International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 p-ISSN: 2395-0072 www.irjet.net 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 ) n1 (9 2 20 ) n1 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 xn1 2 f n 9 yn1 9 f n 40 20 20 gn (4) gn (5) Therefore (3) becomes 20 xn1 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 © 2019, IRJET | 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 ) | ISO 9001:2008 Certified Journal | Page 1566 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 p-ISSN: 2395-0072 20 xn2 36 20 f n 161g n www.irjet.net 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 yn3 xn1 161xn3 0 36 20 (9 f n 2 20 g n ) 161(9 g n 2 20 g n ) 20 xn3 646 20 f n 2889g n 7. yn2 40xn1 9 yn1 0 (8) 8. yn3 720xn1 161yn1 0 For simplicity and clear understanding the other choices of 9.9 yn3 40xn1 161yn2 0 integer solutions are presented below: 20 y n1 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 yn2 161 20 f n 720 g n (10) 20 yn3 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. yn3 40xn2 yn1 0 Some few numerical examples are given in the following table 16.161yn2 40xn3 9 yn1 0 Table 1: Numeric examples 17.xn1 xn3 4 yn2 0 n xn 1 y n 1 -1 4 18 0 72 322 19.40xn3 yn2 9 yn3 0 1 1292 5778 20. 40 yn1 720 yn2 40 yn3 0 2 23184 103682 3 416020 1860498 4 7465176 33385282 18.161yn3 720xn3 yn1 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 161x3n3 483xn1 27xn2 2 3n4 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 ( y3n5 12920x3n3 27 y n3 161 38760xn1 1. xn 3 18xn 2 xn 1 0 1 161x3n5 2889x3n4 483xn3 2 8667 xn2 2. 2 yn 1 9xn 1 xn 2 0 © 2019, IRJET | 1 161y3n 3 40x3n 4 483yn 1 120xn 2 9 161y3n 4 720x3n 4 783yn 2 2160xn 2 Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1567 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 p-ISSN: 2395-0072 www.irjet.net 1 161y3n5 12920x3n4 783 yn3 9 38760xn2 1 2889 y3n3 40 x3n5 yn1 161 40 xn3 1 2889 y3n4 720 x3n5 8667 yn2 9 2160xn3 1 323 y3n3 y3n5 949 yn1 18 3 yn3 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 y3n5 12920x3n5 8667 y n3 38760xn3 18 y3n 3 y3n 4 54 yn 1 3 yn 2 323 y3n 4 18 y3n5 969 y n 2 54 y n3 1 x5n7 321x5n5 5x3n5 1505x3n3 4 10 xn3 3210xn1 1 9 x5n6 161x5n5 805x3n3 45x3n4 2 1610xn1 90 xn2 9 y5n5 40 x5n5 45 y3n3 200 x3n3 400 xn1 90 y n1 2.2. Each of the following expression represents biquadratic integer: y5n6 80 x5n5 5 y3n4 400x3n3 800xn1 10 y n2 1 9 y5n7 12920x5n5 64600x3n3 161 45 y3n5 129200xn1 90 yn3 1 161x5n7 2889x5n6 805 y3n3 2 14445x3n 4 400 xn2 1610 y n1 161y5n6 720 x5n6 3600x3n4 805 y3n4 7200xn 2 10 y n 2 1 x4n 6 321x4n 4 1284x2n 2 4x2n 4 24 4 1 9x4n5 161x4n4 644x2n2 36x2n3 12 2 9 y4n4 40x4n4 36 y2n2 160x2n2 6 y4n 5 80x4n 4 4 y2n 3 320x2n 2 6 1 9 y4n6 12920x4n4 36 y2n4 161 51680x2n2 322 1 161y5n7 12920x5n6 64600x3n4 9 805 y3n5 129200xn2 1610 y n3 1 161x4n6 2889 y4n5 644x2n4 2 11556x2n3 12 1 161y5n5 40 x5n6 200 x3n4 9 805 y3n3 400 xn2 1610 yn1 1 161y4n4 40 x4n5 644 y2n2 9 160 x2n3 54 1 2889 y5n5 40 x5n7 200x3n5 161 14445 y3n3 400xn3 28890 yn1 1 2889 y5n6 720 x5n7 3600x3n5 9 14445 y3n4 7200xn3 28890 y n2 2889 y5n7 12920x5n7 64600x3n5 14445 y3n5 129200xn3 28890 y n3 18 y5n5 y5n7 90 y3n3 5 y3n4 10 y n2 180 y n1 1 323 y5n5 y5n7 1615 y3n3 18 5 y3n5 10 yn3 3230 yn1 323 y5n6 18 y 5n7 1615 y3n4 90 y3n5 180 yn3 3230 yn2 161y 4n5 720 x4n5 644 y 2n3 2880x2n3 6 1 161y4n6 12920x4n5 644 y2n4 51680x2n3 9 54 1 2889 y4n4 40 x4n6 11556 y2n2 161 160x2n4 322 