On the Positive Pell Equation
y
2
35
x
2
14
K. Meena1, M. A. Gopalan2, T. Priya Lakshmi 3 1
Former VC, Bharathidasan University, Trichy-620 024.Tamilnadu, India
2
Professor, Department of Mathematics, SIGC, Trichy, India.
3
M.Phil Scholar, Department of Mathematics, SIGC, Trichy, India
Abstract: The binary quadratic equation represented by the positive Pellian
y
2
35
x
2
14
is analyzed for its distinct integer solutions. A few interesting relations among the solutions are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbola and parabola.Keywords: Binary quadratic, hyperbola, integral solutions, parabola, pell equation. 2010 mathematics subject classification: 11D09
I. INTRODUCTION
A binary quadratic equation of the form 2 2
1
Dx
y
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 [1-2]. For an extensive review of various problems, one may refer [3-12]. In this communication, yet another interesting hyperbola given byy
2
35
x
2
14
is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are obtained.A. Method of Analysis
The Positive Pell equation representing hyperbola under consideration is
14
35
22
x
y
(1)whose smallest positive integer solution is
7
,
1
00
y
x
To obtain the other solutions of (1), consider the Pell equation
1
35
22
x
y
whose general solution is given by
n n
n
n
g
y
f
x
2
1
~
,
35
2
1
~
where
1
135
6
35
6
n nn
f
6
35
1
6
35
1,
1
,
0
,
1
,
2
...
n
g
n n nApplying Brahmagupta lemma between
x
0,
y
0
and
~
x
n,
y
~
n
, the other integer solutions of (1) are given byn n
n
f
g
x
35
2
7
2
1
1
n n
n
f
g
y
2
35
2
7
1
The recurrence relations satisfied by x and y are given by
0
12
2 13
n n
n
x
x
x
0
12
2 13
n n
n
y
y
y
Table: 1 Numerical examples
n
1
n
x
y
n1-1 1 7
0 13 77
1 155 917
2 1847 10927
3 22009 130207
From the above table, we observe some interesting relations among the solutions which are presented below:
1)
x
n1 and yn+1 are always odd.2) Relations among the solutions
a)
x
n2
6
x
n1
y
n1
0
b)
y
n2
6
x
n2
x
n1
0
c)
y
n3
71
x
n2
6
x
n1
0
d)
x
n3
12
x
n2
x
n1
0
e)
x
n3
71
x
n2
y
n1
0
f)
x
n3
6
x
n2
y
n2
0
g)
x
n3
71
x
n1
12
y
n1
0
h)
35
x
n3
6
y
n3
y
n2
0
i)
y
n1
x
n2
6
x
n1
0
j)
12
y
n1
x
n3
x
n1
0
k)
12
y
n1
x
n3
71
x
n1
0
l)
2
y
n2
x
n3
x
n1
0
m)
12
y
n3
71
x
n3
x
n1
0
n)
349
x
n1
72
y
n1
x
n3
0
o)
349
x
n2
781
y
n1
6
x
n3
0
p)
349
y
n2
4614
y
n1
35
x
n3
0
q)
349
y
n3
55019
y
n1
420
x
n3
0
r)
6
y
n1
71
y
n2
35
x
n3
0
s)
6
y
n3
35
x
n3
y
n2
0
t)
y
n1
71
y
n3
420
x
n3
0
u)
y
n3
12
y
n2
y
n1
0
v)
280
x
n1
3
y
n3
353
y
n1
0
w)
y
n2
35
x
n1
6
y
n1
0
x)
y
n3
420
x
n1
71
y
n1
0
z)
y
n1
6
x
n3
71
x
n2
0
aa)
y
n2
x
n3
6
x
n2
0
bb)
y
n3
6
x
n3
x
n2
0
cc)
y
n3
70
x
n2
y
n1
0
dd)
6
y
n2
35
x
n1
y
n1
0
