Linear combinations of factorials in binary recurrence sequences

TL;DR

This paper investigates the solutions to a Diophantine equation involving binary recurrence sequences and factorial sums, proving finiteness of such solutions and explicitly identifying certain terms expressible as sums of factorials.

Contribution

It establishes the finiteness of solutions to the factorial sum equation for binary recurrence sequences and explicitly characterizes terms that are sums of two factorials.

Findings

Finitely many terms of the sequence can be expressed as sums of factorials.

Explicit characterization of sequence terms that are sums of two factorials.

Effective methods to compute all such solutions.

Abstract

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive discriminant. In this paper, we consider the Diophantine equation $u_m + u_n = a_1 n_1! + \cdots + a_k n_k!$ and prove that there are only finitely many effectively computable terms which can be expressed as a sum of factorials. Furthermore, we find the terms of the balancing sequences that can be written as a sum of two factorials.