On Perfect Powers in k-Generalized Pell-Lucas Sequence

TL;DR

This paper investigates perfect powers within the k-generalized Pell-Lucas sequence, providing explicit solutions for when sequence terms are perfect squares or fourth powers for all k>=3.

Contribution

It explicitly solves the Diophantine equation for perfect powers in the sequence, identifying all solutions with y between 2 and 100 for k>=3.

Findings

Solutions include (n,m,y)=(3,2,4) and (3,4,2) for k>=3.

Sequence terms at n=3 are perfect powers 16 and 4.

All solutions for y in [2,100] are classified.

Abstract

Let k>=2 and let (Q_{n}^{(k)})_{n>=2-k} be the k-generalized Pell sequence defined by Q_{n}^{(k)}=2Q_{n-1}^{(k)}+Q_{n-2}^{(k)}+...+Q_{n-k}^{(k)} for n>=2 with initial conditions Q_{-(k-2)}^{(k)}=Q_{-(k-3)}^{(k)}=...=Q_{-1}^{(k)}=0, Q_{0}^{(k)}=2,Q_{1}^{(k)}=2. In this paper, we solve the Diophantine equation Q_{n}^{(k)}=y^{m} in positive integers n,m,y,k with m,y,k>=2. We show that all solutions (n,m,y) of this equation in positive integers n,m,y,k such that 2<=y<=100 are given by (n,m,y)=(3,2,4),(3,4,2) for k>=3. Namely, Q_{3}^{(k)}=16=2^4=4^2 for k>=3.