I recently came up with the following algorithm.
But was given the feedback it was impractical due to too much memory consumption in storing $c!$ it then occurred to me I didn't have to calculate $c!$ and I could get away my simply calculating the last digits of $c!$ and see if that tends to $0$.
The Algorithm So Far
Given: $a<b$ and $ab=c$
We are interested in: $ \frac{c!}{c^\lambda}$
Then its simple to show that:
$$ \frac{c!}{c^a} = \text{integer}$$
whereas,
$$ \frac{c!}{c^{a+1}} \neq \text{integer}$$
The Latest Addition
I realized we don't have to calculate all the digits of $\frac{c!}{c^{\lambda}}$. We can asymptotically expand $c!\sim \sqrt{2 \pi c } (\frac{c}{e})^c $ and then as we are interested in finding if
$$\frac{c!}{c^\lambda} \stackrel{?}{=} \text{integer}$$
$$ \implies \frac{\sqrt{2 \pi c }(c/e)^c}{c^\lambda}(1+ \frac{1}{12(c+1)} + \dots + \text{relevant terms} + \dots) \stackrel{?}{=} \text{integer}$$
where $$ \text{relevant terms} \times \frac{\sqrt{2 \pi c }(c/e)^c}{c^\lambda} = \alpha_0 + \frac{\alpha_{-1}}{10}$$
Where $ \alpha_0 $, $\alpha_{-1}$ are the numbers in the units and first decimal place.
Hence, if:
$$ \text{relevant terms} \times \frac{\sqrt{2 \pi c }(c/e)^c}{c^\lambda} - \alpha_0 = \frac{\alpha_{-1}}{10} \approx 0 $$
Question
Does this make my algorithm viable? What is the running time with the new addition? Does this already exist in the literature?
There was more problems with your previous algorithm than just memory storage. The number of steps it takes to calculate $c!$ -even with Stirling's approximation and only going up to relevant terms- is still tremendous. Numbers get really big farther down the road. For example, the factorial of a number with around $100$ digits (roughly $10^{100}$) should be around the size $10^{10^{100}}$; i.e. larger than comprehension.
On the other hand, consider primality testing algorithms, such as Fermat's test or the AKS test. These compensate for large powers by taking the problem modulo $n$ (or in the case of AKS, modulo $n$ and $x^r-1$). Modular exponentiation is extremely fast, so it dramatically reduces the runtime of these algorithms.
Bottom line, factorials are just impractical for integer factorization.