International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
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INTEGRAL SOLUTIONS OF THE DIOPHANTINE EQUATION Y2=20x2+4
Dr. G. Sumathi1, M. Jaya Bharathi2
1Assistant
Professor, Department of mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
2PG Scholar, Department of Mathematics, Shrimati Indira Gandhi College, Trichy-620 002,
Tamil Nadu, India.
---------------------------------------------------------------------***----------------------------------------------------------------------
Abstract:- The binary quadratic equation y 2  20 x 2  4
Whose smallest positive integer solutions of ( x0 , y0 ) is,
is considered and a few interesting properties among the
solutions are presented .Employing the integral solutions
of the equation under considerations a few patterns of
Pythagorean triangles are observed.
x0  4 , y0  18
(2)
To obtain the other solutions of (1), Consider pellian
equation is
Key Words: Binary, Quadratic, Pyramidal numbers, integral
solutions
y 2  20 x 2  1
The initial solution of pellian equation is
1. INTRODUCTION
(3)
~
x0  2 , ~
y0  9
The binary quadratic equation of the form y 2  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
[1-5]. In this context one may also refer [4, 10]. These
results have motivated us to search for the integral
solutions of yet another binary quadratic equation
y 2  20 x 2  4 representing a hyperbola .A few interesting
properties among the solution are presented. Employing
the integral solutions of the equation consideration a few
patterns of Pythagorean triangles are obtained.
Whose general solution ~
xn , ~
yn  of (2) is given by,
~
xn 
1
1
gn , ~
yn  f n
2
2 20
where,
f n  (9  2 20 ) n 1  (9  2 20 ) n 1
g n  (9  2 20 ) n1  (9  2 20 ) n1
1.1 Notations
Applying Brahmagupta lemma between x0 , y0  and ~
xn , ~
yn 
the other integer solution of (1) are given by,
t m,n : Polygonal number of rank n with size m
Pn m : Pyramidal number of rank n with size m
Prn : Pronic number of rank n
Sn : Star number of rank n
Ctm,n : Centered Pyramidal number of rank n with
size m
GFn (k , s) : Generalized Fibonacci sequence of rank n
GLn (k , s) : Generalized Lucas sequence of rank n
xn1  2 f n 
9
yn1  9 f n 
40
20
20
gn
(4)
gn
(5)
Therefore (3) becomes
20 xn1  2 20 f n  9 g n
2. METHOD OF ANALYSIS
(6)
Replace n by n  1 in (6), we get
Consider the binary quadratic Diophantine equation is
y 2  20x 2  4
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20 xn  2  2 20 f n 1  9 g n 1
(1)
Impact Factor value: 7.211
 2 20 (9 f n  2 20 g n )  9(9 g n  2 20 f n )
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Volume: 06 Issue: 03 | Mar 2019
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20 xn2  36 20 f n  161g n
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3. 2 yn  2  xn 1  9xn  2  0
(7)
n by n  1 in (7), we get
4. 2 yn 3  9xn 1  161xn  2  0
20 xn  3  36 20 f n 1  161g n 1
5. 36 yn 1  161xn 1  xn 3  0
Replace
6.36 yn3  xn1 161xn3  0
 36 20 (9 f n  2 20 g n )  161(9 g n  2 20 g n )
20 xn3  646 20 f n  2889g n
7. yn2  40xn1  9 yn1  0
(8)
8. yn3  720xn1 161yn1  0
For simplicity and clear understanding the other choices of
9.9 yn3  40xn1 161yn2  0
integer solutions are presented below:
20 y n1  9 20 f n  9 g n
10. 2 yn 1  161xn  2  9xn 3  0
(9)
11. 2 yn  2  9xn  2  xn 3  0
20 yn2  161 20 f n  720 g n
(10)
20 yn3  2889 20 f n  12920g n
12. 2 yn 3  xn  2  9xn 3  0
(11)
13. 9 yn 3  720xn  2  9 yn 1  0
These are representing the non-zero distinct integer
14. 9 yn  2  40xn  2  yn 1  0
solutions of (1)
15. yn3  40xn2  yn1  0
Some few numerical examples are given in the following
table
16.161yn2  40xn3  9 yn1  0
Table 1: Numeric examples
17.xn1  xn3  4 yn2  0
n
xn 1
y n 1
-1
4
18
0
72
322
19.40xn3  yn2  9 yn3  0
1
1292
5778
20. 40 yn1  720 yn2  40 yn3  0
2
23184
103682
3
416020
1860498
4
7465176
33385282
18.161yn3  720xn3  yn1  0
2.1. Each of the following expression represents a
cubical integer:
1
x  321x3n 3  3xn 3  963xn 1 )
 4 3n  5
1
9x  161x3n3  483xn1  27xn2 
 2 3n4
 9 y3n 3  40x3n 3  27 yn 1  120xn 1
 y3n  4  80x3n 3  3 yn  2  240xn 1
The recurrence relation satisfied by the values of xn 1 and
yn 1 are respectively
xn 3  18xn  2  xn 1  0
,
n  1,0,1,......
yn 3  18 yn  2  yn 1  0 n  1,0,1,.....
,
A few interesting relations among the solutions are
presented below:

