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 Pellian Like Equation 5 x 2  7 y 2  8
S. Vidhayalakshmi1, A. Sathya2, S. Nivetha3
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 Scholor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002, Tamil Nadu, India.
----------------------------------------------------------------------------***------------------------------------------------------------------2Assisant
Abstract – The binary quadratic equation represented by the
X 2  35T 2  4
pellian like equation 5x  7 y  8 is analyzed for its 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.
2
2
whose smallest positive integer solution is
X 0  12
X 2  35T 2  1
1. INTRODUCTION
~ ~
( X 0 , To )  (6,1)
The general solution of (4) is given by
~
Tn 
This communication concerns with the problem of obtaining
non-zero distinct integer solutions to the binary quadratic

n1  6  35 n1
n 1
n 1
gn  6  35   6  35 
Applying Brahmagupta lemma between ( X 0 , T0 ) and
~ ~
( X n , Tn ) the other integer solutions of (3) are given by
The Diophantine Equation representing the binary quadratic
equation to be solved for its non-zero distinct integral
solution is
X n 1  6 f n  35 g n
Tn 1  f n 
(1)
(2)
35
xn 1  13 f n 
From (1) and (2), we have
Impact Factor value: 7.211
6
gn





(5)
From (2), (4) and (5) the values of x and y satisfying (1) are
given by
Consider the linear transformations
|
~
1
gn , X n  fn
2
f n  6  35
2. Method of Analysis
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1
2 35
where
equation given by 5x 2  7 y 2  8 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.
y  X  5T
(4)
whose smallest positive integer solution is
The binary quadratic Diophantine equation 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].
x  X  7T
T0  2
To obtain the other solutions of (3), consider the pellian
equation is
Key Words: Binary quadratic, Hyperbola, Parabola, Pell
equation, Integer solutions.
5x 2  7 y 2  8
(3)
|
77
35
gn
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65
yn 1  11 f n 
35
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
gn


The recurrence relation satisfied by the solution x and
y are given by


xn3 12 xn 2  xn1  0

yn3 12 yn 2  yn1  0



Some numerical examples of xn and yn satisfying (1) are
given in the Table :1 below
2. Each of the following expressions represents a nasty
number:
Table: 1 Numerical Examples
n
xn
yn
0
26
22
1
310
262
2
3694
3122
3
44018
37202
4
524522
443302
5
6250246
5282422
6
74478430
62945762
7
887490914
750066722
From the above table, we observe some interesting relations
among the solutions which are presented below:

xn and yn values are always even.
1. Relation among the solutions are given below:











xn 2  7 yn1  6 xn1  0
xn3  84 yn1  71xn1  0
yn 2  6 yn1  5xn1  0
yn3  71yn1  60 xn1  0
xn 3  12 xn  2  xn 1  0
xn1  6 xn 2  7 yn 2  0
xn3  xn1  14 yn 2  0
7 yn3  71xn 2  6xn1  0
xn1  71xn3  84 yn3  0
xn3  6 xn 2  7 yn 2  0
71yn  2  5xn 1  6 yn 3  0
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7 yn1  71xn2  6 xn3  0
yn 1  5xn  2  6 yn  2  0
yn1  10 xn 2  yn3  0
yn 2  5xn3  6 yn3  0
71yn 2  6 yn1  5xn3  0
yn1  60xn3  71yn3  0
yn1  12 yn 2  yn3  0
7 yn3  6 xn3  xn 2  0
yn3  6 yn 2  5xn 2  0
|

377 y2n  2  65x2n  2  4

3
77 x2n  3  917 x2n  2  28
7

1
77 x2n  4  10927 x2n  2  336
28

1
77 y2n  3  775x2n  2  24
2

3
77 y2n  4  9235x2n  2  284
71

1
917 y2n  2  65x2n  3  24
2

3
10927 y2n  2  65x2n  4  284
71

3155 y2n  2  13 y2n  3  4

1
1847 y2n  2  13 y2n  4  48
4

3131x2n  4  1561x2n  3  4

3917 y2n  3  775x2n  3  4

1
917 y2n  4  9235x2n  3  24
2

1
10927 y2n  3  775x2n  4  24
2
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
310927 y2n  4  9235x2n  4  4

