An irrationality criterion involving recurring sequences

Fabio Lucchini (1)

1 UNIMI - Università degli Studi di Milano = University of Milan

The main purpose of this paper is to prove an irrationality criterion involving recurring sequences. Let $f\in\Z[X]$ be a polynomial of degree $d>1$ and leading coefficient $c\neq 0$. Suppose that there exists two unbounded sequences $x_n,y_n$ ($n\in\N$) such that \begin{equation*} x_{n+1}=f(x_n),\qquad y_{n+1}=cy_n^d \end{equation*} and $x_n\sim y_n$ as $n\to\infty$. If $x_0$ integer and $y_0$ is rational then there exists $a\in\Q$ such that \begin{equation*} f(X)=c(X-a)^d+a. \end{equation*}

Domaines: Théorie des nombres [math.NT]

Fabio Lucchini. An irrationality criterion involving recurring sequences. 2011. ⟨hal-00651136v2⟩