1 ( y3n5  12920x3n3  27 y n3 


161  38760xn1

1. xn 3  18xn  2  xn 1  0

1 161x3n5  2889x3n4  483xn3 


2  8667 xn2

2. 2 yn 1  9xn 1  xn  2  0

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1
161y3n 3  40x3n  4  483yn 1  120xn  2 
9
 161y3n  4  720x3n  4  783yn  2 2160xn  2
Impact Factor value: 7.211
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Volume: 06 Issue: 03 | Mar 2019
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
1 161y3n5  12920x3n4  783 yn3 


9  38760xn2


1 2889 y3n3  40 x3n5  yn1 


161  40 xn3


1 2889 y3n4  720 x3n5  8667 yn2 


9  2160xn3


1 323 y3n3  y3n5  949 yn1 


18  3 yn3


1
323y4n  4  y4n 6  1292 y2n  2  4 y2n  4  36
18

232 y4n 5  18 y4n  6  1292 y2n 3  72 y2n  4  6
2.3. Each of the following expression represents quintic
integer:
2889 y3n5  12920x3n5  8667 y n3 
 

 38760xn3

 18 y3n 3  y3n  4  54 yn 1  3 yn  2
323 y3n 4  18 y3n5  969 y n 2 
 

 54 y n3


1  x5n7  321x5n5  5x3n5  1505x3n3 


4  10 xn3  3210xn1


1 9 x5n6  161x5n5  805x3n3  45x3n4 


2  1610xn1  90 xn2


9 y5n5  40 x5n5  45 y3n3  200 x3n3 


 400 xn1  90 y n1


2.2. Each of the following expression represents biquadratic integer:

 y5n6  80 x5n5  5 y3n4  400x3n3 


 800xn1  10 y n2

1 9 y5n7  12920x5n5  64600x3n3 


161  45 y3n5  129200xn1  90 yn3 
1 161x5n7  2889x5n6  805 y3n3 


2  14445x3n 4  400 xn2  1610 y n1 
161y5n6  720 x5n6  3600x3n4  805 y3n4 


 7200xn 2  10 y n 2


1
x4n  6  321x4n  4  1284x2n  2  4x2n  4  24
4


1
9x4n5  161x4n4  644x2n2  36x2n3  12
2

9 y4n4  40x4n4  36 y2n2 160x2n2  6
y4n 5  80x4n  4  4 y2n 3  320x2n  2  6

1 9 y4n6  12920x4n4  36 y2n4 


161  51680x2n2  322

1 161y5n7  12920x5n6  64600x3n4 


9  805 y3n5  129200xn2  1610 y n3 

1 161x4n6  2889 y4n5  644x2n4 


2 11556x2n3  12

1 161y5n5  40 x5n6  200 x3n4 


9  805 y3n3  400 xn2  1610 yn1 

1 161y4n4  40 x4n5  644 y2n2 


9  160 x2n3  54


1 2889 y5n5  40 x5n7  200x3n5


161  14445 y3n3  400xn3  28890 yn1 


1 2889 y5n6  720 x5n7  3600x3n5


9  14445 y3n4  7200xn3  28890 y n2 

2889 y5n7  12920x5n7  64600x3n5 


 14445 y3n5  129200xn3  28890 y n3 

18 y5n5  y5n7  90 y3n3  5 y3n4 


 10 y n2  180 y n1


