1. Obtain some properties of Fibonacci numbers, Lucas numbers, the order of appearance, the divisor function, and factorials.
2. Obtain solutions to some Diophantine equations involving the order of appearance and factorials.
3. Obtain solutions to some Diophantine equations involving the divisor function and factorials.
4. Obtain solutions to some Diophantine equations involving Fibonacci numbers, Lucas numbers, and factorials.
5. Obtain some results concerning the divisor function on residue classes, the local behaviors of the order of appearance, and/or variations of Brocard-Ramanujan equation.
6. Give a new research topic to mathematical community.
