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On the Positive Pell Equation y 2  35 x 2  29
M.A. Gopalan1, T. Mahalakshmi2, K. sevvanthi3
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.
----------------------------------------------------------------------***--------------------------------------------------------------------2Assistant
Abstract – The binary quadratic Diophantine equation represented by the positive Pellian
y 2  35x 2  29 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.
Key Words: Binary quadratic, Hyperbola, Parabola, Pell equation, Integer solutions
1. INTRODUCTION
The binary quadratic equation of the form y 2  Dx 2  1 where D is non-square positive integer, has been selected by
various mathematicians for its non-trivial integer solutions when D takes different integral values [1-4]. For an extensive
review of various problems, one may refer [5-17]. In this communication, yet another an interesting equation given by
y 2  35x 2  29 is considered and infinitely many integer solutions are obtained. A few interesting properties among the
solutions are presented.
2. METHOD OF ANALYSIS
The Positive pell equation representing hyperbola under consideration is
y 2  35x 2  29
(1)
The smallest positive integer solutions of (1) are
x0  1 , y0  8
To obtain the other solutions of (1), consider the Pell equation
y 2  35x 2  1
(2)
whose initial solution is
~
x0  1, ~
y0  6
xn , ~
yn ) of (2) is given by
The general Solution ( ~
~
xn 
1
1
gn , ~
yn  f n
2
2 35
where
f n  (6  35 ) n1  (6  35 ) n1
g n  (6  35 ) n1  (6  35 ) n1 , n  1, 0 ,1,...
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Applying Brahmagupta lemma between x0 , y0  and ~
xn , ~
yn  , the other integer solutions of (1) are given by
xn1 
1
4
fn 
gn
2
35
yn1  4 f n 
35
gn
2
The recurrence relations satisfied by the solutions 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 y n satisfying (1) are given in the Table: 1 below:
Table: 1 Examples
n
xn
yn
0
1
8
1
14
83
2
167
988
3
1990
11773
4
23713
140288
5
282566
1671683
6
3367079
19919908
From the above table, we observe some interesting relations among the solutions which are presented below:

Both xn and y n values are alternatively odd and even.
2.1. Relations among the solutions are given below:

xn 2  yn1  6 xn1

xn3  12 yn1  71xn1

y n 2  6 y n1  35xn1

yn3  71yn1  420 xn1

yn 2  6 xn 2  xn1

yn3  71xn 2  6 xn1
 12 xn 2  xn3  xn1
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
xn3  2 yn 2  xn1

6 yn3  71yn 2  35xn1

71xn3  12 yn3  xn1

6 xn3  yn1  71xn 2

6 yn1  71yn 2  35xn3

6 xn 2  xn3  yn 2
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
yn1  71yn3  420 xn3

xn 2  6 xn3  yn3

yn2  6 yn3  35xn3

6 yn2  yn1  35xn2

yn3  yn1  70 xn 2

yn3  12 yn 2  yn1

yn3  6 yn 2  35xn2
2.2. Each of the following expression is a nasty number:















6
58  16 y2n2  70 x2n2 
29
6
58  16 x2n3  166 x2n2 
29
1
348  8x2n4  988x2n2 
29
1
348  16 y2n3  980 x2n2 
29
6
4118  16 y2n4  11690 x2n2 
2059
1
348  166 y2n2  70 x2n3 
29
6
4118  1976 y2n2  70x2n4 
2059
6
58  28 y2n2  2 y2n3 
29
1
348  167 y2n2  y2n4 
29
6
58  166 x2n 4  1976 x2n3 
29
6
58  166 y2n3  980 x2n3 
29
1
348  166 y2n4  11690x2n3 
29
1
348  1976 y2n3  980 x2n4 
29
6
58  1976 y2n4  11690x2n4 
29
6
58  334 y2n3  28 y2n4 
29
2.3. Each of the following expressions is a cubical integer:
1
16 y3n3  70 x3n3  48 yn1  210 xn1 

29
1
 16 x3n 4  166 x3n3  48 xn 2  498 xn1 
29
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1
4 x3n5  494 x3n3  12 xn3  1482 xn1 
87
1
 8 y3n 4  490 x3n3  24 yn 2  1470 xn1 
87
1
16 y3n5  11690 x3n3  48 yn3  35070 xn1 

2059
1
 83 y3n3  35 x3n 4  249 yn1  105 xn 2 
87
1
1976 y3n3  70 x3n5  5928 yn1  210 xn3 

