Cullen numbers in sums of terms of recurrence sequence

TL;DR

This paper investigates the solutions of a Diophantine equation involving sums of terms from a linear recurrence sequence and generalized Cullen numbers, providing effective finiteness results and correcting previous errors in related work.

Contribution

It extends previous research on Cullen numbers in recurrence sequences, correcting earlier proofs and establishing new finiteness results for solutions of related Diophantine equations.

Findings

Proves finiteness of solutions to the Diophantine equation involving recurrence sums and Cullen numbers.

Corrects an error in prior work on generalized Cullen numbers in recurrence sequences.

Demonstrates the method with a concrete example.

Abstract

Let $(U_n)_{n\geq 0}$ 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 \cite{bmt}, generalized Cullen numbers in terms of linear recurrence sequence $(U_n)_{n\geq 0}$ under certain weak assumptions has been studied. However, there is an error in their proof. In this paper, we generalize their work, as well as our result fixes their error. In particular, for a given polynomial $Q(x) \in \mathbb{Z}[x]$ we consider the Diophantine equation $U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell} + Q(x)$, and prove effective finiteness result. Furthermore, we demonstrate our method by an example.