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Observations on the Non-homogeneous binary Quadratic Equation
8 x 2  3 y 2  20
T.R. Usha Rani1, V. Bahavathi2, K. Sridevi3
1,2Assistant
Professor, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
3PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
--------------------------------------------------------------------***----------------------------------------------------------------------
Abstract – A Non-homogeneous binary quadratic equation represents hyperbola given by 8x 2  3 y 2  20 is analyzed for its nonzero distinct integer solutions. A few interesting relation between the solution of the given hyperbola, integer solutions for other
choices of hyperbola and parabola are obtained.
Key Words: Non-homogeneous quadratic, binary quadratic, integer solutions.
1. INTRODUCTION
The binary quadratic Diophantine equations of the form ax 2  by 2  N , (a, b, N  0) are rich in variety and have been analysed
by many mathematicians for their respective integer solutions for particular values of a , b and N. In this context, one may refer
[1-13].
This communication concerns with the problem of obtaining non-zero distinct integer solutions to the binary quadratic
equation given by 8x 2  3 y 2  20 representing hyperbola. A few interesting relations among its solutions are presented.
Knowing an integral solution of the given hyperbola, integer solutions for other choices of hyperbolas and parabolas are
presented.
2. Method of Analysis
The Diophantine equations representing the binary quadratic equation to be solved for its non-zero distinct integer solution is
8x 2  3 y 2  20
(1)
Consider the linear transformations
x  X  3T , y  X  6T
(2)
From(1) and(2), we have
X 2  30T 2  19
(3)
Whose smallest positive integer solution is
X 0  7, T0  1
To obtain the other solutions of (3), consider the Pell equation
X 2  30T 2  1
(4)


~ ~
Whose smallest positive integer solution is X 0 , T0  1,5
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The general solution of (4) is given by
~
Tn 
1
~
gn , X n  fn
2
2 30
1
where
f n  (5  24 ) n 1  (5  24 ) n 1
g n  (5  24 ) n1  (5  24 ) n1 , n  1,0,1,.....,
Applying Brahmaguptha lemma between x0 , y0  and ~
xn , ~
yn  the other integer solutions of (1) are given by,
xn 1  8 f n 
39
y n 1  13 f n 
64
24
24
gn
gn
The recurrence relations satisfied by and y are given by
xn 3  10 xn 2  xn 1  0
yn 3  10 yn  2  yn 1  0
Some numerical examples of x and y satisfying (1) are given in the Table :1 below
Table:1 Numerical example
n
xn
yn
0
16
26
1
158
258
2
1564
2554
3
15482
25282
4
153256
250266
From the above table, we observe some interesting relations among the solutions which are presented below:
 Both xn and y n values are even .
2.1. Relations among the solutions are given below.
 3 yn 1  xn  2  5xn 1  0
 xn 3  10 xn  2  xn 1  0
 3 yn  2  5xn  2  xn 1  0
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 3 yn3  49 xn 2  5xn1  0
 30 yn 1  xn3  49 xn 1  0
 49 yn 1  yn 3  80 xn 1  0
 49 yn 2  5 yn3  8xn1  0
 5xn3  49 xn 2  3 yn 1  0
 5 yn 2  8xn  2  yn 1  0
 yn 3  16 xn  2  yn 1  0
 49 yn  2  8xn 3  5 yn1  0
 49 yn 3  80 xn3  yn1  0
 8xn 1  yn  2  5 yn 1  0
 yn 3  10 yn  2  yn 1  0
 3 yn 3  5xn3  xn  2  0
 xn 3  5xn 2  3 yn  2  0
 yn 3  8xn  2  5 yn  2  0
 5 yn 3  8xn 3  yn  2  0
 6 yn 3  xn 3  xn 1  0

30 yn 3  49 xn 3  xn 1  0
2.2. Each of the following expression is a nasty number:
1
60  384 x2n2  234 y2n2 
5
1
 180  2322 x2n 2  234 x2n3 
15
1

1800  22986x2n2  234 x2n4 
150
1
 300  3792 x2n 2  234 y2n3 
25
1

2940  37536 x2n2 _ 234 y2n4 
245
1
 600  384 x2n3  2322 y2n 2 
25
1

2940  384 x2n4  22986 y2n2 
245
1
 480  384 y2n3  3792 y2n 2 
40
1

4800  384 y2n4  37536 y2n2 
400
1
 180  22986 x2n3  2322 x2n 4 
15
1
 60  3792 x2n3  2322 y2n3 
5
1
 300  37536 x2n 3  2322 x2 n  4 
25
1
 300  3792 x2 n 4  22986 y2n 3 
25

