[Math] the running time of this algorithm of prime factorization

Question

I recently figured out my own algorithm to factorize a number given we know it has $2$ distinct prime factors. Let:

$$ ab = c$$

Where, $a<b$

Then it isn't difficult to show that:

$$ \frac{c!}{c^a}= \text{integer}$$

In fact,

$$ \frac{c!}{c^{a+1}} \neq \text{integer}$$

So the idea is to first asymptotically calculate $c!$ and then keep dividing by $c$ until one does not get an integer anymore.

Edit

I just realized a better algorithm would be to first divide $c^{\lfloor \sqrt {c} /2\rfloor }$. If it is not an integer then divide by $c^{\lfloor \sqrt {c} /4\rfloor }$. However is it is an integer then divide by: $c^{3\lfloor \sqrt {c} /4 \rfloor }$ . And so on …

Question

I was wondering if this already existed in the literature? And what is the running time of this algorithm? Can this algorithm be improved upon?

Answer 1

The best factoring algorithm currently available is the General Number Field Sieve. Numbers of more than 200 decimal digits have been factored using this method.

The factorial of such a number would have more than $10^{200}$ digits $-$ where on earth are you going to put them all? And that's even before you start your trial divisions. I'm afraid your method is completely impractical as a factoring algorithm.