Integral solutions of Quadratic Diophantine equation

International Journal Of Engineering Research And Development e-ISSN: 2278-067X, p-ISSN: 2278-800X, www.ijerd.com Volume 13, Issue 9 (September 2017), PP. 51-56

Integral solutions of Quadratic Diophantine equation đ?&#x;?đ?&#x;Žđ?’˜đ?&#x;? − đ?’™đ?&#x;? − đ?’šđ?&#x;? + đ?’›đ?&#x;? = đ?’•đ?&#x;? with five unknowns R.Anbuselvi1, * S.Jamuna Rani2 1

Associate Professor , Department of Mathematics, ADM College for women, Nagapattinam, Tamilnadu ,India Asst Professor, Department of Computer Applications, Bharathiyar college of Engineering and Technology, Karaikal, Puducherry, India Corresponding Author: R.Anbuselvi

2

ABSTRACT: The quadratic Diophantine equation given by 10đ?‘¤ 2 − đ?‘Ľ 2 − đ?‘Ś 2 + đ?‘§ 2 = đ?‘Ą 2 is analyzed for its patterns of non-zero distinct integral solutions. A few interesting relations between the solutions and special polygonal numbers are exhibited.

Keywords: Ternary quadratic, integral solutions, polygonal numbers.

I.

INTRODUCTION

Quadratic equations are rich in variety [1-3].For an extensive review of sizable literature and various problems, one may refer [1-16]. In this communication, we consider yet another interesting quadratic equation 10đ?‘¤ 2 − đ?‘Ľ 2 − đ?‘Ś 2 + đ?‘§ 2 = đ?‘Ą 2 and obtain infinitely many non-trivial integral solutions.

A few interesting relations between the solutions and special Polygonal numbers are presented.

Notations Used  đ?‘Ąđ?‘š ,đ?‘› - Polygonal number of rank ‘n’ with size ‘m’  đ??śđ?‘ƒđ?‘›6 - Centered hexagonal Pyramidal number of rank n  đ??şđ?‘›đ?‘œđ?‘› – Gnomic number of rank ‘n’  đ??šđ?‘ đ??´4 – Figurative number of rank ‘n’ with size ‘m’  đ?‘ƒđ?‘&#x;đ?‘› - Pronic number of rank ‘n’  đ?‘ƒđ?‘›đ?‘š - Pyramidal number of rank ‘n’ with size ‘m’  đ?‘˜đ?‘Śđ?‘› - Keynea number

II.

METHODS OF ANALYSIS

The Quadratic Diophantine equation with five unknowns to be solved for its non zero distinct integral solutions is 10đ?‘¤ 2 − đ?‘Ľ 2 − đ?‘Ś 2 + đ?‘§ 2 = đ?‘Ą 2 (1) On substituting the linear transformation đ?‘Ľ =đ?‘¤+đ?‘§ đ?‘Ś =đ?‘¤âˆ’đ?‘§

(2)

in 1 , it leads to 8đ?‘¤ 2 − đ?‘Ą 2 = đ?‘§ 2

3

We obtain different patterns of integral solutions to 1 through solving 3 which are illustrated as follows: Pattern I Equation 3 can be written as 4đ?‘¤ 2 − đ?‘Ą 2 = đ?‘§ 2 − 4đ?‘¤ 2 2đ?‘¤ + đ?‘Ą đ?‘§ − 2đ?‘¤ đ??´ = = ,đ??ľ ≠0 đ?‘§ + 2đ?‘¤ 2đ?‘¤ − đ?‘Ą đ??ľ Considering đ?‘§ − 2đ?‘¤ đ??´ = 2đ?‘¤ − đ?‘Ą đ??ľ 51

Integral Solutions Of Quadratic Diophantine Equation Which is equivalent to the system of equations đ?‘¤ = đ?‘¤(đ??´, đ??ľ) = đ??´2 + đ??ľ2 t = t A, B = 2A2 + 4AB − 2B 2 z = z(A, B) = 2B 2 + 4AB − 2A2

4

Substituting 4 and 3 in 2 , the corresponding non-zero distinct integral solutions of 1 are given b x A, B = x = 3B 2 − A2 + 4AB y A, B = y = 3A2 − B 2 − 4AB z A, B = z = 2B 2 + 4AB − 2A2 w A, B = w = A2 + B 2 t A, B = t = 2A2 + 4AB − 2B 2

