Largest Factorial Represented Exactly In Floating Point System

"Find the largest integer $\displaystyle n$ such that $\displaystyle n!$ can be represented exactly in the floating point number system where the base is 2, the precision is 24, and the exponent ranges from -100 to 100.

Well, I don't even know where to start. I think it has something to do with expressing factorials as powers of 2, but other than that I haven't a bloody clue.

Re: Largest Factorial Represented Exactly In Floating Point System

Quote:

Originally Posted by

**mathematicalbagpiper** "Find the largest integer $\displaystyle n$ such that $\displaystyle n!$ can be represented exactly in the floating point number system where the base is 2, the precision is 24, and the exponent ranges from -100 to 100.

Well, I don't even know where to start. I think it has something to do with expressing factorials as powers of 2, but other than that I haven't a bloody clue.

Well the number of binary digits is $\displaystyle b=\lceil \lg(n!) \rceil$

Of these $\displaystyle z=\lfloor n/2 \rfloor + \lfloor n/4 \rfloor + ... + \lfloor n/2^{\lfloor\lg(n)\rfloor} \rfloor $ are zeros.

So you now need $\displaystyle b-z\le 24$ and $\displaystyle z \le 100$

Now trial and error should find the answer pretty quickly (it is less than 20)

CB