31847 y2n  3  155 y2n  4  4
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3. Each of the following expressions represents a cubical
integer:

1 10927 y3n  4  775 x3n  5  32781yn  2 


12   2325 xn  3


1 10927 y3n  5  9235 x3n  5  32781yn  3 


2   27705 xn  3


1 1847 y3n  4  155 y3n  5  5541yn  2 


2   465 yn  3


1
77 y3n  3  65x3n  3  231yn 1  195xn 1 
2

1  77 x3n  4  917 x3n  3  231xn  2 


14   2751xn 1


1  77 x3n  5  10927 x3n  3  231xn  3 


168   32781xn 1


1  77 y4n  4  65 x4n  4  308 y2n  2 


2   260 x2n  2  12


1  77 y3n  4  775 x3n  3  231yn  2 


12   2325 xn 1


1  77 y4n  5  917 x4n  4  308 x2n  3 


14   3668 x2n  2  84


1  77 y3n  5  9235 x3n  3  231yn  3 


142   27705 xn 1


1  77 y4n  6  10927 x4n  4  308 x2n  4 


168   43708 x2n  2  1008


1  917 y3n  3  65 x3n  4  2751yn 1 


12   195 xn  2


1  77 y4n  5  775 x4n  4  308 y2n  3 


12   3100 x2n  2  72


1 10927 y3n  3  65 x3n  5  32781yn 1 


142   195 xn  3


1  77 y4n  6  9235 x4n  4  308 y2n  4 


142   36940 x2n  2  852


1
155 y3n  3  13 y3n  4  465 yn 1  39 yn  2 
2

1  917 y4n  4  65 x4n  5  3668 y2n  2 


12   260 x2n  3  72


1
1847 y3n  3  13 y3n  5  5541yn 1  39 yn  3 
24

1 10927 y4n  4  43708 y2n  2 


142   65 x4n  6  260 x2n  4  852 

1
131x3n  5  1561x3n  4  393xn  3  4683xn  2 
2

1 155 y4n  4  13x4n  5  620 y2n  2 


2   52 y2n  3  12


1  917 y3n  4  775 x3n  4  2751yn  2 


2   2325 xn  2


1 1847 y4n  4  13 y4n  6  7388 y2n  2 


24   52 y2n  4  144


1  917 y3n  5  9235 x3n  4  2751yn  3 


12   27705 xn  2


1  917 x4n  6  10927 x4n  5  3668 x2n  4 


14   43708 x2n  3  84

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4. Each of the following expressions represents a
biquadratic integer:
|
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
1  917 y4n  5  775 x4n  5  3668 y2n  3 


2   3100 x2n  3  12


1 1847 y5n  5  13 y5n  7  9235 y3n  3 


24   65 y3n  5  18470 yn 1  130 yn  3 

1  917 y4n  6  9235 x4n  5  3668 y2n  4 


12   36940 x2n  3  72



1  917 x5n  7  10927 x5n  6  4585 x3n  5


14   54635 x3n  4  9170 xn  3  109270 xn  2 

1 10927 y4n  5  775 x4n  6  43708 y2n  3 


12   3100 x2n  4  72


1  917 y5n  6  775 x5n  6  4585 y3n  4 


2   3875 x3n  4  9170 yn  2  7750 xn  2 

1 10927 y4n  6  9235 x4n  6  43708 y2n  4 


2   36940 x2n  4  12



1  917 y5n  7  9235 x5n  6  4585 y3n  5


12   46175 x3n  4  9170 yn  3  923500 xn  2 

1 1847 y4n  5  155 y4n  6  7388 y2n  3 


2   620 y2n  4  12


1 10927 y5n  6  775 x5n  7  54635 y3n  4 


12   3875 x3n  5  109270 yn  2  7750 xn  3 

1 10927 y5n  7  9235 x5n  7  54635 y3n  5 


2   46175 x3n  5  109270 yn  3  92350 xn  3 

1 1847 y5n  6  155 y5n  7  9235 y3n  4 


2   775 y3n  5  18470 yn  2  1550 yn  3 
5. Each of the following expressions represents a quintic
integer:

