Diophantine Equations in Recursive Difference.

Diophantine equations for second order recursive sequences of polynomials

Let B be a nonzero integer. Let define the sequence of polynomials Gn(x) by G0(x) = 0, G1(x) = 1, Gn+1(x) = xGn(x) +BGn−1(x), n ∈ N. We prove that the diophantine equation Gm(x) = Gn(y) for m,n ≥ 3, m 6= n has only finitely many solutions.

Diophantine approximation and Diophantine equations

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...

Exponential Stability of Difference Equations with Several Delays: Recursive Approach

Diophantine Approximations, Diophantine Equations, Transcendence and Applications

This article centres around the contributions of the author and therefore, it is confined to topics where the author has worked. Between these topics there are connections and we explain them by a result of Liouville in 1844 that for an algebraic number α of degree n ≥ 2, there exists c > 0 depending only on α such that | α− p q |> c qn for all rational numbers p q with q > 0. This inequality i...

Diophantine 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...