ee)
y
n3
35
x
n2
6
y
n2
0
3) Each of the following expressions represents a nasty number
a)
6
x
2n3
66
x
2n2
12
b)
131
24
2
1
2 2 4
2n
x
n
x
c)
6
y
2n2
30
x
2n2
12
d)
y
2n3
65
x
2n2
12
e)
6
4650
852
71
1
2 2 4
2n
x
n
y
f)
66
x
2n4
786
x
2n3
12
g)
11
y
2n2
5
x
2n3
12
h)
66
y
2n3
390
x
2n3
12
i)
11
y
2n4
775
x
2n3
12
j)
131
5
142
71
6
4 2 2
2n
x
n
y
k)
131
y
2n3
65
x
2n4
12
l)
786
y
2n4
4650
x
2n4
12
m)
78
6
84
7
1
3 2 2
2n
y
n
y
n)
930
6
1008
84
1
4 2 2
2n
y
n
y
o)
930
78
84
7
1
4 2 3
2n
y
n
y
4) Each of the following expressions represents a cubical integer
a)
x
3n4
11
x
3n3
33
x
n1
3
x
n2b)
3 5131
3 3393
13
3
12
1
n
n
nn
x
x
x
x
c)
y
3n3
5
x
3n3
15
x
n1
3
y
n1d)
3 465
3 3195
13
2
6
1
n
n
nn
x
x
y
e)
3 5775
3 32325
13
3
71
1
n
n
nn
x
x
y
y
f)
11
x
3n5
131
x
3n4
393
x
n2
33
x
n3g)
11
3 35
3 415
233
1
6
1
n
n
nn
x
x
y
y
h)
11
y
3n4
65
x
3n4
195
x
n2
33
y
n2i)
11
3 5775
3 42325
233
3
6
1
n
n
nn
x
x
y
y
j)
917
3 335
3 52751
1105
3
497
1
n
n
nn
x
y
x
y
k)
131
3 465
3 5195
3393
2
6
1
n
n
nn
x
x
y
y
l)
131
y
3n5
775
x
3n5
2325
x
n3
393
y
n3m)
13
3 3 3 439
13
2
7
1
n
n
nn
y
y
y
y
n)
155
3 3 3 5465
13
3
84
1
n
n
nn
y
y
y
y
o)
155
3 413
3 5465
239
3
7
1
n
n
nn
y
y
y
y
5) Each of the following expressions represents a bi-quadratic integer
a)
x
4n5
11
x
4n4
44
x
2n2
4
x
2n3
6
b)
131
524
4
72
12
1
4 2 2 2 4 4 64n
x
n
x
n
x
n
x
c)
y
4n4
5
x
4n4
20
x
2n2
4
y
2n2
6
d)
65
260
4
36
6
1
3 2 2 2 4 4 54n
x
n
x
n
y
n
y
e)
775
3100
4
426
71
1
4 2 2 2 4 4 64n
x
n
x
n
y
n
y
f)
11
x
4n6
131
x
4n5
524
x
2n3
44
x
2n4
6
g)
11
5
20
44
36
6
1
2 2 3 2 5 4 44n
x
n
x
n
y
n
y
h)
11
y
4n5
65
x
4n5
260
x
2n3
44
y
2n3
6
i)
11
775
3100
44
36
6
1
4 2 3 2 5 4 64n
x
n
x
n
y
n
y
j)
131
5
524
20
426
71
1
4 2 2 2 6 4 44n
x
n
y
n
x
n
y
k)
131
65
260
524
36
6
1
3 2 4 2 6 4 54n
x
n
x
n
y
n
l)
131
y
4n6
775
x
4n6
3100
x
2n4
524
y
2n4
6
m)
13
52
4
42
7
1
3 2 2 2 5 4 44n
y
n
y
n
y
n
y
n)
155
620
4
504
84
1
4 2 2 2 6 4 44n
y
n
y
n
y
n
y
o)
155
13
620
52
42
7
1
4 2 3 2 6 4 54n
y
n
y
n
y
n
y
6) Each of the following expressions represents a quintic integer
a)
x
5n6
11
x
5n5
55
x
3n3
5
x
3n4
110
x
n1
10
x
n2b)
5 7131
5 5655
3 35
3 51310
110
3
12
1
n
n
n
n
nn
x
x
x
x
x
x
c)
y
5n5
5
x
5n5
25
x
3n3
5
y
3n3
50
x
n1
10
y
n1d)
5 665
5 5325
3 35
3 4650
110
2
6
1
n
n
n
n
nn
x
x
y
x
y
y
e)
5 7775
5 53875
3 35
3 57750
110
3
71
1
n
n
n
n
nn
x
x
y
x
y
y
f)
11
x
5n7
131
x
5n6
655
x
3n4
55
x
3n5
1310
x
n2
110
x
n3g)
11
5 55
5 625
3 455
3 350
2110
1
6
1
n
n
n
n
nn
x
x
y
x
y
y
h)
11
y
5n6
65
x
5n6
325
x
3n4
55
y
3n4
650
x
n2
110
y
n2i)
11
5 7775
5 63875
3 455
3 57750
2110
3
6
1
n
n
n
n
nn
x
x
y
x
y
y
j)
655
3 3524
5 520
5 725
3 51965
175
3
71
1
n
n
n
n
nn
y
x
x
y
x
y
k)
131
5 665
5 7325