1  77 y5n  5  65 x5n  5  385 y3n  3  325 x3n  3 


2   770 yn 1  650 xn 1



1  77 x5n  6  917 x5n  5  385 x3n  4


14   4585 x3n  3  770 xn  2  9170 xn 1 


1  77 x5n  7  10927 x5n  5  385 y3n  5


168   54635 x3n  3  770 xn  3  109270 xn 1 

1  77 y5n  6  775 x5n  5  385 y3n  4 


12   3875 x3n  3  770 yn  2  7750 xn 1 


1  77 y5n  7  9235 x5n  5  385 y3n  5


142   46175 x3n  3  770 yn  3  92350 xn 1 



REMARKABLE OBSERVATIONS
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 Hyperbolas
S.NO
Hyperbola
1
Y 2  35 X 2  16
2
Y 2  35 X 2  784
1  917 y5n  5  65 x5n  6  4585 y3n  3 


12   325 x3n  4  9170 yn 1  650 xn  2 
3
Y 2  35 X 2  112896
1 10927 y5n  5  65 x5n  7  54635 y3n  3 


142   325 x3n  5  109270 yn 1  650 xn  3 
4
Y 2  35 X 2  576
5
Y 2  35 X 2  80656
1 155 y5n  5  13 y5n  6  775 y3n  3 


2   65 y3n  4  1550 yn 1  130 yn  2 
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(X,Y)
11xn 1  13 yn 1 ,

 77 yn 1  65 xn 1



155 xn 1  13xn  2 , 


 77 xn  2  917 xn 1 
1847 xn 1  13xn 3 ,

 77 xn 3  10927 xn 1



131xn 1  13 yn 1 , 


 77 yn  2  775 xn 1 
1561xn 1  13 yn  3 ,

 77 yn 3  9235 xn 1
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6
Y 2  35 X 2  576
11xn  2  155 yn 1 , 


 917 yn 1  65 xn  2 
4
12Y  35 X 2  576
7
Y 2  35 X 2  80656
11xn 3  1847 yn 1 , 


 10927 yn 1  65 xn 3 
5

142Y  35 X 2  80656 1561xn 1  13 yn 3 ,

 77 y

9235
x

284
2n  4
2n  2


8
Y 2  35 X 2  400
11yn  2  131yn 1 ,

 775 yn 1  65 yn  2



6
12Y  35 X 2  576
9
Y 2  35 X 2  57600
11yn 3  1561yn 1 , 


 9235 yn 1  65 yn 3 
7

142Y  35 X 2  80656 11xn 3  1847 yn 1 ,

 10927 y
2n  2  65 x2n  4  284 

10
Y 2  35 X 2  784
1847 xn  2  155 xn 3 , 


 917 xn  3  10927 xn  2 
8
2Y  7 X 2  80
131xn  2  155 yn  2 , 


 917 yn  2  775 xn  2 
11yn  2  131yn 1 ,



 775 y2n  2  65 y2n 3  20 
9
120Y  7 X 2  11520
11yn 3  1561yn 1 ,



1847 y2n  2  13 y2n  4  48 
11
12
Y 2  35 X 2  16
Y 2  35 X 2  576
1561xn  2  155 yn 3 ,

 917 yn 3  9235 xn  2



13
Y 2  35 X 2  576
131xn 3  1847 yn  2 , 


 10927 yn  2  775 xn 3 
14
Y 2  35 X 2  16
1561xn 3  1847 yn 3 , 


 10927 yn 3  9235 xn 3 
15
Y 2  35 X 2  400
131yn 3  1561yn  2 , 


 9235 yn  2  775 yn 3 
131xn 1  13 yn 1 ,



77
y

775
x

24
2n  3
2n  2


11xn  2  155 yn 1 ,



 917 y2n  2  65 x2n  3  24 
10
2Y  5 X 2  112
1847 xn  2  155 xn  3 ,



 917 x2n  4  10927 x2n  3  28 
11
2Y  35 X 2  16
131xn  2  155 yn  2 ,



917
y

775
x

4
2n  3
2n  3


12
12Y  35 X 2  576
1561xn  2  155 yn 3 ,



917
y

9235
x

24
2n  4
2n  3


II. Employing linear combinations among the solutions of (1),
one may generate integer solutions for other choices of
parabolas which are presented in Table: 3 below:
13
12Y  35 X 2  576
131xn 3  1847 yn  2 ,



