International Journal of Advanced Science and Research ISSN: 2455-4227; Impact Factor: RJIF 5.12
Received: 17-01-2019; Accepted: 21-02-2019 www.allsciencejournal.com
Volume 4; Issue 2; March 2019; Page No. 25-31
On the Negative Pell equation y
2=21x
2- 5
MA Gopalan1, T Mahalakshmi2, K Radha3
1 Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, India
2 Assistant Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, India
3 M.Phil Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy, India Abstract
The binary quadratic equation represented by the negative Pelliany2 =21x2 - 5 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, parabola.
Keywords: binary quadratic, hyperbola, parabola, pell equation, integral solutions
Introduction
A binary quadratic equation of the form y2 = Dx2 + 1, 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 [3-15]. In this communication, yet another interesting hyperbola given by y2 =21x2 - 5 considered and infinitely many integer solutions are obtained. A few interesting properties among the solutions are obtained.
Further, employing the solutions of the above hyperbola, we have obtained solutions of other choices of hyperbola, parabola.
Method of analysis
The Negative Pell equation representing hyperbola under consideration is
5
21
22
= x −
y
(1)whose smallest positive integer solution is
4 , 1
00
= y =
x
To obtain the other solutions of (1), consider the Pell equation
1
21
22
= x +
y
whose general solution is given by
n n n
n
g y f
x 2
~ 1 1 , 2 2
~ = 1 =
where
( 55 + 12 2 1 ) (
+1+ 55 − 12 2 1 )+1
=
n nf
n( 55 + 12 21 ) (
1− 55 − 12 2 1 ) 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 by
n n
n
f g
x
21 2 2
1
1
= +
+
n n
n
f g
y 2
2 21
1
= +
+
The recurrence relations satisfied by x and y are given by
0
110
2 13
−
++
+=
+ n n
n
x x
x
0 110
2 13
−
++
+=
+ n n
n
y y
y
Some numerical examples of x and y satisfying (1) are given in the Table: 1 below:
Table 1: Numerical examples
n x
n+1y
n+1-1 1 4
0 103 472
1 11329 51916
2 1246087 5710288
3 137058241 628079764
4 15075160423 69083063752
From the above table, we observe some interesting relations among the solutions which are presented below:
1.
x
n+1is always odd and
y
n+1is always even 2. Relations among the solutions
➢
55 x
n+1− x
n+2+ 12 y
n+1= 0
➢
x
n+1− 55 x
n+2+ 12 y
n+2= 0
➢
55 x
n+1− 6049 x
n+2+ 12 y
n+3= 0
➢
6049 x
n+1− x
n+3+ 1320 y
n+1= 0
➢
6049 x
n+2− 55 x
n+3+ 12 y
n+1= 0
➢
55 x
n+2− x
n+3+ 12 y
n+2= 0
➢
x