2059
1
 28 y3n3  2 y3n 4  84 yn1  6 yn 2 
29
1
167 y3n3  y3n5  501yn1  3 yn3 

174
1
166 x3n5  1976 x3n4  498xn3  5928xn2 

29
1
166 y3n4  980 x3n4  498 yn2  2940 xn2 

29
1
 83 y3n5  5845 x3n 4  249 yn3  17535 xn 2 
87
1
 988 y3n 4\  490 x3n5  2964 yn 2  1470 xn3 
87
1
1976 y3n5\  11690 x3n5  5928 yn3  35070 xn3 

29
1
 334 y3n 4\  28 y3n5  1002 yn 2  84 yn3 
29

2.4. Each of the following expressions is a biquadratic integer:
1
16 y4n4  70 x4n4  64 y2n2  280 x2n2  174

29
1
 16 x4 n5  166 x4 n 4  64 x2 n3 664 x2 n 2  174
29
1
 4 x4 n6  494 x4 n 4  16 x2n 4  1976 x2 n 2  522
87
1
 8 y4 n5  490 x4 n 4  32 y2 n3  1951x2 n 2  522
87
1
16 y4n6  11690 x4n4  64 y2n4  46760 x2n2  12354

2059
1
 83 y4 n 4  35x4 n5  332 y2 n 2  140 x2n3  522
87
1
1976 y4n4  70x4n6  7904 y2n2  280x2n4  12354

2059
1
 28 y4 n 4  2 y4 n5  112 y2 n 2  8 y2 n3  174
29
1
167 y4n4  y4n6  668 y2n2  4 y2n4  1044

174
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1
166 x4n6  1976 x4n5  664 x2n4  7904 x2n3  174
29
1
166 y4n5  980 x4n5  664 y2n3  3920 x2n3  174

29
1
 83 y4 n 6  5845x4 n5  332 y2 n 4  23380 x2n3  522
87
1
 988 y4 n5  490 x4 n6  3952 y2 n3  1960 x2n 4  522
87
1
1976 y4n6  11690x4n6  7904 y2n4  46760x2n4  174

29
1
 334 y4 n5  28 y4 n 6  1336 y2 n3  112 y2 n 4  174
29

2.5. Each of the following expression is a quintic integer:
1
16 y5n5  70 x5n5  80 y3n3  350 x3n3  160 yn1  700xn1 
29
1
16 x5n6  166 x5n5  80 x3n4  830 x3n3  160 xn2  1660 xn1 

29
1
 4 x5n 7  494 x5n5  20 x3n5  2470 x3n3  40 xn3  4940 xn1 
87
1
 8 y5n 6  490 x5n5  40 y3n 4  2450 x3n3  80 yn 2  4900 xn1 
87
1
16 y5n7  11690 x5n5  80 y3n5  58450 x3n3  160 yn3  116900xn1 

2059
1
 83 y5n5  35x5n 6  415 y3n3  175x3n 4  830 yn1  350 xn 2 
87
1
1976 y5n5  70 x5n7  9880 y3n3  350 y3n5  19760 yn1  700xn3 

2059
1
 28 y5n5  2 y5n6  140 y3n3  10 y3n 4  280 yn1  20 yn 2 
29
1
167 y5n5  y5n7  835 y3n3  5 y3n5  1670 yn1  10 yn3 

174
1
 166 x5n 7  1976 x5n 6  830 x3n5  9880 x3n 4  1660 xn3  19760 xn 2 
29
1
 166 y5n 6  980 x5n 6  830 y3n 4  4900 x3n 4  1660 yn 2  9880 xn 2 
29
1
 83 y5n 7  5840 x5n 6  415 y3n5  29220 x3n 4  830 yn3  58450 xn 2 
87
1
 988 y5n 6  490 x5n 7  4940 y3n 4  2450 x3n5  9880 yn 2  4900 xn3 
87

1
1976 y5n7  11690x5n7  9880 y3n5  58450x3n5  19760 yn3  116900xn3 
29
1
 334 y5n 6  28 y5n 7  1670 y3n 4  140 y3n5  3340 yn 2  280 yn3 
29

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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
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S.NO
Hyperbolas
(Y,X)
1
Y 2  35 X 2  117740
 16 yn1  70 xn1 ,