 10927 y2n 3  775 x2n  4  24 
Table: 3 Parabolas
14
2Y  35 X 2  16
1561xn 3  1847 yn 3 ,



10927 y2n  4  9235 x2n  4  4 
15
2Y  7 X 2  80
131yn  3  1561yn  2 ,



 9235 y2n 3  775 y2n  4  20 
S.NO
Parabola
(X,Y)
1
2Y  35 X 2  16
11xn 1  13 yn 1 ,



77
y

65
x

4
2n  2
2n  2


2
2Y  5 X 2  112
155 xn 1  13xn  2 ,



77
x

917
x

28
2n  3
2n  2


3
24Y  5 X  16128
1847 xn 1  13xn  3 ,



 77 x2n  4  10927 x2n  2  336 
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III. Let p, q be two non-zero distinct integers such that
p  q  0 . Treat p, q as the generators of the Pythagorean
triangle T ( ,  ,  )
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where   2 pq ,   p 2  q 2 ,   p 2  q 2 , p  q  0
Taking p  xn 1  yn 1 , q  yn 1 , it is observed
T ( ,  ,  ) is satisfied by the following relations:

3  7  10  16

4
A
   
p

2
A
 xn 1 yn 1
p
[7]
S. Vidhyalakshmi,et al., Observations on the hyperbola
ax 2  a  1y 2  3a  1 , Discovery, 4(10) (2013) 22-24.
[8]
M.A. gopalan,et al., Integral points on the hyperbola
2
 ay 2  4ak  1  2k 2 , a, k  0 , Indian Journal of
science, 1(2) (2012) 125-126.
[9]
M.A. Gopalan and S. Vidhyalakshmi and A. Kavitha, on
the integer solutions of binary quadratic
equation, x 2  4k 2  1y 2  4t , k , t  0 , BOMSR, 2(2014)
42-46.
[10]
T.R. Usha Rani and K. Ambika, Observation on the NonHomogeneous
Binary
Quadratic
Diophantine
Equation 5x 2  6 y 2  5 , Journal of Mathematics and
Informatics, Vol-10,Dec (2017), 67-74.
[11]
M.A. Gopalan and Sharadha Kumar, On the Hyperbola
2 x 2  3 y 2  23 , Journal of Mathematics and Informatics,
vol-10,Dec (2017),1-9.
[12]
M.A.Gopalan, S.Vidhyalakshmi and A. Kavitha,
Observationson
the
Hyperbola
10y 2  3x 2  13, Archimedes J. Math., 3(1) (2013), 31-34.
[13]
M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, On the
integral solutions of the binary quadratic equation
x 2  15y 2  11t , Scholars Journal of Engineering and
Technology, 2(2A) (2014), 156-158.
[14]
Shreemathi Adiga, N. Anusheela and M.A. Gopalan,
Observations on the Positive Pell Equation y 2  20x 2  1 ,
International Journal of Pure and Applied Mathematics,
120(6) (2018), 11813-11825
that
Where A, P represent the area and perimeter of T ( ,  ,  ) .
3. CONCLUSION
In this paper, we have presented infinitely many integer
solutions for the Pellian like equation 5x 2  7 y 2  8 . 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.
REFERENCES
[1]
R.D. Carmichael,The Theory of Numbers and
Diophantine Analysis, Dover Publications, New York
(1950).
[2]
L.E. Disckson, History of Theory of Numbers, vol II,
Chelsea publishing Co., New York (1952).
[3]
L.J. Mordell, Diophantine Equations, Academic press,
London (1969).
[4]
M.A. Gopalan and R. Anbuselvi, Integral solutions of
4ay 2  a  1x 2  3a  1 , Acta Ciencia Indica, XXXIV(1)
(2008) 291-295.
[5]
M.A. Gopalan,et al., Integral points on the hyperbola
a  2x2  ay 2  4ak 1  2k 2 , a, k  0 , Indian Journal of
Science, 1(2) (2012) 125-126.
[6]
M.A. Gopalan, S. Devibala and R. Vidhylakshmi, Integral
points on the hyperbola 2 x 2  3 y 2  5 , Amearican Jornal
of Applied Mathematics and Mathematical Sciences,
1(2012) 1-4.
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a  2x
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