1 323 y5n5  y5n7  1615 y3n3 


18  5 y3n5  10 yn3  3230 yn1 

323 y5n6  18 y 5n7 1615 y3n4 


 90 y3n5  180 yn3  3230 yn2 






161y 4n5  720 x4n5  644 y 2n3 


 2880x2n3  6

1 161y4n6  12920x4n5  644 y2n4  51680x2n3 


9  54






1 2889 y4n4  40 x4n6  11556 y2n2 


161  160x2n4  322

2889 y 4 n5  720 x4 n6  11556 y 2 n3 
1

 2880 x2 n 4

9
 54

2889 y 4n6  12920x4n6  11556 y 2n 4 


 51680x2n 4  6

18 y4n  4  y4n 5  72 y2n  2  4 y2n 3  6
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Impact Factor value: 7.211
2.4. Each of the following expression represents Nasty
number:
1

48  x2n  4  321x2n  2 
4
 12  54 y2n  2  240x2n  2
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 12  6 y2n 3  480x2n  2








1
1932  54 y2n  4  77520x2n  2 
161
1
12  161x2n  4  2889x2n 3 
2
1
108  966 y2n  2  240x2n 3 
9
12  966 y2n 3  4320x2n 3
8
X n 2  20Yn 2  4
161yn2  720 xn2 , 


161xn2  36 yn2 
9
X n 2  20Yn 2  324
161yn3  12920xn2 , 


 2889xn2  36 yn3 
10
X n 2  20Yn 2
 2889 yn1  40 xn3 , 


 9 xn3  646 yn1 
 103684
1
108  966 y2n  4  77520x2n 3 
9
1
1932  17334 y2n  2  240x2n  4 
161
1
108  17334 y2n 3  4320x2n  4 
9
12  17334 y2n4  77520x2n4
 12  108y2n  2  6 y2n 3
1

216  1938y2n  2  6 y2n  4 
18
2.5. REMARKABLE OBSERVATIONS
11
X n 2  20Yn 2  324
 2889 yn2  720 xn3 , 


161xn3  646 yn2 
12
X n 2  20Yn 2  4
 2889 yn3  12920xn3 , 


 2889xn3  646 yn3 
13
80 X n 2  Yn 2  320
18 y n1  y n2 , 


 9 yn2  161y n1 
14
80 X n 2  81Yn 2
 323 yn1  yn3 , 


 yn3  321yn1 
 103680
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:
15
80 X n 2  Yn 2  320
 323 yn2  18 yn3 , 


161yn3  2889 yn2 
Table 2: Hyperbola
S.NO
Hyperbola
1
X n 2  40Yn 2  4
 9 xn 2  161xn1 , 


18 xn1  xn2 
2
81X n 2  80Yn 2  5184  xn1  321xn3 , 


 323xn1  xn3 
3
X n  20Yn  4
 9 yn1  40 xn1 , 


 9 xn1  2 yn1 
4
81X n 2  20Yn 2  324
 yn2  80 xn1 , 


161xn1  2 yn2 
5
X n 2  20Yn 2
 9 yn3  12920xn1 , 


 2889 xn1  2 yn3 
2
2
 103684
6
X n  80Yn  16
7
X n  162Yn  324
2
2
2
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2
|
II 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:
(Xn,Yn)
Table 3: Parabola
161xn3  2889xn2 , 


