On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences

Abstract

Let ${\mathcal{P}}_n$ be the $n$-th Padovan number, $E_n$ be the $n$-th Perrin number and $N_n$ be the $n$-th Narayana's cows number. Let $b$ be a positive integer such that $b\ge 2$. In this paper, we study the Diophantine equations \[ \mathcal{P}_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] \[ E_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] and \[ N_{n} = (b \pm 1)\cdot b^{l} \pm 1, \] in non-negative integers $n,b$ and positive integer $l$. As a result, we determine the Padovan, Perrin and Narayana's cows numbers that are Thabit and Williams numbers base $b$. Moreover, we determine all solutions of the above equations within the range $2\le b\le 10$.