On the Negative Pell Equation
y
2
48
x
2
23
S. Mallika1, G. Ramya2
1Assistant Professor, 2M. Phil Scholar, Department of Mathematics, SIGC, Trichy-620002, Tamilnadu, India.
Abstract: The binary quadratic equation represents by negative Pellian
y
2
48
x
2
23
is analyzed for its distinct integersolutions. A few interesting relations among the solution are given. Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbolas and parabolas.
Keywords: Binary quadratic, Hyperbola, Parabola, Integral solutions, Pell equation. AMS Mathematics subject Classification (2010):11D09
I. INTRODUCTION
The binary quadratic Diophantine equations (both homogeneous and non-homogeneous) are rich in variety. In [1-17] the binary quadratic non-homogeneous equations representing hyperbolas respectively are studied for their non-zero integral solutions. This communication concerns with yet another binary quadratic equation given by
y
2
48
x
2
23
.The recurrence relations satisfied by the solutions x and y are given. Also a few interesting properties among the solutions are exhibited.II. METHOD OF ANALYSIS
The negative Pell equation representing hyperbola under consideration is
48
23
2 2
x
y
(1) whose smallest positive integer solution is5
,
1
00
y
x
To obtain the other solutions of (1), consider the Pell equation
1
48
2 2
x
y
whose smallest positive integer solution is
7
~
,
1
~
0
0
y
x
whose general solution is given by
n n
n
n
g
y
f
x
2
1
~
,
48
2
1
~
where
1
148
7
48
7
n nn
f
1
148
7
48
7
n nn
g
Applying 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
3
8
5
2
1
1
n n
n
f
g
y
3
6
2
5
1
The recurrence relations satisfied by x and y are given by
0
14
2 13
n n
n
x
x
x
0
14
y
y
Table: 1 Numerical Examples
n
1 n
x
y
n1-1 1 5
0 12 83 1 167 1157 2 2326 16115 3 32397 224453 4 451232 3126227
From the above table, we observe some interesting relations among the solutions which are presented below:
A.
x
n1 values are alternatively odd and even according as n values &y
n1values are alternatively odd.B. Relations Among The Solutions 1)
x
n3
14
x
n2
x
n1
0
2)7
x
n1
x
n2
y
n1
0
3)x
n1
7
x
n2
y
n2
0
4)7
x
n1
97
x
n2
y
n3
0
5)97
x
n1
x
n3
14
y
n1
0
6)x
n1
x
n3
2
y
n2
0
7)x
n1
97
x
n3
14
y
n3
0
8)y
n2
7
y
n1
48
x
n1
0
9)y
n3
97
y
n1
672
x
n1
0
10)7
y
n3
97
y
n2
48
x
n1
0
11)97
x
n2
7
x
n3
y
n1
0
12)7
x
n2
x
n3
y
n2
0
13)x
n2
7
x
n3
y
n3
0
14)7
y
n3
7
y
n1
672
x
n2
0
15)48
x
n2
y
n1
7
y
n2
0
16)y
n3
7
y
n2
48
x
n2
0
17)97
y
n2
7
y
n1
48
x
n3
0
18)97
y
n3
y
n1
672
x
n3
0
19)7
y
n3
y
n2
48
x
n3
0
20)y
n1
14
y
n2
y
n3
0
C. Each Of The Following Expressions Represents A Nasty Number
1)
996
60
276
23
1
3 2 2
2n
x
n
2)
6942
30
1932
161
1
4 2 2
2n
x
n
x
3)
576
60
276
23
1
2 2 2
2n
y
n
x
4)
6912
60
1932
161
1
3 2 2
2n
y
n
x
5)
96192
60
26772
2231
1
4 2 2
2n
y
n
x
6)
13884
996
276
23
1
4 2 3
2n
x
n
x
7)
576
996
1932
161
1
2 2 3
2n
y
n
x
8)
6912
996
276
23
1
3 2 3
2n
y
n
x
9)
96192
996
1932
161
1
4 2 3
2n
y
n
x
10)
576
13884
26772
2231
1
2 2 4
2n
y
n
x
11)
6912
13884
1932
161
1
3 2 4
2n
y
n
x
12)
96192
13884
