ON THE INTEGRAL SOLUTION OF BINARY QUADRATIC …Diophantine equation with two unknown for finding its infinite integral solutions based on Pell’s equation. Keywords: Diophantine

ON THE INTEGRAL SOLUTION OF BINARY QUADRATIC DIOPHANTINE

EQUATION 22 )1()1( yabxxa

aK. Prabhakaran, bA. Hari Ganesh and cK. Mahalakshmi

a,b Assistant Professor and cResearch Scholar

aAnnai Vailankanni Arts and Science College, Thanjvur – 613 007.

b,c Poompuhar College (Autonomous), Melaiyur – 609 107, Nagapattinam (Dt.).

Abstract:

A Diophantine equation is a polynomial equation, usually in two or more unknown, such

that only the integer solutions are studied. In this paper, we have proposed a second order

Diophantine equation 22 )1()1( yabxxa with two unknown for finding its infinite integral

solutions based on Pell’s equation.

Keywords: Diophantine Equation, Integral Solution, Pell’s Equation.

Introduction:

The homogenous and non homogenous binary quadratic Diophantine equations are being

solved in wide range on now days. Particularly, the binary quadratic non-homogeneous equations

representing hyperbolas are studied by many authors for its non - zero integral solutions [2, 3]. In

this paper, we propose a non - homogenous quadratic Diophantine equation for find its non –

zero integral solutions. Moreover, the recurrence relations for the solutions of the proposed non –

homogenous quadratic Diophantine equation are also to be presented in this paper.

Method of Analysis:

The non - homogenous quadratic Diophantine equation representing the hyperbola to be

solved for its non - zero distinct integral solution is

22 )1()1( yabxxa -------------------- (1)

The equation (1) is to be solved through the following two different cases.

Case I:

The first among the two is

22 )1()1( yabxxa -------------------- (2)

Treating (2) as quadratic in x and solving for x , we have

0)1()1( 22 yabxxa

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 4, 2019 (Special Issue) © Research India Publications. http://www.ripublication.com

Page 108 of 115

To eliminate the square root on the right hand side of the solution of the above equation,

assume

2222)1(4)1(2 byabxa

Now the above equation is in the form of Pellian Equation as follows:

NDyX 22

Where bxaX )1(2 , )1(4 2 aD , 2bN , 1 , ),1( aZmamb

The general solution ),( nn yx of equation (2) is given by

)1(2

)1()1(4)1(4)1(4)1(42

10

2

0

2

000

2

0

2

00

a

amyaXayXyaXayX

x

nn

n

0

2

0

2

000

2

0

2

002

)1(4)1(4)1(4)1(4)1(42

1yaXayXyaXayX

ay

nn

n

Where )1(4 2

00 ayX is the fundamental solution of 1)1(4)1(2 222 yabxa

and )1(4 2

00 ayX is the fundamental solution of 2222)1(4)1(2 byabxa

For the sake of simplicity a few solutions of (2) for 2a are presented through the following

example.

Now equation (2) becomes,

223 ybxx ------------------------------ (3)

Treating the above equation as quadratic in x and find its solution in the form of following

equation

222126 bybx

It is in the form of Pellian equation as

NDyX 22

where bxX 6 ,2bN


Page 109 of 115

The general solution ),( nn yx of equation (3) as follows:

6

3121212122

100000000 myXyXyXyX

x

nn

n

00000000 12121212

122

1yXyXyXyXy

nn

n

Where 1200 yX and 1200 yX are fundamental solutions of 1126 22 ybx

Table 1

The

Values

of n

For m = 1, b = 3, N = 9,

the solutions

),(),( , mnmn yx

For m = 2, b = 6, N = 36

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(1, 0)

(4, 6)

(49, 84)

(676, 1170)

(9409, 16296)

(131044, 226974)

(2, 0)

(8, 12)

(98, 168)

(1352, 2340)

(18818, 32592)

(262088, 453948)

(3, 3)

(27, 45)

(363, 627)

(5043, 8733)

(70227, 121635)

(978123, 1694157)

Table 2

The

Values

of n

For m = 3, b = 9, N = 81,

the solutions

),(),( , mnmn yx

For m = 4, b = 12 , N = 144,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(3, 0)

(12, 18)

