On the Positive Pell Equation
Dr. A. Kavitha1, K. Janani2
1Assistant Professor, 2M. Phil Scholar, Department of Mathematics, Shrimati indira Gandhi college, Trichy-620002, Tamilnadu, India.
Abstract: The binary quadratic Diophantine equation
y
2
17
x
2
8
is analyzed for its non-zero distinct integral solutions. A few interesting relations among the solutions are given. Further, employing the solutions have obtained solutions of other choices of hyperbolas and parabolas.Keywords: Binary quadratic, Hyperbola, Parabola, Integral solutions, Pell equation.
I. INTRODUCTION
The binary quadratic equations 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 variousproblems, one may refer[5-10]. In this communication, yet another interesting equation given by
y
2
17
x
2
8
is considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are presentedII. METHODOFANALYSIS
The positive Pell equation representing hyperbola under consideration is,
8
17
2 2
x
y
(1)The smallest positive integer solutions of (1) are,
x
0
1
,y
0
5
D
17
consider the pellian equation is
17
1
2 2
x
y
(2) The initial solution of pellian equation is
8
~
0
x
,~
y
0
33
,The general solution
x
~
n,
~
y
n
of (2) is given by,
n
n
g
x
17
2
1
~
,n
n
f
y
2
1
~
Where,
1 1
)
17
8
33
(
)
17
8
33
(
n nn
f
1 1
)
17
8
33
(
)
17
8
33
(
n nn
g
Applying Brahmagupta lemma between
x
0,
y
0
and
x
~
n,
~
y
n
the other integer solution of (1) are given by,
n n
n
f
g
x
17
2
5
2
1
1
n n
n
f
g
y
17
2
17
2
5
1
The recurrence relation satisfied by the solution
x
andy
are given by,Where n=0,1,2,3...
[image:2.612.191.419.109.221.2]Some numerical examples of
x
n andy
n satisfying (1) are given in the Table 1 below,Table 1: Examples n
x
ny
n0 1 5 1 73 301 2 4817 19861 3 317849 1310525 4 20973217 86474789
From the above table, we observe some interesting relations among the solutions which are presented below.
A. Both
x
n andy
n values are odd .B. Each Of The Following Expression Is A Nasty Number
1)
8
5
2 217
2 2
2
3
y
nx
n2)
64
5
2 3301
2 2
16
3
x
nx
n3)
4224
5
2 419861
2 2
352
1
x
nx
n4)
264
5
2 31241
2 2
22
1
y
nx
n5)
17416
5
2 481889
2 2
4354
3
y
nx
n6)
264
301
2 217
2 3
22
1
y
nx
n7)
17416
19861
2 217
2 4
4354
3
y
nx
n8)
1088
1241
2 217
2 3
272
3
y
ny
n9)
71808
81889
2 217
2 4
5984
1
y
ny
n10)
8
301
2 31241
2 3
2
3
y
nx
n11)
64
301
2 419861
2 3
16
3
x
nx
n12)
264
301
2 481889
2 3
22
1
y
nx
n13)
264
19861
