Linear combinations of factorials in binary recurrence sequences

Abstract

Let $\{u_n{\}}_{n\ge 0}$ 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.

arXiv comment20 pages some error is there and I need some time to fix it