Cullen Numbers in Sums of Terms of Recurrence Sequence

Abstract

Let $$(U_n)_{n\ge 0}$$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $$\ell $$ , let $$\ell \cdot 2^{\ell } + 1$$ be a Cullen number. Recently in Bilu et al. (J Number Theory 202:412–425, 2019), generalized Cullen numbers in terms of linear recurrence sequence $$(U_n)_{n\ge 0}$$ under certain weak assumptions has been studied. In this paper, we consider the more general Diophantine equation $$U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell } + Q(x)$$ , for a given polynomial $$Q(x) \in \mathbb {Z}[x]$$ and prove an effective finiteness result. Furthermore, we demonstrate our method by an example.