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1
60  37536x2n4  22986x2n3 
5
1
 480  3792 y2 n  4  37536 y2 n3 
40

2.3. Each of the following expressions is a cubical integer.
1
64 x3n3  39 y3n3  192 xn1  117 yn1 
5
1
 387 x3n3  39 x3n 4  1161xn1  117 xn 2 
15
1

3831x3n3  39 x3n5  11493xn1  117 xn3 
150
1
 632 x3n3  39 y3n 4  1896 xn1  117 yn 2 
25
1

6256 x3n3  39 y3n5  18768xn1  117 yn3 
245
1
 64 x3n 4  387 y3n3  192 xn 2  1161yn1 
25
1

64 x3n5  3831y3n3  192 xn3  11493 yn1 
245
1
 64 y3n 4  632 y3n3  192 yn 2  1896 yn1 
40
1

64 y3n5  6256 y3n3  192 yn3  18768 yn1 
400
1
 3831x3n 4  387 x3n 5  11493xn 2  1161xn 3 
15
1
 632 x3n 4  387 y3n 4  1896 xn 2  1161yn 2 
5
1
 6256 x3n 4  387 y3n5  18768xn 2  1161yn 2 
25
1
 632 x3n5\  3831y3n 4  1896 xn3  11493 yn 2 
25
1
 6256 x3n5  3831y3n5  18768 xn3  11493 yn3 
5
1
 632 y3n5  6256 y3n 4  1896 yn3  18768 yn 2 
40

2.4. Each of the following expressions is a biquadratic integer.
1
64 x4n4  39 y4n4  256 x2n2  156 y2n2  30
5
1
 387 x2n 2  39 x4 n5  1548x2n 2  156 x2 n3  90
15
1

3831x4n4  39x4n6  15324x2n2  156 x2n4  900
150
1
 632 x4n 4  39 y4n5  2528 x2n 2  156 y2n3  150
25
1

6256x4n4  39 y4n6  25024 x2n2  156 y2n4  1470
245

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1
64 x4n5  387 y4n4  256x2n3  1548 y2n2  150
25
1

64 x4n6  3831y4n4  256 x2n4  15324 y2n2  1470
245
1
 64 y4n5  632 y4n 4  256 y2n3  2528 y2n 2  240
40
1

64 y4n6  6256 y4n4  256 y2n4  25024 y2n2  2400
400
1
 3831x4n5  387 x4 n6  15324 x2n 3  1548x2n 4  90
5
1
 632 x4n5  387 y4n5  2528 x2n3  1548 y2n3  30
5
1
 6256 x4n5  387 y4n6  25024 x2n3  1548 y2n 4  150
25
1
 632 x4n6  3831y4 n6  25624 x2n 4  15324 y2n  4  30
25
1
 6256 x4n6  3831y4n6  25624 x2n 4  15324 y2n 4  30
5
1
 632 y4n6  6256 y4 n5  2528 y2 n 4  25024 y2n 3  240
40

2.5.Each of the following expression is a quintic integer.

1
64x5n5  39 y5n5  320x3n3  195 y3n3  640xn1  390 yn1 
5

1
387 x5n5  39 x5n6  1935x3n3  195x3n 4  3870 xn1  390 xn 2 
15

1
3831x5n5  39 x5n 7  19155x3n3  195x3n 5  38310xn 1  390 xn3 
150

1
632 x5n5  39 y5n6  3160 x3n3  195 y3n4  6320 xn1  390 yn2 
25

1
6256 x5n 5  39 y5n  7  31280 x3n 3 195 y3n 5  62560 xn 5  390 xn 3 
245




1
64 x5n6  387 y5n5  320 x3n 4  1935 y3n3  640 xn 2  3870 yn1 
25
1
64 x5n  7  632 y5n 5  320 y3n  4  3160 y3n 3  640 yn  2  6320 yn 3 
40
1
64 y5n  7  6256 y5n 5  320 y3n 5  31280 y3n 3  640 yn 3  62560 yn 1 
400
1
3831x5n  6  387 x5n  7  19155x3n  4  1932 y3n  4  38310 xn  2  3870 xn 3 
15

