Powers of Two as Sums of Three Pell Numbers

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 $\{P_n\}_{n \geq 0}$ given by $P_0 = 0$, $P_1 = 1$ and $P_{n+1} = 2P_{n} + P_{n-1}$ for all $n \geq 1$.