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

摘要

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 vertical bar F-k. In this article, we obtain a complete formula for the p-adic valuations of z(m!) for all m is an element of N and apply it to solve some Diophantine equations involving z(m!).