 xn  3 , 


 287298 xn  3  17420 yn  3 
 64
2
n
www.irjet.net
13
2176 X n  272 y n2
 9469952
II. Employing linear combination among the solutions of (1),
one may be generate integer solutions for other choices of
parabolas which are presented in table 3 below:
14
Table 3: Parabola
S.
NO
Parabola
 X n ,Yn 
1
X n  544Yn2
 16  66 x 2n 3  4354 x 2n 2 , 


 66 x n1  x n 2

 16
2
528 X n  272Yn2
X n  68Yn2
8
4
8712 X n  272Y
2
n
 34848
5
2177 X n  63Yn2
 37914632
6
8 X n  272Yn2
 128
7
132 X n  272Yn2
 34848
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|
 17416  287298 y 2 n 2 


  1088 x 2n 4 ,

 66 x  17420 y

n 3
n 1 

 264  287298 y 2 n 3



  71808 x 2 n 4 ,

 4354 x  17420 y 
n 3
n 2 

 8  287298 y 2 n 4




4738240
x
,


2n 4
 287298 x  17420 y 
n 3
n 3 

 4352  71808 y n 2 


  1088 y 2 n 3 ,

 66 y


4354
y
n 2
n 1 

143616 X n  272Yn2  287232  4738240 y 2 n 2 


 41251110912
  1088 y 2 n 4 ,

 66 y  287298 y

n 3
n 1


2
2176 X n  272Yn
 4352  4738240 y 2 n 3



 9469952
  71808 y 2 n 4 ,

 4354 y

2 n  4  287298 y n  3 

Table 4: Straight line
 4  y 2n 3  1088 x 2n 2 , 


 4354 x n1  4 y n 2

 17416  66 y 2n 4




4738240
x
,


n 2
 287298 x  4 y 
n 1
n 3 

 16  4354 x 2 n 4



  287298 x 2 n 3 ,

 17420 x


264
x
n 2
n 3 

 264  4354 y 2 n 2 


  1088 x 2 n 3 ,

 66 x

n  2  264 y n 1 

Impact Factor value: 7.211
 264  4354 y 2n 4



  4738240 x 2n 3 ,

 287298 x

n  2  264 y n  3 

III Employing linear combinations among the solutions of
(1),one may generate integer solutions for other choices of
straight line which are presented in the table below:
 1056  66 x 2 n 4 


  287298 x 2n 2 , 
 17420 x  4 x 
n 1
n 3 

 8  66 y 2n 2  1088 x 2n 2 , 


 66 x n1  4 y n1

 557568
3
15
 8  4354 y 2 n 3



  71808 x 2 n 3 ,

 4354 x

n  2  264 y n  2 

|
S.NO
Straight
line
1
Y  66 X
 66 x n 2  4354 x n1 , 


 66 x n 3  287298 x n1 
2
Y  2X
 66 y n1  1088 x n1 , 


 66 x n 2  4354 x n1 
3
Y  4X
 y n 2  1088 x n1 , 


 66 x n 2  4354 x n1 
4
YX
 4354 x n 3  287298 x n 2 , 


 66 x n 2  4354 x n1

5
Y  2X
 287298 y n 3  4738240 x n 3 , 


 66 x n 2  4354 x n1

X , Y 
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6
Y  272 X
 66 x n 2  4354 x n1 , 


 71808 y n1  1088 y n 2 
7
Y  17952 X
 4354 x n 3  287298 x n 2 , 


 4738240 y n1  1088 y n 3 
8
Y  272 X
 66 x n 2  4354 x n1 ,



4738240
y

71808
y
n 2
n 3 

9
Y  132 X
 66 y n1  1088 x n1 , 


 66 x n 3 ,287238 x n1 
10
Y  2X
 y n 2  1088 x n1 , 


 66 y n1  1088 x n1 
11
Y  66 X
 4354 x n 3  287298 x n 2 , 


 66 x n 3  287298 x n1

12
Y  4X
 4354 y n1  1088 x n 2 , 


 66 x n 3  287298 x n1 
13
YX
 4354 y n 3  4738240 x n 2 , 


 4354 y n1  1088 x n 2

14
Y  33 X
 287298 y n 3  4738240 x n 3 , 


 287298 y n 2  71808 x n 3 
V. Employing the solutions in terms of special integers
sequence namely, generalized Fibonacci sequence , GFn(k,s)
and the generalized Lucas sequence GLn(k,s) are exhibited
below:
xn1  4GLn1 (66,1)  264GFn1 (66,1)
yn1  66GLn1 (66,1)  4352GFn1 (66,1)
VI. Employing the solutions of (1),each of the following
among the special polygonal,pyramidal,star numbers,pronic
numbers ,centered pyramidal number is a congruent to
under modulo 16.
2
i.
ii.
5
 12 p 5y 

