Power of Two as sums of Three Pell Numbers

TL;DR

This paper completely characterizes solutions to the equation involving sums of three Pell numbers equaling a power of two, providing a comprehensive solution set for this specific Diophantine problem.

Contribution

It determines all nonnegative integer solutions to the equation P_ell + P_m + P_n = 2^a, where P_k are Pell numbers, extending understanding of sums of Pell numbers equaling powers of two.

Findings

All solutions are explicitly characterized.

The solution set is finite and fully described.

The result advances the understanding of Pell number sums and powers of two.

Abstract

In this paper, we find all the solutions of the Diophantine equation $P_\ell + P_m +P_n=2^a$, in nonnegative integer variables $(n,m,\ell, a)$ where $P_k$ is the $k$-th term of the Pell sequence $\{P_n\}_{n\ge 0}$ given by $P_0=0$, $P_1=1$ and $P_{n+1}=2P_{n}+ P_{n-1}$ for all $n\geq 1$.