n+1− x
n+3+ 24 y
n+2= 0
➢
x
n+1− 6049 x
n+3+ 1320 y
n+3= 0
➢
x
n+2− 55 x
n+3+ 12 y
n+3= 0
➢
55 y
n+1− y
n+2+ 252 x
n+1= 0
➢
y
n+1− 55 y
n+2+ 252 x
n+2= 0
➢
55 y
n+1− 6049 y
n+2+ 252 x
n+3= 0
➢
6049 y
n+1− y
n+3+ 27720 x
n+1= 0
➢
y
n+1− y
n+3+ 504 x
n+2= 0
➢
y
n+1− 6049 y
n+3+ 27720 x
n+3= 0
➢
6049 y
n+2− 55 y
n+3+ 252 x
n+1= 0
➢
55 y
n+2− y
n+3+ 252 x
n+2= 0
➢
y
n+2− 55 y
n+3+ 252 x
n+3= 0
3. Each of the following expressions represents a nasty number
➢
( 472 4 60 )
5 1
3 2 2
2n+
− x
n++ x
➢
( 12979 1650 )
275 2
4 2 2
2n+
− x
n++ x
➢
( 21 4 5 )
5 12
2 2 2
2n+
− y
n++
x
➢
( 2163 4 275 )
275 12
3 2 2
2n+
− y
n++ x
➢
( 237909 4 30245 )
30245 12
4 2 2
2n+
+ y
n++ x
➢
( 12979 118 15 )
5 4
4 2 3
2n+
− x
n++ x
➢
( 21 472 275 )
275 12
2 2 3
2n+
− y
n++ x
➢
( 2163 472 5 )
5 12
3 2 3
2n+
− y
n++ x
➢
( 237909 472 275 )
275 12
4 2 3
2n+
− y
n++ x
➢
( 21 51916 30245 )
30245 12
2 2 4
2n+
− y
n++
x
➢
( 2163 51916 275 )
275 12
3 2 4
2n+
− y
n++
x
➢
( 237909 51916 5 )
5 12
4 2 4
2n+
− y
n++
x
➢
( 103 60 )
5 1
2 2 3
2n+
− y
n++ y
➢
( 11329 6600 )
550 1
2 2 4
2n+
− y
n++
y
➢
( 103 11329 60 )
5 1
3 2 4
2n+
− y
n++
y
4. Each of the following expressions represents a cubical integer
➢
472
3 34
3 41416
112
2
30 1
+ +
+
+
−
n+
n−
nn
x x x
x
➢
51916
3 34
3 5155748
112
3
3300 1
+ +
+
+
−
n+
n−
nn
x x x
x
➢
42
3 38
3 3126
124
1
5 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
4326
3 38
3 412978
124
2
275 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
475818
3 38
3 51427454
124
3
30245 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
51916
3 4472
3 5155748
21416
3
30 1
+ +
+
+
−
n+
n−
nn
x x x
x
➢
42
3 4944
3 3126
22832
1
275 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
4326
3 4944
3 412978
22832
2
5 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
475818
3 4944
3 51427454
22832
3
275 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
42
3 5103832
3 3126
3311496
1
30245 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
4326
3 5103832
3 412978
3311496
2
275 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
475818
3 5103832
3 51427454
3311496
3
5 1
+ +
+
+
−
n+
n−
nn
y x y
x
➢
3 4103
3 33
2309
1
30 1
+ +
+
+
−
n+
n−
nn
y y y
y
➢
3 511329
3 33
333987
1
3300 1
+ +
+
+
−
n+
n−
nn
y y y
y
➢
103
3 511329
3 4309
333987
2
30 1
+ +
+
+
−
n+
n−
nn
y y y
y
5. Each of the following expressions represents a bi-quadratic integer
➢
472 4 1888 16 180
30 1
3 2 2 2 5
4 4
4n+
− x
n++ x
n+− x
n++ x
➢
51916 4 207664 16 19800