  272 p x
 t 3, x
 s y 1  1 



 3 p 3y  2

 t 3, y  2





2
2
3 2

  272 6 p x 
 prx 1 




2
iii.
 4 pt y 3 
 4 

  272 6 p y 1 
 p3

 t 3, 2 y 1 


 y 3 
iv.
 2 p y 1 



  272 4 pt x 3 
 p3 
 t 4, y 1 
 x 3 


v.
IV. Consider p  xn 1  yn 1 , q  xn 1
3 2
 6 p 4y 1 

  272 3 p x 
 t 3, x 1 
 t 3, 2 y 1 




Note that p  q  0 , treat p, q as the generators of the
Pythagorean triangle T ( X , Y , Z )
vi.
 4 p 5y 
 3

  272 3 p x  2
 t 3, y  2
 ct 4, y 1 



vii.
 3 p 3y 2 
 5

  272 p x
t
t

 3, x
 3, y  2 
15
2
 287238 y n1  1088 x n 3 , 


 66 y n 3  4738240 x n1 
YX
2
where X  2 pq, Y  p  q , Z  p  q , p  q  0
2
2
2
2
2
2
2
2
It is odserved that T ( X , Y , Z ) is satisfied by the following
relations:








2
2
2
X  136Y  135Z  16
5 
 36 p 3y  2 


  272 12 p x 
 s y 1  1 
 s x 1  1 




ii.
2A
 xn 1 yn 1
P
2
5
 6 p 2y 1 

  272 4 p x 
 ct 4, x1 
 t 3, 2 y 1 




iii.
4A
X
Y  Z
P
iv.
3(Z  Y ) is a Nasty number
CONCLUSION
v.
4A
 Y is written as the sum of two
P
squares
In this dissertation , we have presented infinitely many
integer solutions of the binary quadratic Diophantine
equation are rich in variety, one may search for the other
choices of binary quadratic Diophantine equation and
determine their integral solution along with suitable
properties.
i.
viii.
2
2
ix.
2
x.
X
where A, P represents the area and perimeter
of T ( X , Y , Z )
© 2019, IRJET
|
Impact Factor value: 7.211
|
2
5
 4 p 5y 

  272 6 p x 
 ct 6, x 1 
 ct 4, y 1 




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e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
www.irjet.net
REFERENCES
[1]
Dickson.L.E, History of Theory of Numbers and
Diophantine Analysis, Dover Publications, New York
2005.
[2]
Mordell.L.J, Diophantine Equations, Academic Press,
New York 1970.
[3]
Carmicheal.R.D, The theory of Numbers and Diophantine
Analysis,Dover Publications,New York 1959.
[4]
Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation
on the hyperbola x 2  19 y 2  3 ”, Scholars Journal of
the Engineering and Technology 2014;2(2A);152-155.
[5]
Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation
on the hyperbola y 2  26 x 2  1 , Bessel Math,Vol 4,
Issue(1),2014;21-25.
[6]
On
special
families
of
Hyperbola, x 2  (4k 2  k ) y 2  a 2 , a  1 The International
Journal of
Science
and
Technology ,Vol
2,Issue(3),Pg.No.94-97, March 2014.
[7]
Dr.G.Sumathi “Observations on the hyperbola
y 2  182 x 2  14 ”, Journal of Mathematics and
Informatics,Vol 11, 73-81,2017
[8]
Dr.G.Sumathi “Observations on the pell equation
y 2  14 x 2  4 ”, International Journal of Creative
Research Thoughts,Vol 6,Issue 1,Pp1074-1084,March
2018
[9]
Dr.G.Sumathi “Observation
on the Hyperbola
y 2  312 x 2  1 ” International Journal of Mathematics
Trends and Technology,Vol 50,Issue 4,31-34,Oct 2017
[10]
Dr.G.Sumathi
“Observation on the Hyperbola
2
2
y  150 x  16 ” International Journal of recent
Trends in Engineering and Research ,Vol 3,Issue
9,Pp198-206,Sep 2017
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