5

Properties 1. đ?‘Ľ đ??´ − 1 + 3đ?‘Ś đ??´ − 1 − 16 đ?‘Ą 3,đ??´ ≥ 0 2. 2đ?‘Ľ 1, đ??ľ + đ?‘Ą 1, đ??ľ − 8 đ?‘Ą 3 ,đ??´ ≥ 0 (đ?‘šđ?‘œđ?‘‘ 8) 3. đ?‘Ą đ??´, 1 + 2đ?‘¤ đ??´, 1 − 8 đ?‘Ą 3,đ??´ ≥ 0 4. đ?‘Ą đ??´(đ??´ + 2),1 + 2đ?‘¤ đ??´(đ??´ + 2),1 − 24 đ?‘ƒđ??´3 ≥ 0 5. đ?‘Ą đ??´ đ??´ + 2 (đ??´ + 3),1 + 2đ?‘¤ đ??´(đ??´ + 2)(đ??´ + 3),1 − 96 đ?‘ƒ đ??´4 ≥ 0 6. đ?‘Ą đ??´ đ??´ + 1 (đ??´ + 2),1 + 2đ?‘¤ đ??´ đ??´ + 1 (đ??´ + 2),1 − 48 đ??š 4,đ?‘›âˆ’4 ≥ 0 7. đ?‘§ 1, đ??ľ − 4 đ?‘Ą3 ,đ??ľ ≥ −2 đ?‘šđ?‘œđ?‘‘ 2 8. 2đ?‘Ľ 1, đ??ľ + đ?‘Ą 1, đ??ľ − đ?‘Ą 8,đ??ľ = 0(đ?‘šđ?‘œđ?‘‘ 12) 9. đ?‘Ľ đ??´, 1 + đ?‘Ś đ??´, 1 − đ?‘Ą 4 ,đ??´ ≥ 2 đ?‘šđ?‘œđ?‘‘ 3 10. đ?‘Ľ đ??´, đ??´ + đ?‘Ś đ??´, đ??´ can be expressed as a perfect square.

11. Each of the following expressions represents a perfect number (i) đ?‘Ą 2,1 + đ?‘§ 1,2 (ii) đ?‘Ľ 2,1 + đ?‘Ľ 2,2 ]đ?‘Ą(2,2) (iii) đ?‘Ľ 1,1 (iv) đ?‘Ą 1,2 + đ?‘Ś 1,1 (v) đ?‘Ľ 2,2 + đ?‘§(1,1) 12. Each of the following expressions represents a perfect square (i) đ?‘¤ 3,3 + đ?‘Ľ(2,1) (ii) đ?‘Ľ 3,3 + đ?‘Ľ 2,2 + đ?‘Ś(2,1) (iii) 2đ?‘Ľ 3,3 + đ?‘Ľ 1,2 + đ?‘Ś(2,1) 13. Each of the following expressions represents a cube number (i) đ?‘Ľ 3,3 + đ?‘Ľ(1,2) (ii) đ?‘Ą(1,2) (iii) đ?‘§ 1,2 + đ?‘¤ 1,2 + đ?‘Ą(1,2) 1 (iv) đ?‘Ľ 3,3 + 2đ?‘§ 3,3 − đ?‘¤(1,1) 2 (v) 4đ?‘Ľ(3,3) 14. Each of the following expressions represents nasty number (i) đ?‘Ľ 2,2 + đ?‘Ľ 1,1 (ii) đ?‘Ľ(1,1)đ?‘Ľ 2,2 (iii) đ?‘Ľ 1,1 đ?‘¤ 1,2 (iv) đ?‘Ś 2,1 đ?‘§ 2,1 đ?‘¤ 2,1 (v) đ?‘Ľ 2,2 đ?‘¤(1,2) Pattern II The equation (3) can be written as 8đ?‘¤ 2 − đ?‘Ą 2 = đ?‘§ 2 ∗ 1 Then write đ?‘§ = 8đ?‘Ž2 – đ?‘? 2 1 = 2 + 1 ( 2 − 1)

(6) (7)

Substituting 7 into 6 reduces to 52

Integral Solutions Of Quadratic Diophantine Equation 8đ?‘¤ 2 − đ?‘Ą 2 = 2 2 đ?‘¤ + đ?‘Ą (2 2 đ?‘¤ − đ?‘Ą) Equating rational and irrational parts, the coefficient values are đ?‘?2 đ?‘¤ = đ?‘¤ đ?‘Ž, đ?‘? = 4đ?‘Ž2 + + 2đ?‘Žđ?‘?

