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D0366025029

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The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new …

The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.

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  • 1. The International Journal Of Engineering And Science (IJES) || Volume || 3 || Issue || 6 || Pages || 25-29 || 2014 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805 www.theijes.com The IJES Page 25 Integral Solutions of Binary Quadratic Diophantine equation 2 2 x pxy y N   1, M.A.GOPALAN , 2, R.ANBUSELVI Department Of Mathematics, Srimathi Indira Gandhi College, Trichy,Tamil Nadu, India. AND Department Of Mathematics, A.D.M.College For Women(Autonomous), Nagapattinam, Tamil Nadu, India. -------------------------------------------------------ABSTRACT--------------------------------------------------- Non – trivial integral solutions for the binary quadratic diophantine equation 2 2 x p xy y N   , p  2 , N  0 (m o d 4 ) are obtained . The recurrence relations satisfied by the solutions along with a few examples are given. KEY WORDS: Binary quadratic, integral solutions MSC 2000 Subject classification number: 11D09 --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 29 April 2014 Date of Publication: 05 July 2014 --------------------------------------------------------------------------------------------------------------------------------------- I. INTRODUCTION It is well known that binomial quadratic (homogeneous or non-homogeneous) Diophantine equations are rich in variety[1,2]. The authors have considered the equation and analysed for its integer solutions[3]. In [4], non-trivial integral solutions for the binary quadratic diophantine equation Nypxyx 4 22  are obtained. In this communication, the non-trivial integral solutions for the binary quadratic diophantine equation 2 2 x p xy y N   , where p  2 and N  0 (m o d 4 ) have been obtained. Also the recurrence relations among the solutions are given. II. METHOD OF ANALYSIS The equation to be solved is 2 2 x p xy y N   , p  2 , N  0 (m o d 4 ) (1) The substitution of the linear transformations     2 2 u X p T v X p T       inequation (1) leads to 2 2 2 4 ( 4)4X p T N   (2) Again, setting 2 ,X P 2T Q Equation (2) becomes  2 2 2 4P p Q N   (3) where 2 4p  is a square free non zero integer.
  • 2. Integral Solutions Of Binary Quadratic…. www.theijes.com The IJES Page 26 Assume that the initial solution of equation (2) be  0 0 ,Q P . Consider the Pellian  2 2 2 4 1P p Q   (4) whose general solution ( , )s s Q P  is given by   1 2 2 0 0 4 4 Q s s s P p Q P p         , 0 ,1, 2 ,....s  (5) in which  0 0 ,Q P  is the least positive integral solution of (4). Applying Brahmagupta’s lemma, the sequence of solutions of equation (3) are given by   1 0 0 2 1 0 0 4 s s s s s s Q P Q Q P P P P p Q Q           (6) where 0 ,1, 2 ,........s   the sequence of solutions of equation (1) are given by 2 2 1 0 0 0 0 2 1 4 4 2 2 4 s p x P G p Q F P F p Q G p                  (7) where     1 1 2 2 0 0 0 0 4 4 s s F P p Q P p Q               1 1 2 2 0 0 0 0 4 4 , -1, 0,1, 2,... s s G P p Q P p Q s            Also the recurrence relations among the solutions are given by (1) 3 0 2 1 2 0s s s x P x x      (2) 3 0 2 1 2 0s s s y P y y      To analyze the nature of solutions, one has to go in for particular values of p and N . For the sake of simplicity and clear understanding, a few numerical examples are given below: 2 1 0 0 2 1 4 4 s y P F p Q G p        
  • 3. Integral Solutions Of Binary Quadratic…. www.theijes.com The IJES Page 27 Illustration :1.1 p and N are both odd. Table 1.1(a) 3p  5N  Observations : (1) (m o d 5)i i y x (2) Each of the expressions 2 2 2i i y x  and 2 1 2 1 2i i y x    is a perfect square. (3) 1 1 0 (m o d 4 0 )i i i i x y y x    Illustration :1.2 p is even and N is odd Table 1.2 (b) 4p  13N  i i x i y 0 1 4 1 29 76 2 521 1364 3 9349 24476 4 167761 439204 5 3010349 7881196 6 54018521 141422324 7 969323029 2537720636 8 17393796001 45537549124 9 312119004989 817138163596 i i x i y 0 1 2 3 4 5 6 7 8 9 1 9 127 1769 24639 343177 4779839 66574569 927264127 12915123209 2 34 474 6602 91954 1280754 17838602 248459674 3460596834 48199896002
  • 4. Integral Solutions Of Binary Quadratic…. www.theijes.com The IJES Page 28 Observations : (1) 1 (m o d 8)i i y y  (2) 3 1 3 2 0 (m o d 1 0 )i i y y    (3) (m o d 4 )i i i i x y x y   (4) 1 (m o d 2 )i i x x  (5) 1 1 5 2 0i i i i x y y x     Illustration :1.3 p and N are both even. Table 1.3(c) 4p  4N  Observations : (1) 1 1 0 (m o d 1 6 )i i i i x y y x    (2) (m o d 4 )i i i i x y x y   (3) 1 (m o d 4 )i i y y  (4) 3 1 3 2 (m o d 4 )i i y y   i i x i y 0 1 2 3 4 5 6 7 8 9 0 8 1 1 2 1560 21728 302612 4214840 58705148 817657232 11388496100 2 30 418 5822 81090 1129438 15731042 219105150 3051741058 42505269662
  • 5. Integral Solutions Of Binary Quadratic…. www.theijes.com The IJES Page 29 Illustration :1.4 p is odd and N is even. Table 1.4(d) 3p  4N  Observations: (1) 1 1 1 0 4i i i x y y     (2) (m o d 4 )i i i i y x y x   In conclusion, one may search for other patterns of solutions and their corresponding properties. REFERENCES: [1] Dickson,L.E. (1952) History of the theory of numbers, Vol II, Chelsea Publishing Company, New York. [2] Batta B. and Singh A.N. History of Hindu Mathematics, Asia Publishing House, Bombay (1938). [3] Gopalan M.A., Devibala S. and Anbuselvi R. A Remarkable observation on Nyxyx  22 , ActaCienciaIndica, Vol.XXXI M, No.4, 997 (2005). [4] Gopalan M.A.,and Anbuselvi R. Integral Solutions of NyPxyx 4 22  , Proc.Nat.Acad.Sci. India, 77(A), III, 2007. i i x i y 0 1 2 3 4 5 6 7 8 9 2 54 970 17406 312338 5604678 100571866 1804688910 32383828514 581104224342 6 1 1 0 1974 35422 635622 11405774 204668310 3672623806 65902560198 1182573459758

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