 16 35 x  2 35 y 
n1
n1 

2
Y 2  35 X 2  117740
16 xn 2  166 xn1 ,



 28 35 x  2 35 x 
n1
n 2 

3
Y 2  35 X 2  16954560
 16 xn3  1976 xn1 ,



 334 35 x  2 35 x 
n1
n 3 

4
Y 2  35 X 2  4238640
 16 yn 2  980 xn1 ,



 166 35 x  2 35 y 
n1
n 2 

5
Y 2  35 X 2  593527340
 16 yn3  11690 xn1 ,



 1976 35 x  2 35 y 
n1
n 3 

6
Y 2  35 X 2  4238640
 166 yn1  70 xn 2 ,



 16 35 x  28 35 y 
n 2
n1 

7
Y 2  35 X 2  593527340
1976 yn1  70 xn3 ,



16 35 x  334 35 y 
n 3
n1 

8
35Y 2  X 2  4120900
 28 yn1  2 yn 2 ,



 16 35 y  166 35 y 
n 2
n1 

9
35Y 2  X 2  593409600
 334 yn1  2 yn3 ,



 16 35 y  1976 35 y 
n

3
n

1


10
Y 2  35 X 2  117740
 166 xn3  1976 xn 2 ,



 334 35 x  28 35 x 
n 2
n 3 

11
Y 2  35 X 2  117740
 166 yn 2  980 xn 2 ,



 166 35 x  28 35 y 
n 2
n 2 

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12
Y 2  35 X 2  4238640
 166 yn3  11690 xn 2 ,



 1976 35 x  28 35 y 
n 2
n 3 

13
Y 2  35 X 2  4238640
 1976 yn 2  980 xn3 ,



 166 35 x  334 35 y 
n 3
n 2 

14
Y 2  35 X 2  117740
 1976 yn3  11690 xn3 ,



 1976 35 x  334 35 y 
n 3
n 3 

15
Y 2  35 X 2  144231500
 11690 yn 2  980 yn3 ,



 166 35 y  1976 35 y 
n 3
n 2 

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:
Table: 3 Parabolas
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S.NO
Parabolas
(Y,X)
1
29Y  X 2  3364
 16 y2 n 2  70 x2 n 2  58, 


 16 35 x  2 35 y 
n1
n1 

2
29Y  X 2  3364
 16 x2 n3  166 x2 n 2  58, 


 28 35 x  2 35 x

n 1
n 2 

3
348Y  X 2  484416
 16 x2 n 4  1976 x2 n 2  696, 


 334 35 x  2 35 x

n 1
n 3


4
174Y  X 2  121104
 16 y2 n3  980 x2 n 2  348, 


 166 35 x  2 35 y

n1
n 2 

5
2059Y  X 2  16957924
16 y2 n 4  11690 x2 n 2  4118, 


1976 35 x  2 35 y

n1
n 3


6
174Y  X 2  121104
166 y2 n 2  70 x2 n3  348, 


16 35 x  28 35 y

n 2
n1 

7
2059Y  X 2  16957924
 1976 y2 n 2  70 x2 n 4  4118, 


 16 35 x  334 35 y

n 3
n1


8
35525Y  X 2  4120900
 28 y2 n 2  2 y2 n3  58, 


 16 35 y  166 35 y 
n 2
n1 

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 334 y2 n 2  2 y2 n 4  696, 


16 35 y  1976 35 y 
n 3
n1 

426300Y  X 2  593409600
10
29Y  X 2  3364
 166 x2 n 4  1976 x2 n3  58, 


 334 35 x  28 35 x

n 2
n 3 

11
29Y  X 2  3364
 166 y2 n3  980 x2 n3  58, 


 166 35 x  28 35 y 
n 2
n 2 

12
166 y2 n 4  11690 x2 n3  348, 


1976 35 x  28 35 y

n 2
n 3


174Y  X 2  121104
13
174Y  X 2  121104
 1976 y2 n3  980 x2 n 4  348, 


 166 35 x  334 35 y

n 3
n 2 

14
29Y  X 2  3364
1976 y2 n 4  11690 x2 n 4  58, 


1976 35 x  334 35 y

n 3
n 3 

15
1015Y  X 2  4120900
11690 y2 n3  980 y2 n 4  2030, 


166 35 y  1976 35 y

n 3
n 2


3. CONCLUSION
In this paper, we have presented infinitely many integer solutions for the positive Pell equation y 2  35x 2  29 . 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
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(2013),22-24.
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[9] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Integral points on the Hyperbola x 2  4 xy  y 2  15x  0, Diophantus
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[13] K.Meena, M.A.Gopalan, T.Swetha, On the negative Pell equation y 2  40 x 2  4, International Journal of Emerging
Technologies in Engineering Research, 5(1)(2017), 6-11.
[14] S.Ramya and A.Kavitha, On the positive Pell equation y 2  90 x 2  31, Journal of Mathematics and Informatics, (11)(2017),
111-117.
[15] Shreemathi Adiga, N. Anusheela and M.A. Gopalan, On the Negative Pellian Equation y 2  20 x 2  31 , International Journal
of Pure and Applied Mathematics, 119(14) (2018), 199-204.


[16] 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.
[17] M.A.Gopalan, A.Sathya, A. Nandhinidevi, Observation on the Positive Pell Equation y 2  15x 2  10 , International Journal
for Research in Applied Science and Engineering Technology,vol 7,Issue II,Feb 2019,pp-
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