(147, 252)

(2028, 3510)

(28227, 48888)

(393132, 680922)

(4, 0)

(16, 24)

(196, 336)

(2704, 4680)

(37636, 65184)

(524176, 907896)

(6, 6)

(54, 90)

(726, 1254)

(10086, 17466)

(140454, 243270)

(1956246, 3388314)


Page 110 of 115

Table 3

The

Values

of n

For m = 5, b = 15, N = 225,

the solutions

),(),( , mnmn yx

For m = 6, b = 18 , N = 324,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(5, 0)

(20, 30)

(245, 420)

(3380, 5850)

(47045, 81480)

(655220, 1134870)

(6, 0)

(24, 36)

(294, 504)

(4056, 7020)

(56454, 97776)

(786264, 1361844)

(9, 9)

(81, 135)

(1089, 1881)

(15129, 26199)

(210681, 364905)

(2934369, 5082471)

Table 4

The

Values

of n

For m = 7, b = 21, N = 441,

the solutions

),(),( , mnmn yx

For m = 8, b = 24 , N = 576,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(7, 0)

(28, 42)

(343, 588)

(4732, 8190)

(65863, 114072)

(917308, 1588818)

(8, 0)

(32, 48)

(392, 672)

(5408, 9360)

(75272, 130368)

(1048352, 1815792)

(12, 12)

(108, 180)

(1452, 2508)

(20172, 34932)

(280908, 486540)

(3912492, 6776628)

Further the solutions satisfy the following recurrence relation:

(a) Recurrence relations for solution ),(),( , mnmn yx and .2 ,1,, ),(),( iyx

ii mnmn among the

different values of b

(i) 0214 )12,1()12,()12,1( bxxx knknkn where 0n and ,.......2 ,1 ,0k

(ii) 0214 )22,1()22,())22(,1( 1 bxxx knknkn where 0n and ,.......2 ,1 ,0k


Page 111 of 115

(b) Recurrence relations for solution ),(),( , mnmn yx and .2 ,1,, ),(),( iyx

ii mnmn

(i) 02 )32,())22(,()12,( 1 knknkn xxx where ,.......2 ,1 ,0k

(ii) ,0)12,())22(,()1,( 1 knknn xxx where ,.......2 ,1 k

(c) Recurrence relations for solution .2 ,1,, ),(),( iyxii mnmn

(i) myyxx knknknkn ))2(,())2(,())2(,())2(,( 2121 where ,.......2 ,1 k

Case II:

The second among the two is

22 )1()1( yabxxa -------------------------- (4)

Treating (4) as a quadratic in x and solving for x , we have

0)1()1( 22 yabxxa

To eliminate the square root on the right hand side of the solution of the above equation,

assume

2222)1(4)1(2 byabxa

Now the above equation is in the form of Pellian Equation as follows:

NDyX 22

Where bxaX )1(2 , )1(4 2 aD , 2bN , 1 , ),1( aZmamb

The general solution ),( nn yx of equation (4) is given by

)1(2

)1()1(4)1(4)1(4)1(42

10

2

0

2

000

2

0

2

00

a

amyaXayXyaXayX

x

nn

n

0

2

0

2

000

2

0

2

002

)1(4)1(4)1(4)1(4)1(42

1yaXayXyaXayX

ay

nn

n

Where )1(4 2

00 ayX is the fundamental solution of 1)1(4)1(2 222 yabxa

and )1(4 2

00 ayX is the fundamental solution of 2222)1(4)1(2 byabxa

For the sake of simplicity a few solutions of (2) for 2a are presented through the following

example.


Page 112 of 115

Now equation (4) becomes,

22 3ybxx

Treating the above equation as quadratic in x and find its solution in the form of following

equation

222122 bybx

It is in the form of Pellian equation as

NDyX 22

Where bxX 2 ,2bN

The general solution ),( nn yx of equation (3) as follows:

2

121212122

100000000 myXyXyXyX

x

nn

n

00000000 12121212

122

1yXyXyXyXy

nn

n

Where 1200 yX and 1200 yX are fundamental solutions of 1122 22 ybx

Table 5

The

Values

of n

For m = 1, b = 1, N = 1,

the solutions

),(),( , mnmn yx

For m = 2, b = 2, N = 4,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(1, 0)