2 31241
2 4
22
1
y
nx
n14)
8
19861
2 481889
2 4
2
3
15)
1088
81889
2 31241
2 4
272
3
y
ny
nC. Each Of The Following Expressions Is A Cubical Integer
1)
5
3 317
3 315
151
1
4
1
n
n
nn
x
y
x
y
2)
5
3 4301
3 315
2903
1
32
1
n
n
nn
x
x
x
x
3)
5
3 519861
3 315
359583
1
2112
1
n
n
nn
x
x
x
x
4)
5
3 41241
3 315
23723
1
132
1
n
n
nn
x
y
x
y
5)
5
3 581889
3 315
3245667
1
8708
1
n
n
nn
x
y
x
y
6)
301
3 317
3 4903
151
2
132
1
n
n
nn
x
y
x
y
7)
19861
3 317
3 559583
151
3
8708
1
n
n
nn
x
y
x
y
8)
1241
3 317
3 43723
151
2
544
1
n
n
nn
y
y
y
y
9)
81889
3 317
3 5245667
151
3
35904
1
n
n
nn
y
y
y
y
10)
301
3 519861
3 4903
359583
2
32
1
n
n
nn
x
x
x
x
11)
301
3 41241
3 4903
23723
2
4
1
n
n
nn
x
y
x
y
12)
301
3 581889
3 4903
3245667
2
132
1
n
n
nn
x
y
x
y
13)
19861
3 4\1241
3 559583
23723
3
132
1
n
n
nn
x
y
x
y
14)
19861
3 5\81889
3 559583
3245667
3
4
1
n
n
nn
x
y
x
y
15)
81889
3 4\1241
3 5245667
23723
3
544
1
n
n
nn
y
y
y
y
D. Each Of The Following Expressions Is A Biquadratic Integer
1)
5
17
20
68
24
4
1
2 2 2 2 4 4 44n
x
n
y
n
x
n
y
2)
5
301
20
1204
192
32
1
2 2 3 2 4 4 54n
x
n
x
n
x
n
3)
5
19861
20
79444
12672
2112
1
2 2 4 2 4 4 64n
x
n
x
n
x
n
x
4)
5
1241
20
4964
792
132
1
2 2 3 2 4 4 54n
x
n
y
n
x
n
y
5)
5
81889
20
327556
52248
8708
1
2 2 4 2 4 4 64n
x
n
y
n
x
n
y
6)
301
17
1204
68
792
132
1
3 2 2 2 5 4 44n
x
n
y
n
x
n
y
7)
19861
17
79444
68
52248
8708
1
4 2 2 2 6 4 44n
x
n
y
n
x
n
y
8)
1241
17
4964
68
3264
544
1
3 2 2 2 5 4 44n
y
n
y
n
y
n
y
9)
81889
17
327556
68
215424
35904
1
4 2 2 2 6 4 44n
y
n
y
n
y
n
y
10)
301
19861
1204
79444
192
32
1
3 2 4 2 5 4 64n
x
n
x
n
x
n
x
11)
301
1241
1204
4964
24
4
1
3 2 3 2 5 4 54n
x
n
y
n
x
n
y
12)
301
81889
1204
327556
792
132
1
3 2 4 2 5 4 64n
x
n
y
n
x
n
y
13)
19861
1241
79444
4964
792
132
1
4 2 3 2 6 4 54n
x
n
y
n
x
n
y
14)
19861
81889
79444
327556
24
4
1
4 2 4 2 6 4 64n
x
n
y
n
x
n
y
15)
81889
1241
327556
4964
3264
544
1
4 2 3 2 6 4 54n
y
n
y
n
y
n
y
E. Each Of The Following Expression Is A Quintic Integer
1)
5
5 517
5 525
3 385
3 350
1170
1
4
1
n
n
n
n
nn
x
y
x
y
x
y
2)
5
5 6301
5 525
3 41505
3 350
23010
1
32
1
n
n
n
n
nn
x
x
x
x
x
x
3)
5
5 719861
5 525
3 599305
3 350
3198610
1
2112
1
n
n
n
n
nn
x
x
x
x
x
x
4)
5
5 61241
5 525
3 46205
3 350
212410
1
132
1
n
n
n
n
nn
x
y
x
y
x
y
5)
5
5 781889
5 525
3 5409445
3 350
3818890
1
8708
1
n
n
n
n
nn
x
y
x
y
x
y
6)
301
5 517
5 61505
3 385
3 43010
1170
2
132
1
n
n
n
n
nn
x
y