2 đ?‘Ą = đ?‘Ą đ?‘Ž, đ?‘? = 8đ?‘Ž2 + đ?‘? 2 + 8đ?‘Žđ?‘? đ?‘§ = z(a, b) = 2b2 + 4ab − 2a2

8

As our interest is on finding integer solutions, it is seen that the values of x, y and z are integers when both a and b are of the same parity. Thus by taking đ?‘Ž = 2đ??´ , đ?‘? = 2đ??ľ in 7 and substituting the corresponding values of u, v in 2 the non-zero integral solutions of 1 are given by đ?‘Ľ = đ?‘Ľ(đ??´, đ??ľ) = 12đ??´2 − 2đ??ľ2 + 4đ??´đ??ľ đ?‘Ś = đ?‘Ś đ??´, đ??ľ = −4đ??´2 + 6đ??ľ2 + 4đ??´đ??ľ đ?‘§ = đ?‘§ đ??´, đ??ľ = 8đ??´2 − 4đ??ľ2 đ?‘¤ = đ?‘¤ đ??´, đ??ľ = 4đ??´2 + 2đ??ľ2 + 4đ??´đ??ľ đ?‘Ą = đ?‘Ą đ??´, đ??ľ = 8đ??´2 + 4đ??ľ2 + 16đ??´đ??ľ

9

Properties đ?‘¤ đ??´, đ??ľ − đ?‘Ś đ??´, đ??ľ = đ?‘§(đ??´, đ??ľ) đ?‘§ đ??´, 1 + đ?‘Ą đ??´, 1 − 16đ?‘ƒđ?‘&#x;đ??´ ≥ 0 đ?‘¤ đ??´, 1 − đ?‘Ś đ??´, 1 − 8 đ?‘Ą 4,đ??´ + 4 ≥ 0 đ?‘Ľ 1, đ??ľ + đ?‘¤ 1, đ??ľ − 4đ??şđ?‘›0 − 20 ≥ 0 2đ?‘Ś 1, đ??ľ + đ?‘§ 1, đ??ľ − 16đ?‘Ą đ??ľ ≥ 0 đ?‘¤ đ??´, 1 + đ?‘Ą đ??´, 1 − 24đ?‘Ąđ??´ ≥ 6(đ?‘šđ?‘œđ?‘‘ 8) đ?‘Ľ 1, đ??ľ + 3đ?‘Ś 1, đ??ľ − 32 đ?‘Ą 8,đ??ľ = 0 đ?‘Ľ 1, đ??ľ đ??ľ + 2 + 3đ?‘Ś 1, đ??ľ đ??ľ + 2 − 96 đ?‘ƒđ??´3 ≥ 0 đ?‘Ľ 1, đ??ľ2 (đ??ľ + 1) + 3đ?‘Ś 1, đ??ľ2 (đ??ľ + 1) − 32 đ?‘ƒ đ??´5 ≥ 0 đ?‘Ľ 1, đ??ľ đ??ľ + 2 (đ??ľ + 3) + 3đ?‘Ś 1, đ??ľ(đ??ľ + 2)(đ??ľ + 3) − 384 đ?‘ƒ đ??´4 ≥ 0 Each of the following expressions represents a Nasty number (i) đ?‘Ą 1,3 + đ?‘§ 1,1 (ii) đ?‘Ą 2,3 + đ?‘§ 2,2 (iii) đ?‘Ľ 3,3 − đ?‘Ľ(1,3) (iv) đ?‘Ľ 3,1 + đ?‘¤ 1,2 + đ?‘Ľ(1,0) (v) đ?‘¤ 1,1 + đ?‘Ľ 1,0 + đ?‘§ 1,0 (vi) đ?‘Ą 1,1 − đ?‘§(1,1) 12. Each of the following expressions represents a perfect number (i) đ?‘Ą 3,3 + 2đ?‘Ľ 3,2 − đ?‘§(1,1) (ii) 4đ?‘Ą(3,1) (iii) 8 đ?‘Ą 3,3 + đ?‘Ą 2,3 + đ?‘Ą 3,2 + đ?‘Ą 3,1 + đ?‘Ľ 3.1 + đ?‘Ľ 3,3 + đ?‘§ 2,2 đ?‘Ś(2,2) (iv) đ?‘Ą 1,2 − đ?‘¤ 1,2 (v) đ?‘§ 2,2 − đ?‘¤ 1,1 13. Each of the following expressions represents a cube number (i) đ?‘Ą 1,3 + đ?‘Ą(3,1) (ii) đ?‘Ą 3,3 + đ?‘Ą 3,2 + đ?‘§ 3,2 + đ?‘¤(1,2) (iii) 4đ?‘Ą 3,3 − đ?‘§ 1,0 (iv) đ?‘Ą 3,3 − đ?‘§ 3,3 (v) đ?‘§ 3,1 − đ?‘¤(1,0) 14. Each of the following expressions represents perfect square (i) đ?‘¤ 2,2 + đ?‘Ś 2,2 (ii) đ?‘Ą 3,3 + đ?‘Ą 0,1 (iii) đ?‘Ľ 3,1 + đ?‘§ 2,2 + đ?‘¤ 1,1 (iv) 2đ?‘Ą 3,2 + đ?‘Ą 2,3 + đ?‘Ą 3,3 (v) đ?‘Ś 2,3 + đ?‘¤(0,1) Pattern III Consider the linear transformation 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

