Parametric solutions for some Diophantine equations
نویسندگان
چکیده
Under some hypotheses we show that the Diophantine equation (1) has infinitely many solutions described by a family depending on k + 2 parameters. Some applications of the main result are given and some special equations are studied.
منابع مشابه
A Generalized Fibonacci Sequence and the Diophantine Equations $x^2pm kxy-y^2pm x=0$
In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations $x^2pmkxy-y^2pm x=0$, where $k$ is positive integer, and describe the structure of solutions.
متن کاملInteger Solutions of Some Diophantine Equations via Fibonacci and Lucas Numbers
We study the problem of finding all integer solutions of the Diophantine equations x2 − 5Fnxy − 5 (−1) y2 = ±Ln, x2 − Lnxy + (−1) y2 = ±5F 2 n , and x2 − Lnxy + (−1) y2 = ±F 2 n . Using these equations, we also explore all integer solutions of some other Diophantine equations.
متن کاملDiophantine Equations Involving Arithmetic Functions of Factorials
DIOPHANTINE EQUATIONS INVOLVING ARITHMETIC FUNCTIONS OF FACTORIALS Daniel M. Baczkowski We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation:
متن کاملDiophantine Equations Related with Linear Binary Recurrences
In this paper we find all solutions of four kinds of the Diophantine equations begin{equation*} ~x^{2}pm V_{t}xy-y^{2}pm x=0text{ and}~x^{2}pm V_{t}xy-y^{2}pm y=0, end{equation*}% for an odd number $t$, and, begin{equation*} ~x^{2}pm V_{t}xy+y^{2}-x=0text{ and}text{ }x^{2}pm V_{t}xy+y^{2}-y=0, end{equation*}% for an even number $t$, where $V_{n}$ is a generalized Lucas number. This pape...
متن کاملPositive Solutions to Some Systems of Diophantine Equations
We consider a sequence defined by the number of positive solutions to a sequence of systems of Diophantine equations. We derive some bounds on the solutions to demonstrate that the terms of the sequence are finite. We develop an algorithm for 1 computing an arbitrary term of the sequence, and consider a graph-theoretic approach to computing the same.
متن کامل