2889 y 4 n5 720 x4 n6 11556 y 2 n3 1 2880 x2 n 4 9 54 2889 y 4n6 12920x4n6 11556 y 2n 4 51680x2n 4 6 18 y4n 4 y4n 5 72 y2n 2 4 y2n 3 6 © 2019, IRJET | 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 | ISO 9001:2008 Certified Journal | Page 1568 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 p-ISSN: 2395-0072 www.irjet.net 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 161yn2 720 xn2 , 161xn2 36 yn2 9 X n 2 20Yn 2 324 161yn3 12920xn2 , 2889xn2 36 yn3 10 X n 2 20Yn 2 2889 yn1 40 xn3 , 9 xn3 646 yn1 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 y2n4 77520x2n4 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 yn2 720 xn3 , 161xn3 646 yn2 12 X n 2 20Yn 2 4 2889 yn3 12920xn3 , 2889xn3 646 yn3 13 80 X n 2 Yn 2 320 18 y n1 y n2 , 9 yn2 161y n1 14 80 X n 2 81Yn 2 323 yn1 yn3 , yn3 321yn1 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 yn2 18 yn3 , 161yn3 2889 yn2 Table 2: Hyperbola S.NO Hyperbola 1 X n 2 40Yn 2 4 9 xn 2 161xn1 , 18 xn1 xn2 2 81X n 2 80Yn 2 5184 xn1 321xn3 , 323xn1 xn3 3 X n 20Yn 4 9 yn1 40 xn1 , 9 xn1 2 yn1 4 81X n 2 20Yn 2 324 yn2 80 xn1 , 161xn1 2 yn2 5 X n 2 20Yn 2 9 yn3 12920xn1 , 2889 xn1 2 yn3 2 2 103684 6 X n 80Yn 16 7 X n 162Yn 324 2 2 2 © 2019, IRJET 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 161xn3 2889xn2 , 323xn2 18xn3 161yn1 40 xn2 , xn2 4 yn1 Impact Factor value: 7.211 S.NO Parabola 1 X n 40Yn 2 8 2 9 X n 40Yn 2 144 8 x2n4 321x2n2 , 323x2n2 x2n4 3 X n 20Yn 2 4 2 9 y2n2 40 x2n2 , 9 x2 n 2 2 y 2 n 2 4 9 X n 20Yn 2 4 8 y2n3 80 x2n2 , 161x2n2 2 y2n3 5 161X n 20Yn 2 322 9 y2n4 12920x2n2 , 2889 x2n2 2 y2n4 101164 6 | X n 40Yn 2 8 (Xn,Yn) 8 9 x2n3 161x2n2 , 18x2 n2 x2n3 4 161x2n4 2889x2n3 , 323x2n3 18x2n4 ISO 9001:2008 Certified Journal | Page 1569 International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 06 Issue: 03 | Mar 2019 p-ISSN: 2395-0072 www.irjet.net 7 X n 180Yn 2 36 18 161y2n2 40 x2n3 , x2n3 4 y2n2 6 Y 9X 323 yn2 18 yn3 , 2889 yn2 720 xn3 8 X n 20Yn 2 4 2 161y 2n3 720 x2n3 , 161x2n3 36 y 2n3 7 Y 2X 2889 y n2 720xn3 , 323 y n1 yn3 9 9 X n 20Yn 2 324 18 161y2n4 12920x2n3 , 2889x2n3 36 y2n4 8 Y 2X 9 yn1 40 xn1 , 9 xn2 161xn1 10 161X n 20Yn 2 322 2889 y2n2 40 x2n4 , 9 x2n4 646 y2n2 9 Y 4X yn2 80 xn1 , xn3 321xn1 103684 11 X n 20Yn 2 36 18 2889 y2n3 720 x2n4 , 161x2n4 646 y2n3 10 Y 161X 161yn2 720 xn2 , 9 yn3 12920xn1 12 X n 20Yn 2 4 2 2889 y 2n 4 12920x2n 4 , 2889x2n4 646 y 2n 4 11 YX 161yn2 12920xn2 , 161yn1 40 xn2 13 20 X n Yn 2 2 18 y2n2 y2n3 , 161y2n2 9 y2n3 12 Y 2X 2889 yn2 720 xn3 , 323 yn1 yn3 36 323 y2n2 y2n4 , 321y2n2 y2n4 13 Y 9X yn2 80 xn1 , 161yn1 40 xn2 2 323 y2n3 18 y2n4 , 2889 y2n3 161y2n4 14 Y 9X yn1 40 xn1 , 161yn1 40 xn2 15 Y 161X 161yn2 720 xn2 , 2889 yn1 40 xn3 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 xn2 161xn1 , xn3 321xn1 2 Y 4X 9 y n1 40 xn1 , xn3 321xn1 3 YX y n2 80 xn1 , 9 y n1 40 xn1 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: yn2 80 xn1 , 9 yn3 12920xn1 323 yn2 18 yn3 , 323 yn1 yn3 | Impact Factor value: 7.211 | 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 ISO 9001:2008 Certified Journal | Page 1570 v. International Research Journal of Engineering and Technology (IRJET) 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 xn1 2GLn1(18,1) 36GFn1(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) 160GFn1(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 Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 1571