53

Integral Solutions Of Quadratic Diophantine Equation đ?‘¤ = đ?‘˘ − đ?‘Ł, đ?‘Ą = 2đ?‘˘ − 4đ?‘Ł Substituting (9) into (3) the equation reduces to 2đ?‘˘ + đ?‘§ đ?‘Ł đ??´ = = , 8đ?‘Ł 2đ?‘˘ − đ?‘§ đ??ľ

(10) đ??ľâ‰ 0

Simplifying and employing the method of factorization, đ?‘˘ = 8đ??´2 + đ??ľ2 đ?‘Ł = 4đ??´đ??ľ đ?‘§ = 3đ??´2 + đ??ľ2 Substituting the corresponding values of u,v in 2 the non-zero integral solutions of 1 are given by đ?‘Ľ đ??´, đ??ľ = đ?‘Ľ = 24đ??´2 − đ??ľ2 − 4đ??´đ??ľ đ?‘Ś đ??´, đ??ľ = đ?‘Ś = 3đ??ľ2 − 8đ??´2 − 4đ??´đ??ľ đ?‘§ đ??´, đ??ľ = đ?‘§ = 16đ??´2 − 2đ??ľ2 đ?‘¤ đ??´, đ??ľ = đ?‘¤ = 8đ??´2 + đ??ľ2 − 4đ??´đ??ľ đ?‘Ą đ??´, đ??ľ = đ?‘Ą = 16đ??´2 + 2đ??ľ2 − 16đ??´đ??ľ

11

Properties 1. đ?‘§ đ??´, 1 − đ?‘Ą đ??´, 1 ≥ −4(đ?‘šđ?‘œđ?‘‘ 16) 2. đ?‘Ś 1, đ??ľ + đ?‘¤ 1, đ??ľ − 4 đ?‘ƒđ?‘&#x; đ??´ ≥ 0(đ?‘šđ?‘œđ?‘‘ 12) 3. 3đ?‘¤ 1, đ??ľ) − đ?‘Ś 1, đ??ľ + 9đ??şđ?‘›0 + 41 ≥ 0 4. đ?‘Ś đ??´, đ??´2 + đ?‘¤ đ??´, đ??´2 + 8đ??śđ?‘ƒđ??´6 − 4đ?‘Ą4,đ??´ ≥ 0 5. đ?‘Ľ đ??ľ, đ??ľ − đ?‘¤ đ??ľ, đ??ľ − 14đ?‘Ą4,đ??ľ ≥ 0 6. Each of the following expressions represents a Nasty number (i) 2đ?‘Ľ 1,2 (ii) đ?‘Ą 2,1 − đ?‘¤ 1,2 (iii) đ?‘Ľ 2,3 - x(1,3) (iv) đ?‘Ľ 3,2 − đ?‘§ 1,2 (v) đ?‘Ľ 3,3 − 3đ?‘Ś 1,3 7. Each of the following expressions represents a perfect number (i) đ?‘¤ 2,2 + đ?‘Ą 2,2 (ii) đ?‘§ 3,1 + đ?‘Ľ 3,2 + đ?‘Ľ 3,1 + đ?‘Ś 2,1 (iii) 2đ?‘Ľ 3,1 + đ?‘Ľ 3,2 + đ?‘¤ 2,2 + đ?‘Ś(1,3) (iv) đ?‘Ś 1,2 (v) đ?‘§ 1,1 − đ?‘§(1,2) 8. Each of the following expressions represents a cube number (i) đ?‘Ľ 3,2 − đ?‘Ś(1,2) (ii) đ?‘§ 2,2 + đ?‘Ą 2,2 (iii) đ?‘Ľ 3,1 + đ?‘§ 3,1 (iv) 2đ?‘§ 3,1 + đ?‘Ľ 3,1 + đ?‘¤ 2,2 + đ?‘¤(1,1) (v) 3đ?‘§ 3,1 + 2 đ?‘Ľ 3,1 − đ?‘Ś(3,1 + đ?‘¤(1,2)] 9. Each of the following expressions represents perfect square (i) đ?‘Ľ 2,1 − đ?‘Ś 2,1 + đ?‘¤(2,2) (ii) đ?‘Ľ 3,2 + đ?‘§(3,2) (iii) đ?‘Ľ 3,1 − đ?‘Ś 1,1 + đ?‘Ľ 3,2 (iv) đ?‘§ 3,3 − đ?‘¤(1,1) (v) đ?‘Ą 3,3 + đ?‘Ś 1,3 Pattern IV Consider the transformation 2đ?‘˘ + đ?‘§ 2đ?‘˘ − đ?‘§ = 8 ∗ đ?‘Ł 2 Equating positive and negative factor, we get 2đ?‘˘ + đ?‘§ = đ?‘Ł 2 2đ?‘˘ − đ?‘§ = 8 Solving the above equations, we get 54

