Diophantine equations with Bernoulli polynomials

Indecomposability of polynomials and related Diophantine equations

We present a new criterion for indecomposability of polynomials over Z. Using the criterion we obtain general finiteness result on polynomial Diophantine equation f(x) = g(y).

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...

Bernoulli polynomials

Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials f (a,B) n (x) defined by the generalized Dickson-type recursive relation (n ≥ 1), f (a,B) 0 (x) = B, f (a,B) 1 (x) = x, f (a,B) n+1 (x) = xf (a,B) n (x)− af (a,B) n−1 (x), where B, a ∈ Q or R. As a direct application of the uniform decomposition result, we fully settle the finiteness problem for the Diophantine equation f (a,B) n (x) = f (â,B̂)...

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.