The order of appearance of factorials in the Fibonacci sequence and certain Diophantine equations

Abstract

Let $$F_n$$ be the nth Fibonacci number. The order (or the rank) of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that $$m\mid F_k$$ . In this article, we obtain a complete formula for the p-adic valuations of z(m!) for all $$m\in {\mathbb {N}}$$ and apply it to solve some Diophantine equations involving z(m!).