3 5655
3 4650
31310
2
6
1
n
n
n
n
nn
x
x
y
x
y
y
l)
131
y
5n7
775
x
5n7
3875
x
3n5
655
y
3n5
7750
x
n3
1310
y
n3m)
13
5 5 5 665
3 35
3 4130
110
2
7
1
n
n
n
n
nn
y
y
y
y
y
y
n)
155
5 5 5 7775
3 35
3 51550
110
3
84
1
n
n
n
n
nn
y
y
y
y
y
y
o)
155
5 613
5 7775
3 465
3 51550
2130
3
7
1
n
n
n
n
nn
y
y
y
y
y
y
B. Remarkable Observations
Table: 2 Hyperbolas S. No
Hyperbolas
X
,
Y
1
7
25
228
Y
X
x
n2
11
x
n1,
13
x
n1
x
n2
2
7
25
24032
Y
X
x
n3
131
x
n1,
155
x
n1
x
n3
3
7
25
228
Y
X
y
n1
5
x
n1,
7
x
n1
y
n1
4
7
25
21008
Y
X
y
n2
65
x
n1,
77
x
n1
y
n2
5
7
25
2141148
Y
X
y
n3
775
x
n1,
917
x
n1
y
n3
6
7
25
228
Y
X
11
x
n3
131
x
n2,
155
x
n2
13
x
n3
7
7
25
21008
Y
X
11
y
n1
5
x
n2,
7
x
n2
13
y
n1
8
7
25
228
Y
X
11
y
n2
65
x
n2,
77
x
n2
13
y
n2
9
7
25
21008
Y
X
11
y
n3
775
x
n2,
917
x
n2
13
y
n3
10
7
25
2141148
Y
X
131
y
n1
5
x
n3,
7
x
n3
155
y
n1
11
7
25
21008
Y
X
131
y
n2
65
x
n3,
77
x
n3
155
y
n2
12
7
25
228
Y
X
131
y
n3
775
x
n3,
917
x
n3
155
y
n3
13
5
27
2980
Y
X
13
y
n1
y
n2,
y
n2
11
y
n1
14
5
27
2141120
Y
X
155
y
n1
y
n3,
y
n3
131
y
n1
15
5
27
2980
Y
2) Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of parabola which are presented in the Table: 3 below:
Table: 3 Parabolas S. No
Parabolas
X
,
Y
1
7
5
214
Y
X
x
2n3
11
x
2n2,
13
x
n1
x
n2
2
84
5
22016
Y
X
x
2n4
131
x
2n2,
155
x
n1
x
n3
3
7
5
214
Y
X
y
2n2
5
x
2n2,
7
x
n1
y
n1
4
42
5
2504
Y
X
y
2n3
65
x
2n2,
77
x
n1
y
n2
5
497
5
270574
Y
X
y
2n4
775
x
2n2,
917
x
n1
y
n3
6
7
5
214
Y
X
11
x
2n4
131
x
2n3,
155
x
n2
13
x
n3
7
42
5
2504
Y
X
11
y
2n2
5
x
2n3,
7
x
n2
13
y
n1
8
7
5
214
Y
X
11
y
2n3
65
x
2n3,
77
x
n2
13
y
n2
9
42
5
2504
Y
X
11
y
2n4
775
x
2n3,
917
x
n2
13
y
n3
10
497
5
270574
Y
X
131
y
2n2
5
x
2n4,
7
x
n3
155
y
n1
11
42
5
2504
Y
X
131
y
2n3
65
x
2n4,
77
x
n3
155
y
n2
12
7
5
214
Y
X
131
y
2n4
775
x
2n4,
917
x
n3
155
y
n3
13
5
270
Y
X
13
y
2n2
y
2n3,
y
n2
11
y
n1
14
60
210080
Y
X
155
y
2n2
y
2n4,
y
n3
131
y
n1
15
5
270
Y
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x
2
6
xy
y
2
40
x
80
y
40
0
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2
24
x
2
1
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y
2
72
x
2
1
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y
2
24
x
2
1
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y
2
220
x
2
9
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y
2
90
x
2
31
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y
2
150
x
2
16
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2
102
x
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33
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5
x
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4
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