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

TL;DR

The paper investigates Diophantine equations linking Thabit and Williams numbers base $b$ with three classical ternary-recurrence sequences: Padovan, Perrin, and Narayana's cows. It employs Matveev's linear forms in logarithms and the Dujella–Pethő reduction method to derive explicit, large-n bounds on $n$ in terms of $b$ for equations of the form $X_n=(b\pm1)b^l\pm1$, where $X_n$ is one of $P_n$, $E_n$, or $N_n$. The authors prove finiteness of all solutions and provide complete solution sets for all $2\le b\le10$, listing the exact tuples $(n,b,l)$ for each equation. The results advance the Diophantine analysis of intersections between Thabit/Williams numbers and ternary-recurrence sequences, offering precise bounds and concrete classifications useful for computational searches and further theoretical work.

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 $2 \leq b \leq 10$.

Theorems & Definitions (3)

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