276
23
1
4 2 4
2n
y
n
x
13)
72
864
1656
138
1
2 2 3
2n
y
n
y
14)
6
1002
1932
161
1
2 2 4
2n
y
n
y
15)
432
6012
828
69
1
3 2 4
2n
y
n
y
D. Each Of The Following Expressions Represents A Cubical Integer
1)
166
3 310
3 4498
130
2
23
1
n
n
nn
x
x
x
x
2)
1157
3 35
3 53471
115
3
161
1
n
n
nn
x
x
x
x
3)
96
3 310
3 3228
130
1
23
1
n
n
nn
y
x
y
x
4)
1152
3 310
3 43456
130
2
161
1
n
n
nn
y
x
y
x
5)
16032
3 310
3 548096
130
3
2231
1
n
n
nn
y
x
y
6)
2314
3 4166
3 56942
2498
3
23
1
n
n
nn
x
x
x
x
7)
96
3 4116
3 3288
2498
1
161
1
n
n
nn
y
x
y
x
8)
1152
3 4166
3 43456
2498
2
23
1
n
n
nn
y
x
y
x
9)
16032
3 4166
3 548096
2498
3
161
1
n
n
nn
y
x
y
x
10)
96
3 52314
3 3288
36942
1
2231
1
n
n
nn
y
x
y
x
11)
1152
3 52314
3 43456
36942
2
161
1
n
n
nn
y
x
y
x
12)
16032
3 52314
3 548096
36942
3
23
1
n
n
nn
y
x
y
x
13)
12
3 4144
3 336
2432
1
138
1
n
n
nn
y
y
y
y
14)
3 5167
3 33
3501
1
161
1
n
n
nn
y
y
y
y
15)
72
3 51002
3 4216
33006
2
69
1
n
n
nn
y
y
y
y
E. Each Of The Following Expressions Represents A Bi-Quadratic Integer
1)
166
10
664
40
138
23
1
3 2 2 2 5 4 44n
x
n
x
n
x
n
x
2)
1157
5
4628
20
966
161
1
4 2 2 2 6 4 44n
x
n
x
n
x
n
x
3)
96
10
384
40
138
23
1
2 2 2 2 4 4 44n
y
n
x
n
y
n
x
4)
1152
10
4608
40
966
161
1
3 2 2 2 5 4 44n
y
n
x
n
y
n
x
5)
16032
10
64128
40
13386
2231
1
4 2 2 2 6 4 44n
y
n
x
n
y
n
x
6)
2314
166
9256
664
138
23
1
4 2 3 2 6 4 54n
x
n
x
n
x
n
x
7)
96
166
384
664
966
161
1
2 2 3 2 4 4 54n
y
n
x
n
y
n
x
8)
1152
166
4608
664
138
23
1
3 2 3 2 5 4 54n
y
n
x
n
y
n
x
9)
16032
166
64128
664
966
161
1
4 2 3 2 6 4 54n
y
n
x
n
y
n
10)
96
2314
384
9256
13386
2231
1
2 2 4 2 4 4 64n
y
n
x
n
y
n
x
11)
1152
2314
4608
9256
966
161
1
4 3 5 3 5 4 64n
y
n
x
n
y
n
x
12)
16032
2314
64128
9256
138
23
1
4 2 4 2 6 4 64n
y
n
x
n
y
n
x
13)
12
144
48
576
828
138
1
2 2 3 2 4 4 54n
y
n
y
n
y
n
y
14)
167
4
668
966
161
1
2 2 4 2 4 4 64n
y
n
y
n
y
n
y
15)
72
1002
288
4008
414
69
1
3 2 4 2 5 4 64n
y
n
y
n
y
n
y
F. Each Of The Following Expressions Represents A Quintic Integer
1)
166
5 510
5 6830
3 350
3 41660
1100
2
23
1
n
n
n
n
nn
x
x
x
x
x
x
2)
1157
5 55
5 75785
3 325
3 511570
150
3
161
1
n
n
n
n
nn
x
x
x
x
x
x
3)
96
5 510
5 5480
3 350
3 3960
1100
1
23
1
n
n
n
n
nn
y
x
y
x
y
x
4)
1152
5 510
5 65760
3 350
3 411520
1100
2
161
1
n
n
n
n
nn
y
x
y
x
y
x
5)
16032
5 510
5 780160
3 350
3 5160320
1100
3
2231
1
n
n
n
n
nn
y
x
y
x
y
x
6)
2314
5 6166
5 711570
3 4830
3 523140
21660
3
23
1
n
n
n
n
nn
x
x
x
x
x
x
7)
96
5 6166
5 5480
3 4830
3 3960
21660
1
161
1
n
n
n
n
nn
y
x
y
x
y
x
8)
1152
5 6166
5 65760
4 5830
4 511520
21660
2
23
1
n
n
n
n
nn
y
x
y
x
y
x
9)
16032
5 6166
5 780160
3 4830
3 5160320
21660
3
161
1
n
n
n
n
nn
y
x
y
x
y
x
10)
96
5 72314
5 5480
3 511570
3 3960
323140
1
1
`
223
1
n
n
n
n
nn
y
x
y
x
y
x
11)
1152
5 72314
5 65760
3 511570
3 411520
323140
2
161
1
n
n
n
n
nn
y
x
y
x
y
x
12)
16032
5 72314
5 780160
3 511570
3 5160320
323140
3
23
1
n
n
n
n
nn
y
x
y
x
y
x
13)
12
5 6144
5 560
3 4720
3 3120
21440
1
138
1
n
n
n
n
nn
y
y
y
y
y
14)
5 7167
5 55
3 5835
3 310
31670
1
161
1
n
n
n
n
nn
y
y
y
y
y
y