3300 1
4 2 2 2 6
4 4
4n+
− x
n++ x
n+− x
n++
x
➢
42 8 168 32 30
5 1
2 2 2
2 4
4 4
4n+
− y
n++ x
n+− y
n++ x
➢
4326 8 17304 32 1650
275 1
3 2 2
2 5
4 4
4n+
− y
n++ x
n+− y
n++
x
➢
475818 8 1903272 32 181470
30245 1
4 2 2
2 6
4 4
4n+
− y
n++ x
n+− y
n++
x
➢
51916 472 207664 1888 180
30 1
4 2 3
2 6
4 5
4n+
− x
n++ x
n+− x
n++
x
➢
42 944 168 3776 1650
275 1
2 2 3
2 4
4 5
4n+
− y
n++ x
n+− y
n++
x
➢
4326 944 17304 3776 30
5 1
3 2 3
2 5
4 5
4n+
− y
n++ x
n+− y
n++
x
➢
475818 944 1903272 3776 1650
275 1
4 2 3
2 6
4 5
4n+
− y
n++ x
n+− y
n++
x
➢
42 103832 168 415328 181470
30245 1
2 2 4
2 4
4 6
4n+
− y
n++ x
n+− y
n++
x
➢
4326 103832 17304 415328 1650
275 1
3 2 4
2 5
4 6
4n+
− y
n++ x
n+− y
n++
x
➢
475818 103832 1903272 415328 30
5 1
4 2 4
2 6
4 6
4n+
− y
n++ x
n+− y
n++
x
➢
103 4 412 180
30 1
2 2 3
2 4 4 5
4n+
− y
n++ y
n+− y
n++ y
➢
11329 4 45316 19800
3300 1
2 2 4
2 4 4 6
4n+
− y
n++ y
n+− y
n++
y
➢
103 11329 412 45316 180
30 1
3 2 4
2 5
4 6
4n+
− y
n++ y
n+− y
n++
y
6. Each of the following expressions represents a quintic integer
➢
472
5 54
5 62360
3 320
3 44720
140
2
30 1
+ +
+ +
+
+
−
n+
n−
n+
n−
nn
x x x x x
x
➢
− +
− +
−
+ +
+ +
+ +
3 1
5 3 3
3 7
5 5 5
40 519160
20 259580
4 51916
3300 1
n n
n n
n n
x x
x x
x x
➢
42
5 58
5 5210
3 340
3 3420
180
1
5 1
+ +
+ +
+
+
−
n+
n−
n+
n−
nn
x x y x y
x
➢
−
+
− +
−
+
+ +
+ +
+ 2
1 4
3 3
3 6
5 5 5
80
43260 40
21630 8
4326 275
1
n
n n
n n
n
y
x y
x y
x
➢
− +
− +
−
+ +
+ +
+ +
3 1
5 3 3
3 7
5 5 5
80 4758180
40 2379090
8 475818
30245 1
n n
n n
n n
y x
y x
y x
➢
− +
− +
−
+ +
+ +
+ +
3 2
5 3 4
3 7
5 6
5
4720 519160
2360 259580
472 51916
30 1
n n
n n
n n
x x
x x
x x
➢
−
+
− +
−
+
+ +
+ +
+ 1
2 3
3 4
3 5
5 6
5
9440
420 4720
210 944
42 275
1
n
n n
n n
n
y
x y
x y
x
➢
− +
− +
−
+ +
+ +
+ +
2 2
4 3 4
3 6
5 6
5
8440 43260
4720 21630
944 4326
5 1
n n
n n
n n
y x
y x
y x
➢
− +
− +
−
+ +
+ +
+ +
3 2
5 3 4
3 7
5 6
5
9440 4758180
4720 2379090
944 475818
275 1
n n
n n
n n
y x
y x
y x
➢
− +
− +
−
+ +
+ +
+ +
2 3
4 3 5
3 6
5 7
5
1038320 43260
519160 21630
103832 4326
275 1
n n
n n
n n
y x
y x
y x
➢
− +
− +
−
+ +
+ +
+ +
3 3
5 3 5
3 7
5 7
5
1038320 4758180
519160 2379090
103832 475818
5 1
n n
n n
n n
y x
y x
y x
➢
5 6103
5 55
3 4515
3 310
21030
1
30 1
+ +
+ +
+
+
−
n+
n−
n+
n−
nn
y y y y y
y
➢
−
+
− +
−
+
+ +
+ +
+
1
3 3
3 5
3 5 5 7
5
113290
10 56645
5 11329
3300 1
n
n n
n n
n
y
y y
y y
y
➢
−
+
− +
−
+
+ +
+ +
+ 2
3 4
3 5
3 6
5 7
5
113290
1030 56645
515 11329
103 30
1
n
n n
n n
n
y
y y
y y
