Why is there no explicit formula for the factorial? I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by:
$$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$
However, I know that no such formula is known for the most simple product (in my opinion) - the factorial:
$$n!=\prod_{k=1}^n k = \prod_{k=2}^n k$$
I don't know if that is how products work, but I would really like to know!
So my question is why is there no explicit formula (in terms of n, other than n(n-1)...2*1) for the product of the first n integers? Is there a proof that says that one cannot exist or is it that one has not been discovered?
By explicit formula I mean a non-functional equation that does not require n amount of steps to calculate - just like the summation formula does not require n additions.



These two formulas give n! This was discovered by Euler. Reference this link for further reading. http://eulerarchive.maa.org/hedi/HEDI-2007-09.pdf
