How can I prove that that $2024!$ cannot be written as product of (not necessarily distinct) Fibonacci Numbers?

Question

How can I prove that $2024!$ cannot be written as product of (not necessarily distinct) Fibonacci Numbers?

I know that some factorials can be written as product of different Fibonacci Numbers, such as:

• $2!=2\cdot1$
• $3!=3\cdot2$
• $4!=8\cdot3$
• $5!=8\cdot5\cdot3$
• $6!=144\cdot5$
• $9!=144\cdot21\cdot8\cdot5\cdot3$
• $11!=144\cdot55\cdot21\cdot8\cdot5\cdot3$

But how can I prove that $2024$ cannot be written as product of (not necessarily distinct) Fibonacci Numbers?

I don’t have any ideas other than above to prove this.

For instance, since $377, 611, 987, 1597, 2584,$ and $4181$ are all divisors of $2024!$, then $2024!$ must can be written as product of (not necessarily distinct) Fibonacci Numbers.

I’m wrong or correct?

Any help is appreciated.

Answer 1

Prime factor $2017$ is necessary , a fibonacci-number $F_n$ is divisible by $2017$ iff $1009\mid n$ , but in this case we also have $732533\mid F_n$ , hence we cannot write $2024!$ as a product of fibonacci-numbers.