y
➢
− +
− +
−
+ +
+ +
+ +
1 3
3 3 5
3 5
5 7
5
1038320 420
519160 210
103832 42
30245 1
n n
n n
n n
y x
y x
y x
Remarkable observations
1. 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 Hyperbolas
( X , Y )
1
X
2− Y 21
2= 3600 ( 472 x
n+1− 4 x
n+2, x
n+2− 103 x
n+1)
2
X
2− Y 21
2= 43560000 ( 51916 x
n+1− 4 x
n+1, x
n+3− 11329 x
n+1)
3
X
2− Y 21
2= 100 ( 42 x
n+1− 8 y
n+1, 2 y
n+1− 8 x
n+1)
4
X
2− Y 21
2= 302500 ( 4326 x
n+1− 8 y
n+2, 2 y
n+2− 944 x
n+1)
5
X
2− Y 21
2= 3659040100
(475818 x
n+1− 8 y
n+3, 2 y
n+3− 103832 x
n+1) 6
X
2− Y 21
2= 3600 ( 51916 x
n+2− 472 x
n+3, 103 x
n+3− 11329 x
n+2)
7
X
2− Y 21
2= 302500 ( 42 x
n+2− 944 y
n+1, 206 y
n+1− 8 x
n+2)
8
X
2− Y 21
2= 100 ( 4326 x
n+2− 944 y
n+2, 206 y
n+2− 944 x
n+2)
9
X
2− Y 21
2= 302500 ( 475818 x
n+2− 944 y
n+3, 206 y
n+3− 103832 x
n+2)
10
X
2− Y 21
2= 3659040100 ( 42 x
n+3− 103832 y
n+1, 22658 y
n+1− 8 x
n+3)
11
X
2− Y 21
2= 302500 ( 4326 x
n+3− 103832 y
n+2, 22658 y
n+2− 944 x
n+3)
12
X
2− Y 21
2= 100 ( 475818 x
n+3− 103832 y
n+3, 22658 y
n+3− 103832 x
n+3)
13
18900 X
2− 900 Y
2= 68040000 ( y
n+2− 103 y
n+1, 472 y
n+1− 4 y
n+2)
14
21 X
2−Y
2= 914760000 ( y
n+3− 11329 y
n+1, 51916 y
n+1− 4 y
n+3)
15
21 X
2−Y
2= 75600 ( 103 y
n+3− 11329 y
n+2, 51916 y
n+2− 472 y
n+3)
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
30 X − Y 21
2= 1800 ( 472 x
2n+2− 4 x
2n+3, x
n+2− 103 x
n+1)
2
3300 X − Y 21
2= 21780000 ( 5196 x
2n+2− 4 x
2n+4, x
n+3− 11329 x
n+1)
3
5 X − Y 21
2= 50 ( 42 x
2n+2− 8 y
2n+2, 2 y
n+1− 8 x
n+1)
4
275 X − Y 21
2= 151250 ( 4326 x
2n+2− 8 y
2n+3, 2 y
n+2− 944 x
n+1)
5
30245 X − Y 21
2= 1829520050 ( 475818 x
2n+2− 8 y
2n+4, 2 y
n+3− 103832 x
n+1)
6
30 X − Y 21
2= 1800 ( 51916 x
2n+3− 472 x
2n+4, 103 x
n+3− 11329 x
n+2)
7
275 X − Y 21
2= 151250 ( 42 x
2n+3− 944 y
2n+2, 206 y
n+1− 8 x
n+2)
8
5 X − Y 21
2= 50 ( 4326 x
2n+3− 944 y
2n+3, 206 y
n+2− 944 x
n+2)
9
275 X − Y 21
2= 151250 ( 475818 x
2n+3− 944 y
2n+4, 206 y
n+3− 103832 x
n+2)
10
30245 X − Y 21
2= 1829520050 ( 42 x
2n+4− 103832 y
2n+2, 22658 y
n+1− 8 x
n+3)
11
275 X − Y 21
2= 151250 ( 4326 x
2n+4− 103832 y
2n+3, 22658 y
n+2− 944 x
n+3)
12
5 X − Y 21
2= 50 ( 475818 x
2n+4− 103832 y
2n+4, 22658 y
n+3− 103832 x
n+3)
13
18900 X − Y 30
2= 1134000 ( y
2n+3− 103 y
2n+2, 472 y
n+1− 4 y
n+2)
14
69300 X −Y
2= 457380000 ( y
2n+4− 11329 y
2n+2, 51916 y
n+1− 4 y
n+3)
15
630 X −Y
2= 37800 ( 103 y
2n+4− 11329 y
2n+3, 51916 y
n+2− 472 y
n+3)
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