 323xn2  18xn3 
161yn1  40 xn2 , 


 xn2  4 yn1

Impact Factor value: 7.211
S.NO
Parabola
1
X n  40Yn 2  8
2
9 X n  40Yn 2  144  8  x2n4  321x2n2 , 


 323x2n2  x2n4

3
X n  20Yn 2  4
 2  9 y2n2  40 x2n2 , 


 9 x2 n  2  2 y 2 n 2

4
9 X n  20Yn 2  4
 8  y2n3  80 x2n2 , 


161x2n2  2 y2n3 
5
161X n  20Yn 2
 322  9 y2n4  12920x2n2 , 


 2889 x2n2  2 y2n4

 101164
6
|
X n  40Yn 2  8
(Xn,Yn)
 8  9 x2n3  161x2n2 , 


18x2 n2  x2n3

 4  161x2n4  2889x2n3 , 


 323x2n3  18x2n4

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7
X n  180Yn 2  36
18  161y2n2  40 x2n3 , 


 x2n3  4 y2n2

6
Y  9X
 323 yn2  18 yn3 , 


 2889 yn2  720 xn3 
8
X n  20Yn 2  4
 2  161y 2n3  720 x2n3 , 


161x2n3  36 y 2n3

7
Y  2X
 2889 y n2  720xn3 , 


 323 y n1  yn3

9
9 X n  20Yn 2  324 18  161y2n4  12920x2n3 , 


 2889x2n3  36 y2n4

8
Y  2X
 9 yn1  40 xn1 , 


 9 xn2  161xn1 
10
161X n  20Yn 2
 322  2889 y2n2  40 x2n4 , 


 9 x2n4  646 y2n2

9
Y  4X
 yn2  80 xn1 , 


 xn3  321xn1 
 103684
11
X n  20Yn 2  36
18  2889 y2n3  720 x2n4 , 


161x2n4  646 y2n3

10
Y  161X
161yn2  720 xn2 , 


 9 yn3  12920xn1 
12
X n  20Yn 2  4
 2  2889 y 2n 4  12920x2n 4 , 


 2889x2n4  646 y 2n 4

11
YX
161yn2  12920xn2 , 


161yn1  40 xn2

13
20 X n  Yn 2
 2  18 y2n2  y2n3 , 


161y2n2  9 y2n3 
12
Y  2X
 2889 yn2  720 xn3 , 


 323 yn1  yn3

 36  323 y2n2  y2n4 , 


 321y2n2  y2n4

13
Y  9X
 yn2  80 xn1 , 


161yn1  40 xn2 
 2  323 y2n3  18 y2n4 , 


 2889 y2n3  161y2n4 
14
Y  9X
 yn1  40 xn1 , 


161yn1  40 xn2 
15
Y  161X
161yn2  720 xn2 , 


 2889 yn1  40 xn3 
 320
14
160 X n  20Yn 2
 11520
15
80 X n  Yn 2
 320
III Employing linear combinations among the solutions of
(1), one may generate integer solutions for other choices of
straight line which are presented in the table below:
IV Consider p  xn 1  yn 1 , q  xn 1
Table: 4 Straight lines
S.NO
Straight line
1
Y  2X
 9 xn2  161xn1 , 


 xn3  321xn1 
2
Y  4X
 9 y n1  40 xn1 , 


 xn3  321xn1 
3
YX
 y n2  80 xn1 , 


 9 y n1  40 xn1 
4
5
Y  161X
Y  18 X
© 2019, IRJET
Note that p  q  0 ,treat p, q as the generators of
the Pythagorean triangle T ( X , Y , Z )
(X,Y)
Where X  2 pq, Y  p 2  q 2 , Z  p 2  q 2 , p  q  0
It is observed that T ( X , Y , Z ) is satisfied by the following
relations:
 yn2  80 xn1 ,



