Diophantine Equations Related with Linear Binary Recurrences
Authors
Abstract:
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 paper continues and extends a previous work of Bahramian and Daghigh.
similar resources
Diophantine Equations with Linear Recurrences an Overview of Some Recent Progress
We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called ”quotient problem” (resp. ”d-th root problem”), which in short asks whether the integrality of the values of the quotient (resp. d-th root) of two (resp. one) linear re...
full textDiophantine Problems with Linear Recurrences via the Subspace Theorem
In this paper we give an overview over recent developments (initiated by P. Corvaja and U. Zannier in [3]) on Diophantine problems where linear recurring sequences are involved and which were solved by using W.M. Schmidt’s Subspace Theorem. Moreover, as a new application, we show: let (Gn) and (Hn) be linear recurring sequences of integers defined by Gn = c1α n 1 + c2α n 2 + · · ·+ ctα t and Hn...
full textSolving Linear Diophantine Equations
An overview of a family of methods for nding the minimal solutions to a single linear Diophantine equation over the natural numbers is given. Most of the formal details were dropped, some illustrations that might give some intuition on the methods being presented instead.
full textCombinatorial Recurrences and Linear Difference Equations
In this work we introduce the triangular arrays of depth greater than 1 given by linear recurrences, that generalize some well-known recurrences that appear in enumerative combinatorics. In particular, we focussed on triangular arrays of depth 2, since they are closely related to the solution of linear three–term recurrences. We show through some simple examples how these triangular arrays appe...
full textSmall solutions of linear Diophantine equations
Let Ax = B be a system of m x n linear equations with integer coefficients. Assume the rows of A are linearly independent and denote by X (respectively Y) the maximum of the absolute values of the m x m minors of the matrix A (the augmented matrix (A, B)). If the system has a solution in nonnegative integers, it is proved that the system has a solution X = (xi) in nonnegative integers with xi <...
full textOptical solutions for linear Diophantine equations
Determining whether a Diophantine equation has a solution or not is the most important challenge in solving this type of problems. In this paper a special computational device which uses light rays is proposed to answer this question, namely check the existence of nonnegative solutions for linear Diophantine equations. The way of representation for this device is similar to an directed graph, h...
full textMy Resources
Journal title
volume 17 issue 1
pages 11- 26
publication date 2022-04
By following a journal you will be notified via email when a new issue of this journal is published.
Hosted on Doprax cloud platform doprax.com
copyright © 2015-2023