(4, 2)

(49, 28)

(676, 390)

(9409, 5432)

(131044, 75658)

(2, 0)

(8, 4)

(98, 56)

(1352, 780)

(18818, 10864)

(262088, 151316)

(3, 1)

(27, 15)

(363, 209)

(5043, 2911)

(70227, 40545)

(978123, 564719)


Page 113 of 115

Table 6

The

Values

of n

For m = 3, b = 3, N = 9,

the solutions

),(),( , mnmn yx

For m = 4, b = 4 , N = 16,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(3, 0)

(12, 6)

(147, 84)

(2028, 1170)

(28227, 16296)

(393132, 226974)

(4, 0)

(16, 8)

(196, 112)

(2704, 1560)

(37636, 21728)

(524176, 302632)

(6, 2)

(54, 30)

(726, 418)

(10086, 5822)

(140454, 81090)

(1956246, 1129438)

Table 7

The

Values

of n

For m = 5, b = 5, N = 25,

the solutions

),(),( , mnmn yx

For m = 6, b = 6 , N = 36,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(5, 0)

(20, 10)

(245, 140)

(3380, 1950)

(47045, 27160)

(655220, 378290)

(6, 0)

(24, 12)

(294, 168)

(4056, 2340)

(56454, 32592)

(786264, 453948)

(9, 3)

(81, 45)

(1089, 627 )

(15129, 8733)

(210681, 121635)

(2934369, 1694157)

Table 8

The

Values

of n

For m = 7, b = 7, N = 49,

the solutions

),(),( , mnmn yx

For m = 8, b = 8 , N = 64,

the solutions

.2 ,1,, ),(),( iyxii mnmn

0

1

2

3

4

5

(7, 0)

(28, 14)

(343, 196)

(4732, 2730)

(65863, 38024)

(917308, 529606)

(8, 0)

(32, 16)

(392, 224)

(5408, 3120)

(75272, 43456)

(1048352, 605264)

(12, 4)

(108, 60)

(1452, 836)

(20172, 11644)

(280908, 162180)

(3912492, 2258876)


Page 114 of 115

Further the solutions satisfy the following recurrence relation:

(c) Recurrence relations for solution ),(),( , mnmn yx and .2 ,1,, ),(),( iyx

ii mnmn among the

different values of b

(i) 0614 )12,1()12,()12,1( bxxx knknkn where 0n and ,.......2 ,1 ,0k

(ii) 0614 )22,1()22,())22(,1( 1 bxxx knknkn where 0n and ,.......2 ,1 ,0k

(d) Recurrence relations for solution ),(),( , mnmn yx and .2 ,1,, ),(),( iyx

ii mnmn

(i) 02 )32,())22(,()12,( 1 knknkn xxx where ,.......2 ,1 ,0k

(i) ,0)12,())22(,()1,( 1 knknn xxx where ,.......2 ,1 k

(c) Recurrence relations for solution .2 ,1,, ),(),( iyxii mnmn

(i) 0))2(,())2(,())2(,())2(,( 2121 knknknkn yyxx where ,.......2 ,1 k

Reference:

1. Carmichael, R.D., The Theory of Numbers and Diophantine Analysis, Dover

Publications, New York, 1950.

2. Gopalan, M.A., Vidhyalakahmi, S., and Devibala, S., On The Diophantine Equation

143 2 xyx , Acta Ciencia Indica,Vol. XXXIII M.No2, pp.645-646, 2007.

3. Gopalan, M.A., Devibala, S., & Vidhyalakahmi, R., Integral points on the hyperbola

532 22 yx , American Journal of Applied Mathematics and Mathematical

Sciences,Vol. I, No. 1, pp.1- 4, 2012.

4. Mollion, RA., All Solutions of the Diophatine Equations, nDYX 22, Far East

Journal of Mathematical Sciences,Vol. III, pp. 257 – 293, 1998.

5. Mordell, L.J., Diophantine Equations, Acadamic Press, London, 1969.

***********


Page 115 of 115

Documents

ON THE INTEGRAL SOLUTION OF BINARY QUADRATIC …Diophantine equation with two unknown for finding its infinite integral solutions based on Pell’s equation. Keywords: Diophantine