(12)

Integral Solutions Of Quadratic Diophantine Equation đ?‘Ł2 + 8 4 đ?‘Ł2 − 8 đ?‘§= 2

�=

(13)

As our interest on finding integer solutions, choose v so that u and z are integers Then write đ?‘˘ = đ??´2 + 2đ??´ + 3 đ?‘Ł = 2đ??´ + 2

(14)

Substituting 14 in (13), the solutions are đ?‘Ľ đ??´, đ??ľ = đ?‘Ľ = 3đ??´2 + 4đ??´ − 1 đ?‘Ś đ??´, đ??ľ = đ?‘Ś = −đ??´2 − 4đ??´ + 3 đ?‘§ đ??´, đ??ľ = đ?‘§ = 2đ??´2 + 4đ??´ − 2 đ?‘¤ đ??´, đ??ľ = đ?‘¤ = đ??´2 + 1 đ?‘Ą đ??´, đ??ľ = đ?‘Ą = 2đ??´2 − 4đ??´ − 2

(15)

Properties 1.

2.

3.

4.

5.

6.

7.

8.

đ?‘¤ 2 − đ?‘Ś 7 is a Kynea number đ?‘Ś đ??´ − đ?‘Ľ đ??´ + 2đ?‘§(đ??´) = 0 đ?‘Ś đ??´2 + đ?‘¤ đ??´2 + 4đ??śđ?‘ƒđ??´6 ≥ 4 đ?‘§ đ??´ + 2đ?‘¤ đ??´ − 2đ??şđ?‘› 0 ≥ 2 Each of the following expression represents a perfect Square (i) đ?‘§(9) (ii) đ?‘Ľ 7 + đ?‘Ś(7) (iii) đ?‘§ 9 + đ?‘Ą(9) (iv) đ?‘Ľ 9 + đ?‘§ 9 + 2đ?‘Ą(3) (v) đ?‘Ľ 8 + đ?‘Ľ 10 + đ?‘Ą 7 − đ?‘Ą(3) Each of the following expression represents a perfect number (i) đ?‘Ľ 9 + đ?‘Ľ 8 − đ?‘Ą(3) (ii) đ?‘§(3) (iii) 2 đ?‘Ľ 8 + đ?‘¤(7) (iv) đ?‘§ 7 + đ?‘Ś 7 = đ?‘¤(7) Each of the following expression represents a cube number (i) đ?‘Ľ 10 + đ?‘§(1) (ii) đ?‘Ľ 10 + đ?‘§ 10 − đ?‘¤(8) (iii) đ?‘Ľ 9 + đ?‘¤(8) (iv) đ?‘¤ 6 + đ?‘§ 4 − đ?‘Ś(5) (v) đ?‘Ľ 8 + đ?‘Ľ 10 − đ?‘¤(7) Each of the following expression represents a nasty number (i) đ?‘¤ 8 − đ?‘¤ 2 (ii) đ?‘§ 7 − 2đ?‘¤ 1 (iii) đ?‘¤ 10 − đ?‘¤ 2 (iv) đ?‘Ľ 3 + đ?‘§ 4 + đ?‘Ľ 5 + đ?‘¤(1) (v) đ?‘Ľ 7 − 6đ?‘§(1)

III.

CONCLUSION

To conclude, one may search for other patterns of solutions and their corresponding properties.

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R.Anbuselvi.“Integral solutions of Quadratic Diophantine equation 〖10w〗^2-x^2-y^2 z^2=t^2 with five unknowns.â€? International Journal Of Engineering Research And Development , vol. 13, no. 09, 2017, pp. 51–56.

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