The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the operands have. Intuitively, it would seem that this cost could not be improved.

However, how can we be sure? What if there is some clever equivalent algorithm with complexity $\mathcal{O}(\log(N))$, or $\mathcal{O}(N^{0.998})$? If so, how could we know it is possible, or find an example? If not, why not?