x
y
x
7)
1241
5 517
5 66205
3 385
3 412410
1170
2
544
1
n
n
n
n
nn
y
y
y
y
y
y
8)
4817
5 5 5 724085
3 35
3 548170
110
3
2112
1
n
n
n
n
nn
y
y
y
y
y
y
9)
301
5 719861
5 61505
3 599305
3 43010
3198610
2
32
1
n
n
n
n
nn
x
x
x
x
x
x
10)
301
5 61241
5 61505
3 46205
3 43010
212410
2
4
1
n
n
n
n
nn
x
y
x
y
x
y
11)
301
5 781889
5 61505
3 5409445
3 43010
3818890
2
132
1
n
n
n
n
nn
x
y
x
y
x
y
12)
19861
5 61241
5 799305
3 46205
3 5198610
212410
3
132
1
n
n
n
n
nn
x
y
x
y
x
y
13)
19861
5 781889
5 799305
3 5409445
3 5198610
3818890
3
4
1
n
n
n
n
nn
x
y
x
y
x
y
14)
81889
5 61241
5 7409445
3 46205
3 5818890
212410
3
544
1
n
n
n
n
nn
y
y
y
y
y
y
F. Relations Among The Solutions Are Given Below 1)
x
n2
8
y
n1
33
x
n12)
x
n3
528
y
n1
2177
x
n13)
y
n2
33
y
n1
136
x
n14)
y
n3
2177
y
n1
8976
x
n15)
x
n3
66
x
n2
x
n16)
8
y
n2
33
x
n2
x
n17)
8
y
n3
2177
x
n2
33
x
n18)
16
y
n2
x
n3
x
n19)
528
y
n3
2177
x
n3
x
n110)
33
y
n3
2177
y
n2
136
x
n111)
33
x
n3
8
y
n1
2177
x
n212)
33
y
n2
y
n1
136
x
n213)
y
n3
y
n1
272
x
n214)
2177
y
n2
33
y
n1
136
x
n315)
2177
y
n3
y
n1
8976
x
n316)
y
n3
66
y
n2
y
n117)
8
y
n2
x
n3
33
x
n219)
y
n3
33
y
n2
136
x
n220)
33
y
n3
y
n2
136
x
n3III. REMARKABLE OBSERVATIONS
[image:6.612.61.554.180.722.2]A. 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.NO Hyperbola (X,Y)
1 2
17
264
X
Y
5
x
n1
y
n1,
5
y
n1
17
x
n1
2 2
17
24096
X
Y
73
x
n1
x
n2,
5
x
n2
301
x
n1
3 2
17
217842176
X
Y
4817
x
n1
x
n3,
5
x
n3
19861
x
n1
4 2
17
269696
X
Y
301
x
n1
y
n2,
5
y
n2
1241
x
n1
5 2
17
2303317056
X
Y
19861
x
n1
y
n3,
5
y
n3
81889
x
n1
6 2
17
269696
X
Y
5
x
n2
73
y
n1,
301
y
n1
17
x
n2
7 2
17
2303317056
X
Y
5
x
n3
4817
y
n1,
19861
y
n1
17
x
n3
8 2
17
21183744
X
Y
5
y
n2
301
y
n1,
1241
y
n1
17
y
n2
9 2
17
25156388864
X
Y
5
y
n3
19861
y
n1,
81889
y
n1
17
y
n3
10 2
17
24096
X
Y
4817
x
n2
73
x
n3,
301
x
n3
19861
x
n2
11 2
17
264
X
Y
301
x
n2
73
y
n2,
301
y
n2
1241
x
n2
12 2
17
269696
X
Y
19861
x
n2
73
y
n3,
301
y
n3
81889
x
n2
13 2
17
269696
X
Y
301
x
n3
4817
y
n2,
19861
y
n2
1241
x
n3
14 2
17
264
X
Y
19861
x
n3
4817
y
n3,
19861
y
n3
81889
x
n3
15 2
17
21183744
X
B. Employing linear combination among the solutions of (1), one may generate integer solutions for other choices of parabolas which are presented in table 3 below:
Table 3: Parabola S.NO Parabola (X,Y)
1
4
17
264
X
Y
5
x
n1
y
n1,
5
y
2n2
17
x
2n2
8
2
32
17
24096
X
Y
73
x
n1
x
n2,
5
x
2n3
301
x
2n2
64