 9 yn3  12920xn1 
 323 yn2  18 yn3 , 


 323 yn1  yn3 
|
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|
i.
X  10Y  9Z  4
ii.
2A
 xn 1 yn 1
P
iii.
X
iv.
3(Z  Y ) is a Nasty number
4A
Y  Z
P
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v.
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e-ISSN: 2395-0056
Volume: 06 Issue: 03 | Mar 2019
p-ISSN: 2395-0072
X
www.irjet.net
CONCLUSION
4A
 Y is written as the sum of two squares
P
In this dissertation , we have presented infinitely many
integer solutions of the binary quadratic Diophantine
equation are rich in variety, one may search for the other
choices of binary quadratic Diophantine equation and
determine their integral solutions along with suitable
properties.
Where A, P represents the area and perimeter of T ( X , Y , Z )
V Employing the solutions in terms of special integers
sequence namely, generalized Fibonacci sequence, GFn(k,s)
and the generalized Lucas sequence GLn(k,s) are exhibited
below:
REFERENCES
xn1  2GLn1(18,1)  36GFn1(18,1)
1.
Dickson.L.E, History of Theory of Numbers and
Diophantine Analysis, Dover Publications, New York
2005.
2.
Mordell.L.J, Diophantine Equations, Academic Press,
New York 1970.
3.
Carmicheal.R.D, The theory of Numbers and Diophantine
Analysis, Dover Publications, New York 1959.
4.
Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation
on the hyperbola x 2  19 y 2  3 ”, Scholars Journal of
the Engineering and Technology 2014;2(2A);152-155.
5.
Gopalan.M.A, Sumathi.G, Vidhyalakshmi.S, “Observation
on the hyperbola y 2  26x 2  1 ,Bessel Math, Vol 4,
Issue(1),2014;21-25.
6.
On
special
families
of
Hyperbola, x 2  (4k 2  k ) y 2  a 2 , a  1 The International
Journal of
Science
and
Technology ,Vol
2,Issue(3),Pg.No.94-97,March 2014.
7.
Dr.G.Sumathi “Observations on the hyperbola
y 2  182x 2  14 ”, Journal of Mathematics and
Informatics,Vol 11, 73-81,2017
8.
Dr.G.Sumathi “Observations on the pell equation
y 2  14x 2  4 ”, International Journal of Creative
Research Thoughts,Vol 6,Issue 1,Pp1074-1084,March
2018
9.
Dr.G.Sumathi “Observation
on the Hyperbola
y 2  312x 2  1 ” International Journal of Mathematics
Trends and Technology,Vol 50,Issue 4,31-34,Oct 2017
yn 1  9GLn 1(18,1)  160GFn1(18,1)
VI Employing the solutions of (1),each of the following
among the special polygonal, pyramidal, star numbers,
pronic numbers, centered pyramidal number is a congruent
to under modulo 4.
2

3
 12 p 5 y 



  20 36 p x  2 
 s 1 
 s y 1  1 
 x 1



2

5
 6 p 4 y 1 



  20 4 p x 
 ct  1 
 t3,2 y 1 
 4, x



2
2

5
 4 p5 y 



  20 6 p x 
 ct  1 
 ct 4, y  1 
 6, x 


2

5
 3 p3 y  2 



  20 4 p x 
 ct  1 
 t3, y  2 
 4, x 


2

 2 p 5 y 1 
 3 

  20 3 p x 
t

 t4, y 1 
 3, x 1 


2

5 
 p5 y 


  20 12 p x 
 s 1 
 t3, y 
 x 1 


2


 6 p 4 y 1 



  20 4 pt x 3 
3
 p x 3 
 t3, 2 y 1 





 3 p3 y  2 
 5 

  20 p x 
t 
 t3, y  2 
 3, x 


2

© 2019, IRJET
2
2
2
2
2
2
 3 p3 y  2 
 3 

  20 6 p x 
 pr 
 t3, y  2 
 x 1 


2
2
2
2
4
3
 6 p 4 y 1 



  20 3( p x 1  p x 1 ) 


 t3,2 y 1 
t4, x 1




|
10. Dr.G.Sumathi
“Observation on the Hyperbola
2
2
y  150x  16 ” International Journal of recent
Trends in Engineering and Research ,Vol 3,Issue
9,Pp198-206,Sep 2017
2
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