On Diophantine equations involving intersection of Thabit and Williams numbers base $b$ and some ternary recurrent sequences
Authors
Bibhu Prasad Tripathy
Asutosh Satapathy
Utkal Keshari Dutta
Bijan Kumar Patel
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 $b \geq 2$. 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 $2 \leq b \leq 10$.