3
2112
17
217842176
X
Y
4817
x
n1
x
n3,
5
x
2n4
19861
x
2n2
4224
4
132
17
269696
X
Y
301
x
n1
y
n2,
5
y
2n3
1241
x
2n2
264
5
8708
17
2303317056
X
Y
19861
x
n1
y
n3,
5
y
2n4
81889
x
2n2
17416
6
132
17
269696
X
Y
5
x
n2
73
y
n1,
301
y
2n2
17
x
2n3
264
7
8708
17
2303317056
X
Y
5
x
n3
4817
y
n1,
19861
y
2n2
17
x
2n4
17416
8
544
17
21183744
X
Y
5
y
n2
301
y
n1,
1241
y
2n2
17
y
2n3
1088
9
35904
17
25156388864
X
Y
5
y
n3
19861
y
n1,
81889
y
2n2
17
y
2n4
71808
10
32
17
24096
X
Y
4817
x
n2
73
x
n3,
301
x
2n4
19861
x
2n3
64
11
4
17
264
X
Y
301
x
n2
73
y
n2,
301
y
2n3
1241
x
2n3
8
12
132
17
269696
X
Y
19861
x
n2
73
y
n3,
301
y
2n4
81889
x
2n3
264
13
132
17
269696
X
Y
301
x
n3
4817
y
n2,
19861
y
2n3
1241
x
2n4
264
14
4
17
264
X
Y
19861
x
n3
4817
y
n3,
19861
y
2n4
81889
x
2n4
8
15
544
17
21183744
X
Y
301
y
n3
19861
y
n2,
81889
y
2n3
1241
y
2n4
1088
G. Some Special Cases Of The Solutions Are Given Below
1)
P
y10
t
3,x1
2
153
P
x6
t
3,y 2
8
t
3,y 2
t
3,x1
22)
9
P
y6
t
3,x 2
17
P
x10
t
3,y1
2
8
t
3,x 2
t
3,y1
23)
P
y10
t
3,2x2
2
17
6
P
x41
2
t
3,y 2
8
t
3,y 2
t
3,2x2
24)
36
P
y81
t
3,x 2
17
P
x10
t
3,2y2
2
8
t
3,x 2
t
3,2y2
25)
9
P
y6
t
3,2x2
2
17
36
P
x81
t
3,y1
2
8
t
3,2x2
2
t
3,y1
26)
6
P
y41
2
t
3,x1
2
17
3
P
x3
2
t
3,2y2
2
8
t
3,x1
2
t
3,2y2
2III.CONCLUSIONS
REFERENCES
[1] L.E.Dickson, History of theory of number, Chelsa publishing company, vol-2, (1952) New York. [2] L.J.Mordel, Diophantine equations, Academic Press, (1969) New York.
[3] S.J.Telang, Number Theory, Tata McGraw Hill Publishing company Limited,(2000) New Delhi.
[4] D.M.Burton, Elementary Number Theory, Tata McGraw Hill Publishing company Limited,(2002) New Delhi.
[5] M.A.Gopolan, s.vidhyalakshmi and A.Kavitha, on the integral solutions of the Binary quadratic Equation
X
2
4
K
2
1
Y
2
4
t, Bulletin of Mathematics and Statistics Research, 2(1)(2014)42-46.[6] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, On the binary quadratic Diophantine equation
x
2
3
xy
y
2
33
x
0
, International Journal of Scientific Engineering and Applied Science, 1(4) (2015) 222-225.[7] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, observations on the Hyperbola
10
y
2
3
x
2
13
, Archimedes J.Math.,3(1) (2013) 31-34. [8] S.Vidhyalakshmi, A.Kavitha and M.A.Gopalan, Observations on the Hyperbolaax
2
a
1
y
2
3
a
1
, Discovery, 4(10) (2013) 22-24[9] A.Kavitha and P.Lavanya, on the positive Pell equation