SOLUSI PERSAMAAN PELL NEGATIF BERBENTUK x^2=〖13y〗^2- 3^t
Abstract
ABSTRAK
Persamaan diophantin terdiri atas dua kelompok yaitu persamaan diophantin linier dan nonlinier. Solusi persamaan diophantin nonlinier ini lebih kompleks jika dibandingkan dengan persamaan diophantin linier. Oleh karena itu, untuk mencari solusi persamaan diophantin nonlinier, harus memperhatikan bentuk persamaannya. Salah satu bentuk dari persamaan diophantin nonlinier adalah persamaan Pell yang berbentuk dimana merupakan bilangan bulat positif bukan kuadrat sempurna, persamaan ini pertama kali ditemukan oleh Fermat. Persamaan Pell yang berbentuk mempunyai solusi tak berhingga dengan adalah bilangan bulat positif bukan kuadrat sempurna. Masalah lain yang akan muncul yaitu untuk syarat eksistensi dari solusi persamaan yang berbentuk dengan bilangan bulat positif bukan kuadrat sempurna. Persamaan tersebut dikenal dengan nama Persamaan Pell Negatif. Tujuan dari penelitian ini adalah untuk menemukan eksistensi solusi persamaan Pell negatif berbentuk dengan , dan . Hasil dari penelitian ini adalah Lemma brahmagupta dapat digunakan untuk mencari solusi lain dari persamaan Pell dengan menggunakan rumus:
Kata kunci : Persamaan Diophantin, Persamaan Pell, Persamaan Pell negatif.
ABSTRACT
The diophantin equation consists of two groups is linear and nonlinear diophantin equations. The solution of this nonlinear diophantin equation is more complex when compared with linear diophantin equations. Therefore, to find a solution of nonlinear diophantin equations, must pay attention to the form of equation. One form of nonlinear diophantin equation is the Pell equation in the form of where is a positive integer not a perfect square, this equation was first discovered by Fermat. The Pell equation in the form of has an infinite solution with is positive integers are not perfect squares. Another problem that will arise is to condition the existence of an equation solution in the form with a positive integer d not a perfect square. The equation is known as the Negative Pell Equation. The purpose of this research was to find the existence of a solution of Pell negative equations in the form with , and . The result of this research is Lemma brahmagupta can be used to find other solution of equation Pell by using formula :
Keywords : Diophantin equation, Pell equation, Pell negative equation.
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Eynden, C. V. 2001. Elementary Number Theory Second Edition. Illinois State University. Singapore.
Ahmet Tekcan, Betul Gezer and Osman Bizin “On the integer solutions of the Pell equation ”, World Academy of Science,Engineering and Technology, 1(2007) 522-526.
Gopalan. M. A, V.Sangeetha and Manju Somanath “ On the integer solutions of the Pell equation ”International Journal of Engineering and Science Invention (IJESI),2 (12) (2013) 01-03.
Burton, D. M. 2007. Elementary Number Theory Fifth Edition. McGraw Hill: New York.
Rosen. K. H. 1993. Elementary Number Theory And Its Aplication. Murray Hills: New Jersey.
Bartle & Sherbet. 2011. Introduction to Real Analysis 4th Edition. Hamilton: United State of Amerika.
Arnold, Jimmy. 2005. An Introduction to Mathematical Proofs. http://www.math.vt.edu/people/elder/Math3034.
Yang, Seung Hyun. 2008. Continued Fraction and Pell’s Equation. http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Yang.pdf.
Nathanson. M. B. 1999. Elementary Method in Number Theory. Springer: New Jersey.
Gopalan. M. A.,V.Sangeetha and Manju Somanath “ On the integer solutions of the Pell equation x^2=〖13y〗^2-3^t’’International Journal of Applied Mathematical Research,3 (1) (2014) 58-61.
R. A. Mollin. 2007. Fundamental Number Theory with Applications Second Edition. University of Calgary Alberta. Canada.
T. S. Bhanu Murthy. 2009. A Modern Introduction to Ancient Indian Mathematics Second Edition. University of Madras. Chennai.
Niven dkk. 1991. An Introduction to The Theory of Numbers Fifth Edition. New York : John Wiley & sons.
DOI: https://doi.org/10.20527/jm.v1i2.555
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