On the Problem of Pillai with Fibonacci numbers and powers of 33

International audienceConsider the sequence {Fn} n≥0 of Fibonacci numbers defined by F 0 = 0, F 1 = 1 and F n+2 = F n+1 + Fn for all n ≥ 0. In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 3

Repdigits as sums of three Padovan numbers

International audienceLet $\{P_{n}\}_{n\geq 0}$ be the sequence of Padovan numbers defined by $P_0=0$, $P_1 =1=P_2$, and $P_{n+3}= P_{n+1} +P_n$ for all $n\geq 0$. In this paper, we find all repdigits in base $10$ which can be written as a sum of three Padovan numbers

On the xx--coordinates of Pell equations that are products of two Padovan numbers

Let $\{P_{n}\}_{n\geq 0}$ be the sequence of Padovan numbers defined by $P_0=0$, $P_1 = P_2=1$, and $P_{n+3}= P_{n+1} +P_n$ for all $n\geq 0$. In this paper, we find all positive square-free integers $d \ge 2$ such that the Pell equations $x^2-dy^2 = \ell$, where $\ell\in\{\pm 1, \pm 4\}$, have at least two positive integer solutions $(x,y)$ and $(x^{\prime}, y^{\prime})$ such that each of $x$ and $x^{\prime}$ is a product of two Padovan numbers