Parametric solutions diophantine equation pdf

We discuss the rational solutions of the Diophantine equations (f(x)^2 pm f(y)^2=z^2). This problem can be solved either by the theory of elliptic curves or by elementary number theory. Inspired by the work of Ulas and Togbé (Publ Math Debrecen 76(1-2):183-201, 2010 ) and following the approach of Zhang and Zargar (Period Math Hung, 2018 . When n =3orn = 4 the parametric generation of 2000 Mathematics Subject Classification: 11D09, 20H99 Keywords: JORDAN O. TIRRELL and CLIFFORD A. REITER the Diophantine solutions to Equation (1) is also known [3,4]. Recently, matrix generators for the case n = 3 were determined [3]. Our interest in Diophantine solutions to sums of A similar Diophantine equation f (x)f (y)=f (z^2) was studied by Zhang and Cai [ 14, 15 ]. We call a solution ( x , y , z) of ( 1.2) or ( 1.5) nontrivial if f (x)f (y) (f (x)+f (y)) e 0. Using the theory of Pell's equations, we have Theorem 1.1 Let f=x^2+bx+c be a quadratic polynomial without multiple roots, where b , c are integers. from which we can express t as a function in a and b. We thus find a parametric solution for any symmetric diophantine equation of degree 5 in 6 variables: x i = f i(a,b) for which we have x1 +···+x6 = 0. 3. General results In this section we generalize Theorem 2.4 to the case of a form of an arbitrary odd degree. Theorem 3.1. The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b. Abstract. A new formulation of the subject equation is presented. Several parametric and semi-parametric solutions are derived. Originally presented in 1972, two of the then new parametric solutions for a=-1 were later published in a comprehensive survey of the a=1 case. Full PDF. 2 w = p2 - bq 2- cu z = 2pu II. Results In [4] we describe a new method for solving quaternary equations using the notion of "quadratic combination". If by G2 2 we denote the complete set of solutions of the equation: x2 + y 2 = z 2and by G 3 the complete set of solutions of the equation: x + y 2 + z 2 = w , we can make the following definition: A parametric family of quartic Thue equations Andrej Dujella and Borka Jadrijevic ´ Abstract In this paper we prove that the Diophantine equation x4 − 4cx3 y + (6c + 2)x2 y 2 + 4cxy 3 + y 4 = 1, where c ≥ 3 is an integer, has only the trivial solutions (±1, 0), (0, ±1). About this book. This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method Reducible Diophantine Equations and Their Parametric Representations - Volume 7. Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings. We use this solution to obtain parametric solutions of two diophantine systems concerning fifth powers, namely, the system of simultaneous equations x 1 +x 2 +x 3 =y 1 +y 2 +y 3 =0, x 1 5 +x 2 5 H.