15)
72
5 71002
5 6360
3 55010
3 4720
310020
2
69
1
n
n
n
n
nn
y
y
y
y
y
y
III. REMARKABLE OBSERVATIONS
A. Employing linear combinations among the solutions of (1), one may generate integer solutions for other choices of hyperbola which are presented in the Table: 2 below:
Table: 2 Hyperbolas
S. No Hyperbola
X
,
Y
1 2
48
2529
Y
X
83
x
n1
5
x
n2,
x
n2
12
x
n1
2 2
48
2103684
Y
X
1157
x
n1
5
x
n3,
x
n3
167
x
n1
3 2
48
2529
Y
X
48
x
n1
5
y
n1,
y
n1
5
x
n1
4 2
48
225921
Y
X
576
x
n1
5
y
n2,
y
n2
83
x
n1
5 2
48
24977361
Y
X
8016
x
n1
5
y
n3,
y
n3
1157
x
n1
6 2
48
2529
Y
X
1157
x
n2
83
x
n3,
12
x
n3
167
x
n2
7 2
48
225921
Y
X
48
x
n2
83
y
n1,
12
y
n1
5
x
n2
8 2
48
2529
Y
X
576
x
n2
83
y
n2,
12
y
n2
83
x
n2
9 2
48
225921
Y
X
8016
x
n2
83
y
n3,
12
y
n3
1157
x
n2
10 2
48
24977361
Y
X
48
x
n3
1157
y
n1,
167
y
n1
5
x
n3
11 2
48
225921
Y
X
576
x
n3
1157
y
n2,
167
y
n2
83
x
n3
12 2
48
2529
Y
X
8016
x
n3
1157
y
n3,
167
y
n3
1157
x
n3
13
16
23
276176
Y
X
3
y
n2
36
y
n1,
83
y
n1
5
y
n2
14
48
2 24976832
Y
X
y
n3
167
y
n1,
1157
y
n1
5
y
n3
B. 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 Parabola
X
,
Y
1
23
96
2529
Y
X
83
x
2n2
5
x
2n3,
x
n2
12
x
n1
2
322
96
2103684
Y
X
1157
x
2n2
5
x
2n4,
x
n3
167
x
n1
3
23
96
2529
Y
X
48
x
2n2
5
y
2n2,
y
n1
5
x
n1
4
161
96
225921
Y
X
576
x
2n2
5
y
2n3,
y
n2
83
x
n1
5
2231
96
24977361
Y
X
8016
x
2n2
5
y
2n4,
y
n3
1157
x
n1
6
23
96
2529
Y
X
1157
x
2n3
83
x
2n4,
12
x
n3
167
x
n2
7
161
96
225921
Y
X
48
x
2n3
83
y
2n2,
12
y
n1
5
x
n2
8
23
96
2529
Y
X
576
x
2n3
83
y
2n3,
12
y
n2
83
x
n2
9
161
96
225921
Y
X
8016
x
2n3
83
y
2n4,
12
y
n3
1157
x
n2
10
2231
96
24977361
Y
X
48
x
2n4
1157
y
2n2,
167
y
n1
5
x
n3
11
161
96
225921
Y
X
576
x
2n4
1157
y
2n3,
167
y
n2
83
x
n3
12
23
96
2529
Y
X
8016
x
2n4
1157
y
2n4,
167
y
n3
1157
x
n3
13
552
3
238088
Y
X
3
y
2n3
36
y
2n2,
83
y
n1
5
y
n2
14
23184
3
27465248
Y
X
y
2n4
167
y
2n2,
1157
y
n1
5
y
n3
15
552
3
238088
Y
X
36
y
2n4
501
y
2n3,
1157
y
n2
83
y
n3
IV. CONCLUSION
In this paper, we have presented infinitely many integer solutions for al hyperbola represented by the negative Pell Equation
23
48
2 2
x
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x
2
Ay
2
4
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x
2
dy
2
2
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x
2
(
k
2
k
)
y
2
2
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x
2
13
y
2
3
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x
2
19
y
2
3
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x
2
15
y
2
11
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y
2
80
x
2
31
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y
2
10
x
2
6
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y
2
45
x
2
11
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y
2
31
x
2
6
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y
2
60
x
2
15
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y
2
180
x
2
11
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y
2
72
x
2
8
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