1
632 x5n 6  387 y5n 6  3160 x3n 4  1932 y3n 4  6320 xn  2  3870 yn  2 
5

1
6256 x5n  6  387 y5n  7  31280 x3n  4 1935 y3n 5  62560 xn  2  3870 yn 3 
25

1
632 x5n  7  3831y5n  6  3160 x3n 5 19155 y3n  4  6320 xn 3  38310 yn  2 
25
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
1
6256 x5n  7  3831y5n  7  31280 x3n 5  19155 y3n 5  62560 xn  2  38310 yn 3 
5

1
632 y5n  7  6256 y5n  6  3160 y3n 5  31280 y3n  4  6320 yn 3  62650 yn  2 
40

1
64 x5n  7  3831y5n 5  320 x3n 5  19155 y3n 3  640 xn 3  38310 yn 1 
245
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
II.
S.NO
Hyperbola
(X,Y)
1
Y 2  24 X 2  100
8 yn1  13xn1,64xn1  39 yn1 
2
Y 2  24 X 2  900
8xn2  79xn1 ,387 xn1  39xn2 
3
Y 2  24 X 2  90000
8xn3  782xn1,3831xn1  39xn3 
4
Y 2  24 X 2  2500
8 yn2 129xn1 ,632xn1  39 yn2 
5
Y 2  24 X 2  240100
8 yn3  1277 xn1,6256xn1  39 yn3 
6
Y 2  24 X 2  2500
79 yn1 13xn2 ,64xn2  387 yn3 
7
Y 2  24 X 2  240100
782 yn1 13xn3 ,64 yn2  3831yn1 
8
Y 2  24 X 2  6400
129 yn1 13 yn2 ,64 yn2  632 yn1 
9
Y 2  24 X 2  640000
1277 yn1  13 yn3 ,64 yn3  6256 yn1 
10
Y 2  24 X 2  900
79xn3  782xn2 ,3831xn2  387 xn3 
11
Y 2  24 X 2  100
79 yn2  129xn2 ,632xn2  387 yn2 
12
Y 2  24 X 2  2500
79 yn3 1277 xn2 ,6256xn2  387 yn3 
13
Y 2  24 X 2  2500
782 yn2 129xn3 ,632xn3  3831yn2 
14
Y 2  24 X 2  100
782 yn3  1277 xn3 ,6256xn3  3831yn3 
15
Y 2  24 X 2  6400
1277 yn2  129 yn3 ,632 yn3  6256 yn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:
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Table: 3 Parabolas
S.N0
Parabola
(X,Y)
1
5Y  24 X 2  100
8 yn1  13xn1 ,64x2n2  39 y2n2  10
2
15Y  24 X 2  900
8xn2  79xn1 ,387 x2n2  39x2n3  30
3
150Y  24 X 2  90000
8xn3  782xn1,3831x2n2  39x2n4  300
4
25Y  24 X 2  2500
8 yn2 129xn1 ,632x2n2  39 y2n3  50
5
245Y  24 X 2  240100
8 yn3 1277 xn1 ,6256x2n2  39 y2n4  490
6
25Y  24 X 2  2500
79 yn1  13xn2 ,64x2n3  387 y2n2  50
7
245Y  24 X 2  240100
782 yn1 13xn3 ,64x2n4  3831y2n2  490
8
40Y  24 X 2  6400
129 yn1  13 yn2 ,64 y2n3  632 y2n2  80
9
400Y  24 X 2  640000
1277 yn1 13 yn3 ,64 y2n4  6256 y2n2  800
10
15Y  24 X 2  90
79xn3  782xn2 ,3831x2n3  387 x2n4  30
11
5Y  24 X 2  100
79 yn2  129xn2 ,632x2n3  387 y2n3  10
12
25Y  24 X 2  2500
79 yn2 1277 xn ,6256x2n3  387 y2n4 50
13
25Y  24 X 2  2500
782 yn2 129xn3 ,632x2n4  3831y2n3  50
14
5Y  24 X 2  100
782 yn3  1277 xn3 ,6256x2n4  3831y2n4  10
15
40Y  24 X 2  6400
1277 yn2  129 yn3 ,632 y2n4  6256 y2n4  1610
3. CONCLUSION
In this paper, we have presented infinitely many integer solutions for the Diophantine equation, represented by hyperbola is
given by 8x 2  3 y 2  20 . As the binary quadratic Diophantine equations are rich in variety, one may search for the other
choices of equations and determine their integer solutions along with suitable properties.
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