Common terms of generalized Pell and Narayana's cows sequences

TL;DR

This paper determines all solutions to the equation relating generalized Pell sequences and Narayana's cows sequence, using advanced number theory techniques including linear forms in logarithms and continued fractions.

Contribution

It completely characterizes solutions to the Diophantine equation connecting generalized Pell and Narayana's cows sequences, a novel result in this area.

Findings

Explicit solutions for the Diophantine equation are found.

The methods combine linear forms in logarithms with continued fractions.

The results extend understanding of relationships between these special sequences.

Abstract

For an integer $k \geq 2$, let $\{ P_{n}^{(k)} \}_{n}$ be the $k$-generalized Pell sequence which starts with $0, \dots,0,1$($k$ terms) and each term afterwards is the sum of $k$ preceding terms. In this paper, we find all the solutions of the Diophantine equation $P_{n}^{(k)} = N_{m}$ in non-negative integers $(n, k, m)$ with $k \geq 2$, where $\{ N_{m} \}_m$ is the Narayana's cows sequence. Our approach utilizes the lower bounds for linear forms in logarithms of algebraic numbers established by Matveev, along with key insights from the theory of continued fractions.